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| ap_physics_c_electricity_and_magnetism [2024/04/06 01:24] – mrdough | ap_physics_c_electricity_and_magnetism [2024/04/20 05:37] (current) – external edit 127.0.0.1 |
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| ====== AP Physics C: E&M Study Guide ====== | ====== AP Physics C: E&M Study Guide ====== |
| <html> | * There is soooo much content for this class, it would likely get ruined through formatting etc, so I have linked the respective pdfs I found online. |
| | * If you have a full AP Physics C: E&M Study Guide, please email me with your file. More info in [[How to Import my Notes into Cramify]] |
| |
| <head> | ==== Unit 1: Electrostatics ==== |
| <meta http-equiv=Content-Type content="text/html; charset=utf-8"> | [[https://uploads-ssl.webflow.com/605fe570e5454a357d1e1811/60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics.pdf|Unit 1: Electrostatics PDF link]] |
| <meta name=Generator content="Microsoft Word 15 (filtered)"> | ==== Unit 2: Conductors, Capacitors, Dielectrics ==== |
| | [[https://uploads-ssl.webflow.com/605fe570e5454a357d1e1811/60a033edf9cc8f596c44e9b8_AP-Physics-C-EM-Conductors-Capacitors-Dielectrics.pdf|Unit 2: Conductors, Capacitors, Dielectrics PDF link]] |
| | ==== Unit 3: Electric Circuits ==== |
| | [[https://uploads-ssl.webflow.com/605fe570e5454a357d1e1811/60a0340e754e6ba021783194_AP-Physics-C-EM-Electric-Circuits.pdf|Unit 3: Electric Circuits PDF link]] |
| | ==== Unit 4: Magnetism ==== |
| | [[https://uploads-ssl.webflow.com/605fe570e5454a357d1e1811/60a0342f9d01ec5525037ee6_AP-Physics-C-EM-Magnetism.pdf|Unit 4: Magnetism PDF link]] |
| | ==== Unit 5: Electromagnetism ==== |
| | [[https://uploads-ssl.webflow.com/605fe570e5454a357d1e1811/60a0344bf3b25f7260ce0a15_AP-Physics-C-EM-Electromagnetism.pdf|Unit 5: Electromagnetism PDF link]] |
| |
| <p class=MsoNormal align=center style='text-align:center'><b><span | |
| style='font-size:21.5pt'>UNIT 1: ELECTROSTATICS</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895840768;margin-left:183px;margin-top:90px;width:259px;height:232px'><img | |
| width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image001.jpg"></span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection3> | |
| |
| <p class=MsoNormal><a name=page2></a><b><span style='font-size:18.0pt'>Before | |
| You Begin</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:8.55pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:156%'><span style='font-size:11.5pt;line-height:156%'>This study | |
| guide will guide you through each of the topics covered on the AP Physics C: | |
| Electricity and Magnetism exam, and cover core concepts, formulas, and other | |
| important info. However, it's important to note that there's a lot of stuff you | |
| should know before starting on E&M.</span></p> | |
| |
| <p class=MsoNormal style='line-height:8.1pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:156%'><span style='font-size:11.5pt;line-height:156%'>Firstly, | |
| you're going to need a decent grasp of basic calculus. Both Physics C exams are | |
| <b>calculus-based</b>, rather than algebra-based, like Physics 1 and 2 are. You | |
| will need to know how to do basic<b> </b>integration and differentiation, as | |
| well as solve separable differential equations. It's not like Calculus BC: | |
| there won't be any integration by parts or anything, so don't fret about not | |
| being able to comprehend any super-advanced calculus concepts, because you | |
| won't need them here.</span></p> | |
| |
| <p class=MsoNormal style='line-height:8.2pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:148%'><span style='font-size:12.0pt;line-height:148%'>You should | |
| also have a decent grasp of regular high school curriculum-level physics | |
| concepts. If you've taken Physics 1 or 2, that works as well. The course and | |
| exam expand on some of the electricity and magnetism concepts covered there in | |
| greater detail, so if you already have a grasp of the basics, it'll really help | |
| you when trying to wrap your head around E&M concepts.</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895839744'><span style='position:absolute;left:183px;top:-74px; | |
| width:259px;height:232px'><img width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image001.jpg"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:7.95pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:156%'><span style='font-size:11.5pt;line-height:156%'>Also, something | |
| else of note: I will <b>BOLD</b> any variables in formulas that are vector | |
| quantities. That’s how I’ll be notating vectors in formulas in this study | |
| guide. Keep that in mind as you use this guide: some people bold variables | |
| instead, to mark them as vectors, but I’ll be using the arrows.</span></p> | |
| |
| <p class=MsoNormal style='line-height:7.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>With all that said, let's get | |
| started!</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection4> | |
| |
| <p class=MsoNormal><a name=page3></a><b><span style='font-size:18.0pt'>What Is | |
| Electrostatics?</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:18.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:148%'><span style='font-size:12.0pt;line-height:148%'>Electrostatics | |
| is the study of charges <i>at rest</i>. It is weighted at about 26-34% of the | |
| exam. We study charges at rest (that aren't moving relative to each other) in | |
| this unit only because it simplifies the learning a bit. When charges move, | |
| they generate a <i>magnetic field</i> as well. This complicates stuff beyond | |
| what we're concerned with for the moment, so for this unit you only consider | |
| charges that are at rest or moving <b>very slowly</b>.</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:12.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:14.0pt'>Charged Particles Review</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:7.9pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>Here’s a quick review on | |
| charged particles:</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895838720;margin-left:183px;margin-top:7px;width:259px;height:232px'><img | |
| width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image001.jpg"></span></p> | |
| |
| <p class=MsoNormal style='line-height:13.25pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in'><span | |
| style='font-size:12.0pt;font-family:"Calibri",sans-serif'>●<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><span | |
| style='font-size:12.0pt'>Protons carry a positive charge.</span></p> | |
| |
| <p class=MsoNormal style='line-height:6.2pt'><span style='font-size:12.0pt; | |
| font-family:"Calibri",sans-serif'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in'><span | |
| style='font-size:12.0pt;font-family:"Calibri",sans-serif'>●<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><span | |
| style='font-size:12.0pt'>Electrons carry a negative charge.</span></p> | |
| |
| <p class=MsoNormal style='line-height:6.35pt'><span style='font-size:12.0pt; | |
| font-family:"Calibri",sans-serif'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in'><span | |
| style='font-size:12.0pt;font-family:"Calibri",sans-serif'>●<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><span | |
| style='font-size:12.0pt'>An object with a <i>net charge</i> has an excess of | |
| protons or electrons.</span></p> | |
| |
| <p class=MsoNormal style='line-height:6.2pt'><span style='font-size:12.0pt; | |
| font-family:"Calibri",sans-serif'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in'><span | |
| style='font-size:12.0pt;font-family:"Calibri",sans-serif'>●<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><span | |
| style='font-size:12.0pt'>Similar charges repel, opposite charges attract.</span></p> | |
| |
| <p class=MsoNormal style='line-height:15.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:79%'><span style='font-size:10.5pt;line-height:79%'>Charge is | |
| measured in the unit of <b>coulombs</b>. A single proton has a charge of +e, | |
| while an electron has a charge of -e. e is the <i>elementary charge</i>, | |
| defined as the electric charge a proton carries, and is about 1.602</span><span | |
| style='font-size:10.5pt;line-height:79%;font-family:"Cambria Math",serif'> ×</span><span | |
| style='font-size:10.5pt;line-height:79%'> 10</span><sup><span style='font-size: | |
| 14.0pt;line-height:79%'>-19</span></sup><span style='font-size:10.5pt; | |
| line-height:79%'> coulombs. Charged particles exert force on other charged | |
| particles, with the direction determined by their charge (same or opposite to | |
| each other).</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection5> | |
| |
| <p class=MsoNormal><a name=page4></a><b><span style='font-size:18.0pt'>The | |
| Electrostatic Force</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895837696;margin-left:165px;margin-top:51px;width:293px;height:470px'><img | |
| width=293 height=470 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image002.jpg"></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:11.05pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center'><a | |
| href="https://physics.stackexchange.com/questions/286907/what-is-the-truth-of-the-electrostatic-force"><span | |
| style='font-size:12.0pt;color:#1155CC'>Picture Credit</span></a></p> | |
| |
| <p class=MsoNormal style='line-height:15.05pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>The <b>electrostatic force</b> | |
| that a point charge of <i>q</i></span><span style='font-size:8.0pt'>1</span><span | |
| style='font-size:12.0pt'> would exert on another charge <i>q</i></span><span | |
| style='font-size:8.0pt'>2</span><span style='font-size:12.0pt'> is given by</span></p> | |
| |
| <p class=MsoNormal style='line-height:6.7pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:12.0pt'>Coulomb’s Law for Electrostatic | |
| Force:</span></b></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:184.0pt;border-collapse:collapse'> | |
| <tr style='height:35.75pt'> | |
| <td width=29 rowspan=2 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:35.75pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=25 rowspan=2 valign=bottom style='width:19.0pt;padding:0in 0in 0in 0in; | |
| height:35.75pt'> | |
| <p class=MsoNormal align=right style='margin-right:.2pt;text-align:right; | |
| line-height:40.65pt'><span style='font-size:9.5pt;font-family:"Cambria Math",serif'>=</span></p> | |
| </td> | |
| <td width=52 colspan=2 valign=bottom style='width:39.0pt;border:none; | |
| border-bottom:solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:35.75pt'> | |
| <p class=MsoNormal align=center style='text-align:center'><i><sub><span | |
| style='font-size:23.0pt'>1 2</span></sub></i></p> | |
| </td> | |
| <td width=28 rowspan=2 valign=bottom style='width:21.0pt;padding:0in 0in 0in 0in; | |
| height:35.75pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:40.65pt'><span | |
| style='font-size:9.5pt;font-family:"Cambria Math",serif'> ̂</span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:35.75pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:3.95pt'> | |
| <td width=27 rowspan=2 valign=bottom style='width:20.0pt;padding:0in 0in 0in 0in; | |
| height:3.95pt'> | |
| <p class=MsoNormal style='margin-left:2.0pt;line-height:14.75pt'><span | |
| style='font-size:3.5pt;font-family:"Cambria Math",serif'>|</span></p> | |
| </td> | |
| <td width=25 rowspan=2 valign=bottom style='width:19.0pt;padding:0in 0in 0in 0in; | |
| height:3.95pt'> | |
| <p class=MsoNormal align=right style='margin-right:1.8pt;text-align:right; | |
| line-height:14.75pt'><span style='font-size:2.5pt;font-family:"Cambria Math",serif'>|</span><i><sup><span | |
| style='font-size:3.5pt'>2</span></sup></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:3.95pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:.15in'> | |
| <td width=29 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:.15in'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:10.8pt'><i><span | |
| style='font-size:11.5pt'>12</span></i></p> | |
| </td> | |
| <td width=25 valign=bottom style='width:19.0pt;padding:0in 0in 0in 0in; | |
| height:.15in'> | |
| <p class=MsoNormal><span style='font-size:9.0pt'> </span></p> | |
| </td> | |
| <td width=28 valign=bottom style='width:21.0pt;padding:0in 0in 0in 0in; | |
| height:.15in'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:10.8pt'><i><span | |
| style='font-size:11.5pt'>12</span></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:.15in'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:12.6pt'> | |
| <td width=29 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:12.6pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=25 valign=bottom style='width:19.0pt;padding:0in 0in 0in 0in; | |
| height:12.6pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=52 colspan=2 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:12.6pt'> | |
| <p class=MsoNormal align=right style='margin-right:13.8pt;text-align:right; | |
| line-height:12.6pt'><i><span style='font-size:11.5pt'>12</span></i></p> | |
| </td> | |
| <td width=28 valign=bottom style='width:21.0pt;padding:0in 0in 0in 0in; | |
| height:12.6pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:12.6pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:7.8pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>Here’s what the symbols mean:</span></p> | |
| |
| <p class=MsoNormal style='line-height:14.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in'><span | |
| style='font-size:12.0pt;font-family:"Calibri",sans-serif'>●<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><b><i><span | |
| style='font-size:12.0pt'>F</span></i></b><span style='font-size:8.0pt'>12</span><span | |
| style='font-size:12.0pt'> is the force that charge <i>q</i></span><span | |
| style='font-size:8.0pt'>1</span><span style='font-size:12.0pt'> exerts on | |
| charge <i>q</i></span><span style='font-size:8.0pt'>2</span><span | |
| style='font-size:12.0pt'>, in Newtons.</span></p> | |
| |
| <p class=MsoNormal style='line-height:.3pt'><span style='font-size:12.0pt; | |
| font-family:"Calibri",sans-serif'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;line-height:75%'><span | |
| style='font-size:6.0pt;line-height:75%;font-family:"Calibri",sans-serif'>●<span | |
| style='font:7.0pt "Times New Roman"'> | |
| </span></span><i><span style='font-size:6.0pt;line-height:75%'>k</span></i><span | |
| style='font-size:6.0pt;line-height:75%'> is <b>Coulomb’s constant</b>, about | |
| 8.99</span><span style='font-size:6.0pt;line-height:75%;font-family:"Cambria Math",serif'> | |
| ×</span><span style='font-size:6.0pt;line-height:75%'> 10</span><sup><span | |
| style='font-size:7.5pt;line-height:75%'>9</span></sup><span style='font-size: | |
| 6.0pt;line-height:75%'> N m</span><sup><span style='font-size:7.5pt;line-height: | |
| 75%'>2</span></sup><span style='font-size:6.0pt;line-height:75%'>/C</span><sup><span | |
| style='font-size:7.5pt;line-height:75%'>2</span></sup><span style='font-size: | |
| 6.0pt;line-height:75%'>.</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.85pt'><span style='font-size:6.0pt; | |
| font-family:"Calibri",sans-serif'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in'><span | |
| style='font-size:12.0pt;font-family:"Calibri",sans-serif'>●<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><i><span | |
| style='font-size:12.0pt'>q</span></i><span style='font-size:8.0pt'>1</span><span | |
| style='font-size:12.0pt'> and <i>q</i></span><span style='font-size:8.0pt'>2</span><span | |
| style='font-size:12.0pt'> are charges 1 and 2 being considered, in coulombs.</span></p> | |
| |
| <p class=MsoNormal style='line-height:6.2pt'><span style='font-size:12.0pt; | |
| font-family:"Calibri",sans-serif'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in'><span | |
| style='font-size:12.0pt;font-family:"Calibri",sans-serif'>●<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><b><i><span | |
| style='font-size:12.0pt'>r</span></i></b><span style='font-size:8.0pt'>12</span><span | |
| style='font-size:12.0pt'> is the distance vector from <i>q</i></span><span | |
| style='font-size:8.0pt'>1</span><span style='font-size:12.0pt'> to <i>q</i></span><span | |
| style='font-size:8.0pt'>2</span><span style='font-size:12.0pt'>.</span></p> | |
| |
| <p class=MsoNormal style='line-height:6.3pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:1.0in;text-indent:-.25in'><span | |
| style='font-size:12.0pt;font-family:"Courier New"'>o<span style='font:7.0pt "Times New Roman"'> | |
| </span></span><span style='font-size:12.0pt'>|<b><i>r</i></b></span><span | |
| style='font-size:8.0pt'>12</span><span style='font-size:12.0pt'>|</span><sup><span | |
| style='font-size:16.0pt'>2</span></sup><span style='font-size:12.0pt'> is the | |
| square of the scalar distance between charges 1 and 2.</span></p> | |
| |
| <p class=MsoNormal style='margin-left:.75in;line-height:83%'><span | |
| style='font-size:10.0pt;line-height:83%;font-family:"Courier New"'>o</span><span | |
| style='font-size:10.0pt;line-height:83%'> </span><span style='font-size: | |
| 2.0pt;line-height:83%;font-family:"Cambria Math",serif'> ̂</span><span | |
| style='font-size:2.0pt;line-height:83%'> is the unit vector pointing from | |
| charge 1 to 2. This partially determines the</span></p> | |
| |
| <p class=MsoNormal style='margin-left:76.0pt;line-height:92%'><i><span | |
| style='font-size:8.5pt;line-height:92%'>12</span></i></p> | |
| |
| <p class=MsoNormal style='line-height:5.4pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:1.0in'><span style='font-size:12.0pt'>direction | |
| of the electrostatic force that charge 1 will exert.</span></p> | |
| |
| <p class=MsoNormal style='line-height:14.9pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>If the vector notation scares | |
| you, here’s the scalar form that often also works:</span></p> | |
| |
| <p class=MsoNormal style='line-height:8.6pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:235.0pt'><i><span style='font-size:13.0pt'>1</span></i><span | |
| style='font-size:10.0pt'> </span><i><span style='font-size:12.0pt'>2</span></i></p> | |
| |
| <p class=MsoNormal style='margin-left:189.0pt'><i><span style='font-size:8.5pt'>12</span></i><sup><span | |
| style='font-size:10.5pt;font-family:"Cambria Math",serif'>=</span></sup><span | |
| style='font-size:10.0pt'> </span><i><sub><span | |
| style='font-size:20.5pt'>2</span></sub></i></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895836672'><span style='position:absolute;left:314px;top:-41px; | |
| width:60px;height:2px'><img width=60 height=2 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image003.png"></span></span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection6> | |
| |
| <p class=MsoNormal><a name=page5></a><b><span style='font-size:18.0pt'>Electrostatic | |
| Force With Multiple Charges</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895835648;margin-left:147px;margin-top:51px;width:329px;height:189px'><img | |
| width=329 height=189 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image004.jpg"></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:13.2pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center'><a | |
| href="https://www.khanacademy.org/science/electrical-engineering/ee-electrostatics/ee-electric-force-and-electric-field/a/ee-electric-force"><span | |
| style='font-size:12.0pt;color:#1155CC'>Picture Credit</span></a></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895834624;margin-left:183px;margin-top:11px;width:259px;height:232px'><img | |
| width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image001.jpg"></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:13.15pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:148%'><span style='font-size:12.0pt;line-height:148%'>When you have | |
| multiple point charges that are close to each other, each of them is exerting | |
| an electrostatic force on each other. To find the total electrostatic force <b><i>F</i></b> | |
| exerted on charge <i>Q</i> by <i>N</i> charges <i>q</i></span><span | |
| style='font-size:8.0pt;line-height:148%'>i</span><span style='font-size:12.0pt; | |
| line-height:148%'>, we simply do a vector sum of all the forces.</span></p> | |
| |
| <p class=MsoNormal style='line-height:9.55pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:125.0pt;border-collapse:collapse'> | |
| <tr style='height:12.45pt'> | |
| <td width=83 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:12.45pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=157 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:12.45pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=20 rowspan=2 valign=bottom style='width:15.0pt;border:none; | |
| border-bottom:solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:12.45pt'> | |
| <p class=MsoNormal align=right style='text-align:right'><i><sub><span | |
| style='font-size:23.0pt'>1</span></sub></i></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:12.45pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:12.45pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:16.55pt'> | |
| <td width=83 rowspan=2 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal style='line-height:21.3pt'><span style='font-size:5.0pt; | |
| font-family:"Cambria Math",serif'>= ∑</span></p> | |
| </td> | |
| <td width=157 rowspan=2 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal align=right style='margin-right:.25in;text-align:right; | |
| line-height:21.3pt'><span style='font-size:5.0pt;font-family:"Cambria Math",serif'>= | |
| (∑</span></p> | |
| </td> | |
| <td width=32 rowspan=2 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal align=center style='margin-left:.6pt;text-align:center; | |
| line-height:21.3pt'><span style='font-size:5.0pt;font-family:"Cambria Math",serif'> | |
| ̂)</span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:3.95pt'> | |
| <td width=20 rowspan=2 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:3.95pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:7.2pt'><i><span | |
| style='font-size:8.0pt'>2</span></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:3.95pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:3.25pt'> | |
| <td width=83 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:3.25pt'> | |
| <p class=MsoNormal><span style='font-size:2.5pt'> </span></p> | |
| </td> | |
| <td width=157 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:3.25pt'> | |
| <p class=MsoNormal><span style='font-size:2.5pt'> </span></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:3.25pt'> | |
| <p class=MsoNormal><span style='font-size:2.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:3.25pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:8.65pt'> | |
| <td width=83 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:7.5pt'> </span></p> | |
| </td> | |
| <td width=157 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:7.5pt'> </span></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:7.5pt'> </span></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:7.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:8.5pt'> | |
| <td width=83 rowspan=2 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=157 rowspan=2 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal align=right style='margin-right:19.0pt;text-align:right; | |
| line-height:12.6pt'><span style='font-size:3.0pt;font-family:"Cambria Math",serif'> | |
| =</span><i><span style='font-size:3.0pt'>1</span></i></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:4.05pt'> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:4.05pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:4.05pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:4.05pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:12.6pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:145%'><span style='font-size:12.0pt; | |
| line-height:145%'>If the summation notation looks scary, don’t fret. This is | |
| just a fancy way of writing “sum up each of the force vectors from each charge | |
| to find the net force.”</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection7> | |
| |
| <p class=MsoNormal><a name=page6></a><b><span style='font-size:18.0pt'>Electrostatic | |
| Force With Objects That Take Up Space</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:18.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:148%'><span style='font-size:12.0pt;line-height:148%'>The | |
| electrostatic force equations that we just went over only cover interactions | |
| between <b>point charges</b>. A point charge is a charge that doesn’t take up | |
| any space: it’s a single point with some charge. When you have a charged object | |
| with <b>spatial extent</b> (that takes up space), however, you have to approach | |
| it a little differently. Let’s go back to our sum of forces from before.</span></p> | |
| |
| <p class=MsoNormal style='line-height:9.8pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:125.0pt;border-collapse:collapse'> | |
| <tr style='height:12.45pt'> | |
| <td width=83 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:12.45pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=157 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:12.45pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=20 rowspan=2 valign=bottom style='width:15.0pt;border:none; | |
| border-bottom:solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:12.45pt'> | |
| <p class=MsoNormal align=right style='text-align:right'><i><sub><span | |
| style='font-size:23.0pt'>1</span></sub></i></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:12.45pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:12.45pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:16.55pt'> | |
| <td width=83 rowspan=2 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal style='line-height:21.2pt'><span style='font-size:5.0pt; | |
| font-family:"Cambria Math",serif'>= ∑</span></p> | |
| </td> | |
| <td width=157 rowspan=2 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal align=right style='margin-right:.25in;text-align:right; | |
| line-height:21.2pt'><span style='font-size:5.0pt;font-family:"Cambria Math",serif'>= | |
| (∑</span></p> | |
| </td> | |
| <td width=32 rowspan=2 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal align=center style='margin-left:.6pt;text-align:center; | |
| line-height:21.2pt'><span style='font-size:5.0pt;font-family:"Cambria Math",serif'> | |
| ̂)</span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:3.95pt'> | |
| <td width=20 rowspan=2 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:3.95pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:7.25pt'><i><span | |
| style='font-size:8.0pt'>2</span></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:3.95pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:3.25pt'> | |
| <td width=83 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:3.25pt'> | |
| <p class=MsoNormal><span style='font-size:2.5pt'> </span></p> | |
| </td> | |
| <td width=157 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:3.25pt'> | |
| <p class=MsoNormal><span style='font-size:2.5pt'> </span></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:3.25pt'> | |
| <p class=MsoNormal><span style='font-size:2.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:3.25pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:8.65pt'> | |
| <td width=83 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:7.5pt'> </span></p> | |
| </td> | |
| <td width=157 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:7.5pt'> </span></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:7.5pt'> </span></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:7.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:8.65pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:8.5pt'> | |
| <td width=83 rowspan=2 valign=bottom style='width:62.0pt;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=157 rowspan=2 valign=bottom style='width:118.0pt;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal align=right style='margin-right:19.0pt;text-align:right; | |
| line-height:12.6pt'><span style='font-size:3.0pt;font-family:"Cambria Math",serif'> | |
| =</span><i><span style='font-size:3.0pt'>1</span></i></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:8.5pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:4.05pt'> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:4.05pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=32 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:4.05pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:4.05pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:13.8pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:78%'><span style='font-size:10.5pt;line-height:78%'>We can think of | |
| charged objects that take up space as a massive collection of an infinite | |
| number of small charges. Now, what would happen if we had an infinite amount of | |
| tiny charges, e.g.</span><span style='font-size:10.5pt;line-height:78%; | |
| font-family:"Cambria Math",serif'> →</span><i><span style='font-size:10.5pt; | |
| line-height:78%'> ∞</span></i><span style='font-size:10.5pt;line-height:78%'>? | |
| Well, we get:</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895833600'><span style='position:absolute;left:183px;top:-55px; | |
| width:259px;height:232px'><img width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image005.jpg"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:7.75pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal align=center style='margin-right:-5.95pt;text-align:center'><i><span | |
| style='font-size:11.5pt'>∞</span></i></p> | |
| |
| <p class=MsoNormal align=center style='margin-right:-5.95pt;text-align:center'><span | |
| style='font-size:3.5pt;font-family:"Cambria Math",serif'>=</span><span | |
| style='font-size:10.0pt'> </span><span style='font-size:3.5pt; | |
| font-family:"Cambria Math",serif'>∑</span><span style='font-size:10.0pt'> </span><i><sup><span | |
| style='font-size:5.0pt'>1</span></sup></i><i><sub><span style='font-size:5.0pt'>2</span></sub></i><span | |
| style='font-size:3.0pt;font-family:"Cambria Math",serif'> ̂</span></p> | |
| |
| <p class=MsoNormal style='margin-left:227.0pt;line-height:78%'><sub><span | |
| style='font-size:7.0pt;line-height:78%;font-family:"Cambria Math",serif'> =</span></sub><i><sub><span | |
| style='font-size:7.0pt;line-height:78%'>1</span></sub></i></p> | |
| |
| <p class=MsoNormal style='line-height:15.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>If you’ve | |
| done some calculus, you’ll know that this looks suspiciously like it can be | |
| represented with an integral. If you thought that, you’d be right!</span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:131.0pt;border-collapse:collapse'> | |
| <tr style='height:29.6pt'> | |
| <td width=236 rowspan=3 valign=bottom style='width:177.0pt;padding:0in 0in 0in 0in; | |
| height:29.6pt'> | |
| <p class=MsoNormal style='line-height:41.55pt'><span style='font-size:9.5pt; | |
| font-family:"Cambria Math",serif'>=∫=∫</span></p> | |
| </td> | |
| <td width=24 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:29.6pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=27 rowspan=3 valign=bottom style='width:20.0pt;padding:0in 0in 0in 0in; | |
| height:29.6pt'> | |
| <p class=MsoNormal style='margin-left:3.0pt;line-height:41.55pt'><span | |
| style='font-size:9.5pt;font-family:"Cambria Math",serif'> ̂</span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:29.6pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:3.65pt'> | |
| <td width=24 valign=bottom style='width:.25in;border:none;border-bottom:solid windowtext 1.0pt; | |
| padding:0in 0in 0in 0in;height:3.65pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:3.65pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:27.25pt'> | |
| <td width=24 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:27.25pt'> | |
| <p class=MsoNormal style='margin-left:1.0pt'><i><sup><span style='font-size: | |
| 23.0pt'>2</span></sup></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:27.25pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:9.7pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:145%'><span style='font-size:12.0pt; | |
| line-height:145%'>So, to solve a problem like the one below, all you need to do | |
| is set up the right equations and integral and solve. Try finding the force | |
| exerted by the charged rod <i>Q</i> on <i>q.</i></span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895832576;margin-left:206px;margin-top:12px;width:212px;height:119px'><img | |
| width=212 height=119 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image006.jpg"></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:12.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center'><a | |
| href="https://cdn.kastatic.org/googleusercontent/nn_2RwqeiBcnniVlV1U570wKnJaYs_G3-h9hU8ZSgkGJpFGPYtqwlNy_hLmodD_rM4RT1czP7PFFORM1ip3-DLFQ"><b><span | |
| style='font-size:12.0pt;color:#1155CC'>Picture Credit</span></b></a></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection8> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><a name=page7></a><span | |
| style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:18.6pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Electric Field</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:17.95pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:14.0pt'>What Is An Electric Field?</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:8.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:147%'><span style='font-size:12.0pt;line-height:147%'>The electric | |
| field <b><i>E</i></b> created by a charge <i>q</i></span><i><span | |
| style='font-size:8.0pt;line-height:147%'>1</span></i><span style='font-size: | |
| 12.0pt;line-height:147%'> is a vector function called a <i>vector field</i>, | |
| that shows how the charge affects other charges around it. It is very similar | |
| to the concept of the gravitational field generated by objects with mass.</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:12.7pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:14.0pt'>Electric Field Near A | |
| Point Charge</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:7.9pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>The electric field a distance | |
| <i>r</i> away from point charge <i>q</i> is given by:</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895831552'><span style='position:absolute;left:183px;top:-8px; | |
| width:259px;height:232px'><img width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image007.jpg"></span></span></p> | |
| |
| <p class=MsoNormal style='margin-left:176.0pt;line-height:75%'><span | |
| style='font-size:10.5pt;line-height:75%;font-family:"Cambria Math",serif'>( , | |
| ) =</span><span style='font-size:14.0pt;line-height:75%;font-family:"Cambria Math",serif'> | |
| <sub>| |</sub></span><i><sub><span style='font-size:12.0pt;line-height:75%'>2</span></sub></i><span | |
| style='font-size:10.5pt;line-height:75%;font-family:"Cambria Math",serif'> ̂</span></p> | |
| |
| <p class=MsoNormal style='line-height:19.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>The direction | |
| of <b><i>E</i></b> is radially outward from a positive point charge and | |
| radially inward towards a negative charge.</span></p> | |
| |
| <p class=MsoNormal style='line-height:9.2pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>Once again, | |
| if you don’t like or need the vector notation, you can use the scalar form of | |
| the equation.</span></p> | |
| |
| <p class=MsoNormal align=right style='margin-right:188.0pt;text-align:right; | |
| line-height:76%'><span style='font-size:9.0pt;line-height:76%;font-family:"Cambria Math",serif'>( | |
| , )=</span> <i><sub>2</sub></i></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895830528'><span style='position:absolute;left:350px;top:-20px; | |
| width:28px;height:2px'><img width=28 height=2 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image008.png"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.8pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='margin-right:8.0pt;line-height:103%'><span | |
| style='font-size:12.0pt;line-height:103%'>To find the electrostatic force | |
| exerted on a charge due to the electric field it’s in, multiple <i>E</i> and | |
| the charge <i>q</i>, just like you would with gravity.</span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center;line-height:77%'><span | |
| style='font-size:10.5pt;line-height:77%;font-family:"Cambria Math",serif'>=</span><span | |
| style='font-size:10.0pt;line-height:77%'> </span><span | |
| style='font-size:10.5pt;line-height:77%;font-family:"Cambria Math",serif'>⟶</span><span | |
| style='font-size:10.0pt;line-height:77%'> </span><span | |
| style='font-size:10.5pt;line-height:77%;font-family:"Cambria Math",serif'>=</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:11.15pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>See the similarity?</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection9> | |
| |
| <p class=MsoNormal><a name=page8></a><b><span style='font-size:18.0pt'>Electric | |
| Field Lines</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:18.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:145%'><span style='font-size:12.0pt; | |
| line-height:145%'>Electric field lines visually describe an electric field. The | |
| lines in an electric field line diagram describe the direction a positive test | |
| charge would accelerate if placed where the line was.</span></p> | |
| |
| <p class=MsoNormal style='line-height:8.6pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>They begin from a charge and | |
| end at infinity, and never intersect. They also don’t have any ends:</span></p> | |
| |
| <p class=MsoNormal style='line-height:6.85pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>they extend out to infinity.</span></p> | |
| |
| <p class=MsoNormal style='line-height:14.9pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>They point outwards from | |
| positive charges and inwards from negatives charges.</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895829504;margin-left:0px;margin-top:21px;width:635px;height:482px'><img | |
| width=635 height=482 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image009.jpg"></span></p> | |
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| |
| <p class=MsoNormal style='line-height:10.75pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center'><a | |
| href="https://www.toppr.com/content/concept/electric-field-lines-248765/"><span | |
| style='font-size:12.0pt;color:#1155CC'>Picture Credit</span></a></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Electric Field With More | |
| Than One Charge</span></b></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection10> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><a name=page9></a><span | |
| style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:10.35pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:147%'><span style='font-size:12.0pt;line-height:147%'>The notation | |
| and thinking for electric fields with multiple charges is the same as with | |
| electrostatic force: simply sum up the fields created by each charge. This is | |
| the equation for the electric field at any position <b><i>r.</i></b></span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:117.0pt;border-collapse:collapse'> | |
| <tr style='height:43.9pt'> | |
| <td width=93 rowspan=2 valign=bottom style='width:70.0pt;padding:0in 0in 0in 0in; | |
| height:43.9pt'> | |
| <p class=MsoNormal style='line-height:49.5pt'><span style='font-size:11.5pt; | |
| font-family:"Cambria Math",serif'>= ∑</span></p> | |
| </td> | |
| <td width=163 rowspan=2 valign=bottom style='width:122.0pt;padding:0in 0in 0in 0in; | |
| height:43.9pt'> | |
| <p class=MsoNormal style='margin-left:12.0pt;line-height:49.5pt'><span | |
| style='font-size:11.5pt;font-family:"Cambria Math",serif'>= {∑</span></p> | |
| </td> | |
| <td width=21 valign=bottom style='width:16.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:43.9pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=36 rowspan=2 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:43.9pt'> | |
| <p class=MsoNormal align=center style='text-align:center;line-height:49.5pt'><span | |
| style='font-size:11.5pt;font-family:"Cambria Math",serif'> ̂}</span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:43.9pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
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| <td width=21 rowspan=2 valign=bottom style='width:16.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:8.25pt'><i><span | |
| style='font-size:9.5pt'>2</span></i></p> | |
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| height:4.55pt'> | |
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| height:9.7pt'> | |
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| height:3.2pt'> | |
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| height:3.2pt'> | |
| <p class=MsoNormal><span style='font-size:2.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:3.2pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:19.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Electric Field Of Charges | |
| With Spatial Extend</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895828480'><span style='position:absolute;left:183px;top:-13px; | |
| width:259px;height:232px'><img width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image001.jpg"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:17.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:147%'><span style='font-size:12.0pt;line-height:147%'>Like with | |
| electrostatic force, we can conceptualize charged objects that take up space as | |
| an infinite amount of small point charges. Therefore, we can apply the same | |
| method to turn the infinite sum into an integral. We get the following | |
| equation.</span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:124.0pt;border-collapse:collapse'> | |
| <tr style='height:32.05pt'> | |
| <td width=249 rowspan=3 valign=bottom style='width:187.0pt;padding:0in 0in 0in 0in; | |
| height:32.05pt'> | |
| <p class=MsoNormal style='line-height:45.7pt'><span style='font-size:10.5pt; | |
| font-family:"Cambria Math",serif'>=∫=∫</span></p> | |
| </td> | |
| <td width=28 valign=bottom style='width:21.0pt;padding:0in 0in 0in 0in; | |
| height:32.05pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=29 rowspan=3 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:32.05pt'> | |
| <p class=MsoNormal style='margin-left:3.0pt;line-height:45.7pt'><span | |
| style='font-size:10.5pt;font-family:"Cambria Math",serif'> ̂</span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:32.05pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:4.2pt'> | |
| <td width=28 valign=bottom style='width:21.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:30.7pt'> | |
| <td width=28 valign=bottom style='width:21.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal align=right style='text-align:right'><i><sup><span | |
| style='font-size:26.0pt'>2</span></sup></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:9.85pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:147%'><span style='font-size:12.0pt;line-height:147%'>This integral | |
| isn’t always trivial. However, they're often ways you can simplify your | |
| equations and your integral to make your life easier, like looking for symmetry. | |
| Check out<span style='color:#0563C1'> </span></span><a | |
| href="http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html"><span | |
| style='font-size:12.0pt;line-height:147%;color:#0563C1'>this HyperPhysics</span></a><u><span | |
| style='font-size:12.0pt;line-height:147%;color:#0563C1'> </span></u><a | |
| href="http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html"><span | |
| style='font-size:12.0pt;line-height:147%;color:#0563C1'>page</span><span | |
| style='font-size:12.0pt;line-height:147%;color:black;text-decoration:none'> </span></a><span | |
| style='font-size:12.0pt;line-height:147%;color:black'>to see an example of this | |
| integral in use.</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:17.15pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Something To Remember</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:18.55pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:147%'><span style='font-size:12.0pt;line-height:147%'>Electric | |
| fields, like gravitational fields, don’t do anything <i>until a charge enters | |
| the field.</i> The field is a description of how a charge will influence other | |
| charges, so if there aren’t any other charges, then the electric field isn’t | |
| actually doing anything at all.</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection11> | |
| |
| <p class=MsoNormal><a name=page10></a><b><span style='font-size:18.0pt'>Gauss’s | |
| Law And Electric Flux</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:17.95pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:14.0pt'>What Is Flux?</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:8.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>Flux is a | |
| concept that is important to many areas of physics. The flux of a vector | |
| quantity <b>X</b> is the amount of the quantity flowing through a surface.</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895827456;margin-left:162px;margin-top:11px;width:300px;height:372px'><img | |
| width=300 height=372 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image010.jpg"></span></p> | |
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| <p class=MsoNormal style='line-height:13.7pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center'><a | |
| href="http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html"><span | |
| style='font-size:12.0pt;color:#1155CC'>Picture Credit</span></a></p> | |
| |
| <p class=MsoNormal style='line-height:14.9pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>The direction of | |
| infinitesimal area <i>d</i><b>A</b> is <b>outward normal</b> to the surface.</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:19.15pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:14.0pt'>Electric Flux</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:18.4pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:147%'><span style='font-size:12.0pt;line-height:147%'>Flux can be | |
| of something physical, like water, or of something abstract, like an electric | |
| field, which is what we are looking at right now. You can compute a flux with a | |
| surface and a vector field <b>X</b> = <b>X</b>(<i>x,y,z</i>). With electric | |
| flux, our vector field <b>X</b> is just referring to the electric field <b>E</b>.</span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center;line-height:75%'><span | |
| style='font-size:9.0pt;line-height:75%;font-family:"Cambria Math",serif'>= ∫</span><span | |
| style='font-size:10.0pt;line-height:75%'> </span><span | |
| style='font-size:9.0pt;line-height:75%;font-family:"Cambria Math",serif'>⋅</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:19.75pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:77%'><span style='font-size:8.5pt;line-height:77%'>Here, we are | |
| taking the dot product of the field with the “area vector”, to get the amount | |
| of the vector pointing in the direction perpendicular to the surface. (This is | |
| equivalent to the</span><span style='font-size:10.0pt;line-height:77%; | |
| font-family:"Cambria Math",serif'> ( )</span><span style='font-size: | |
| 8.5pt;line-height:77%'> in the diagram above: that one just uses the angle | |
| instead of the dot product.)</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection12> | |
| |
| <p class=MsoNormal><a name=page11></a><b><span style='font-size:14.0pt'>Gauss’s | |
| Law</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:18.4pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>Gauss’s Law | |
| tells us the electric flux if we have a closed surface, like a sphere or cube. | |
| The formula is as follows:</span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:136.0pt;border-collapse:collapse'> | |
| <tr style='height:42.05pt'> | |
| <td width=260 colspan=2 valign=bottom style='width:195.0pt;padding:0in 0in 0in 0in; | |
| height:42.05pt'> | |
| <p class=MsoNormal style='line-height:32.3pt'><span style='font-size:7.5pt; | |
| font-family:"Cambria Math",serif'>=∮⋅=</span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:42.05pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:12.9pt'> | |
| <td width=124 valign=bottom style='width:93.0pt;padding:0in 0in 0in 0in; | |
| height:12.9pt'> | |
| <p class=MsoNormal> </p> | |
| </td> | |
| <td width=137 rowspan=2 valign=bottom style='width:103.0pt;padding:0in 0in 0in 0in; | |
| height:12.9pt'> | |
| <p class=MsoNormal style='margin-left:76.0pt'><i><sub><span style='font-size: | |
| 26.0pt'>0</span></sub></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:12.9pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
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| <tr style='height:17.8pt'> | |
| <td width=124 valign=bottom style='width:93.0pt;padding:0in 0in 0in 0in; | |
| height:17.8pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:17.8pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895826432'><span style='position:absolute;left:391px;top:-48px; | |
| width:54px;height:2px'><img width=54 height=2 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image011.png"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:4.4pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:76%'><span style='font-size:9.5pt;line-height:76%'>The scary | |
| integral symbol with a circle in it is a <i>surface integral</i>: it means you | |
| are adding up the infinitesimal bits over the surface you are considering. <i>Q</i></span><i><span | |
| style='font-size:6.5pt;line-height:76%'>encl</span></i><span style='font-size: | |
| 9.5pt;line-height:76%'> is the charge enclosed by the closed surface, while</span><span | |
| style='font-size:14.0pt;line-height:76%'> <i><sub>0</sub></i></span><span | |
| style='font-size:9.5pt;line-height:76%'> is the permittivity of free space, | |
| about</span><span style='font-size:10.5pt;line-height:76%'> <i>8</i></span><span | |
| style='font-size:10.5pt;line-height:76%;font-family:"Cambria Math",serif'>.</span><i><span | |
| style='font-size:10.5pt;line-height:76%'>85</span></i><span style='font-size: | |
| 10.5pt;line-height:76%;font-family:"Cambria Math",serif'> ×</span><span | |
| style='font-size:10.5pt;line-height:76%'> <i>10</i></span><sup><span | |
| style='font-size:14.0pt;line-height:76%;font-family:"Cambria Math",serif'>−</span></sup><i><sup><span | |
| style='font-size:14.0pt;line-height:76%'>12 2</span></sup></i><span | |
| style='font-size:10.5pt;line-height:76%;font-family:"Cambria Math",serif'>/ </span><i><sup><span | |
| style='font-size:14.0pt;line-height:76%'>2</span></sup></i><span | |
| style='font-size:10.5pt;line-height:76%'>.</span><span style='font-size:9.5pt; | |
| line-height:76%'> This is basically how easily an electric field can permeate | |
| in a vacuum.</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895825408'><span style='position:absolute;left:183px;top:-21px; | |
| width:259px;height:232px'><img width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image012.jpg"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:14.9pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>You remember | |
| that constant mentioned in the previous 2 chapters, <i>k</i>? Well, it’s quite | |
| closely related to the permittivity of free space:</span></p> | |
| |
| <p class=MsoNormal style='line-height:2.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:247.0pt'><i><span style='font-size:18.0pt'>1</span></i></p> | |
| |
| <p class=MsoNormal style='margin-left:199.0pt;line-height:77%'><sup><span | |
| style='font-size:9.0pt;line-height:77%;font-family:"Cambria Math",serif'>=</span></sup><sup><span | |
| style='font-size:9.0pt;line-height:77%'> </span></sup><i><span | |
| style='font-size:9.0pt;line-height:77%'>4</span></i><i><span style='font-size: | |
| 8.5pt;line-height:77%'> <sub>0</sub></span></i></p> | |
| |
| <p class=MsoNormal style='line-height:9.9pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>Now let’s | |
| figure out <b>why</b>. Imagine a charge <i>q</i> that is surrounded by a sphere | |
| of radius <i>r</i>. Let’s try calculating the electric flux.</span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:106.0pt;border-collapse:collapse'> | |
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| <p class=MsoNormal style='margin-left:30.0pt;line-height:41.9pt'><span | |
| style='font-size:10.0pt;font-family:"Cambria Math",serif'>= ∮</span></p> | |
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| <td width=21 valign=bottom style='width:16.0pt;padding:0in 0in 0in 0in; | |
| height:36.4pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=100 colspan=4 rowspan=2 valign=bottom style='width:75.0pt; | |
| padding:0in 0in 0in 0in;height:36.4pt'> | |
| <p class=MsoNormal style='margin-left:5.0pt;line-height:41.9pt'><span | |
| style='font-size:10.0pt;font-family:"Cambria Math",serif'>⋅ =</span></p> | |
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| <td width=53 colspan=5 valign=bottom style='width:40.0pt;border:none; | |
| border-bottom:solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:36.4pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=11 rowspan=2 valign=bottom style='width:8.0pt;padding:0in 0in 0in 0in; | |
| height:36.4pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
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| <td width=29 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:36.4pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
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| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:36.4pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
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| height:4.55pt'> | |
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| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
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| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
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| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
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| <td width=29 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
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| height:12.9pt'> | |
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| padding:0in 0in 0in 0in;height:12.9pt'> | |
| <p class=MsoNormal style='margin-left:4.0pt'><i><sub><span style='font-size: | |
| 26.0pt'>0</span></sub></i></p> | |
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| height:17.8pt'> | |
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| height:17.8pt'> | |
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| height:17.8pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
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| height:17.8pt'> | |
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| height:17.8pt'> | |
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| height:17.8pt'> | |
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| <td width=128 rowspan=3 valign=bottom style='width:96.0pt;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal align=right style='margin-right:.8pt;text-align:right; | |
| line-height:33.5pt'><span style='font-size:8.0pt;font-family:"Cambria Math",serif'>=</span></p> | |
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| <td width=21 valign=bottom style='width:16.0pt;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=89 colspan=2 rowspan=3 valign=bottom style='width:67.0pt; | |
| padding:0in 0in 0in 0in;height:19.85pt'> | |
| <p class=MsoNormal style='margin-left:5.0pt;line-height:33.5pt'><span | |
| style='font-size:8.0pt;font-family:"Cambria Math",serif'>⟶ =</span></p> | |
| </td> | |
| <td width=5 valign=bottom style='width:4.0pt;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=5 valign=bottom style='width:4.0pt;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=37 colspan=4 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=16 valign=bottom style='width:12.0pt;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=11 valign=bottom style='width:8.0pt;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=29 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:19.85pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:4.2pt'> | |
| <td width=21 valign=bottom style='width:16.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=5 valign=bottom style='width:4.0pt;border:none;border-bottom:solid windowtext 1.0pt; | |
| padding:0in 0in 0in 0in;height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=31 colspan=3 valign=bottom style='width:23.0pt;border:none; | |
| border-bottom:solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=8 valign=bottom style='width:6.0pt;padding:0in 0in 0in 0in; | |
| height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=4 valign=bottom style='width:3.0pt;padding:0in 0in 0in 0in; | |
| height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=16 valign=bottom style='width:12.0pt;padding:0in 0in 0in 0in; | |
| height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=11 valign=bottom style='width:8.0pt;padding:0in 0in 0in 0in; | |
| height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=29 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:30.7pt'> | |
| <td width=21 valign=bottom style='width:16.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal align=right style='text-align:right'><i><sub><span | |
| style='font-size:26.0pt'>0</span></sub></i></p> | |
| </td> | |
| <td width=48 colspan=6 valign=bottom style='width:.5in;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><i><sub><span style='font-size:26.0pt'>0</span></sub></i></p> | |
| </td> | |
| <td width=16 valign=bottom style='width:12.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=11 valign=bottom style='width:8.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=29 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:31.55pt'> | |
| <td width=197 colspan=3 rowspan=2 valign=bottom style='width:148.0pt; | |
| padding:0in 0in 0in 0in;height:31.55pt'> | |
| <p class=MsoNormal align=center style='text-align:center;line-height:37.0pt'><span | |
| style='font-size:7.0pt;font-family:"Cambria Math",serif'>=</span><i><span | |
| style='font-size:7.0pt'>4</span></i><i><span style='font-size:8.5pt'> <sup>2</sup></span></i><span | |
| style='font-size:7.0pt;font-family:"Cambria Math",serif'>⟶ =(</span></p> | |
| </td> | |
| <td width=41 valign=bottom style='width:31.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:31.55pt'> | |
| <p class=MsoNormal align=right style='text-align:right'><i><span | |
| style='font-size:18.0pt'>1</span></i></p> | |
| </td> | |
| <td width=5 valign=bottom style='width:4.0pt;border:none;border-bottom:solid windowtext 1.0pt; | |
| padding:0in 0in 0in 0in;height:31.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=16 colspan=2 rowspan=2 valign=bottom style='width:12.0pt; | |
| padding:0in 0in 0in 0in;height:31.55pt'> | |
| <p class=MsoNormal align=right style='margin-right:3.0pt;text-align:right; | |
| line-height:37.0pt'><span style='font-size:8.5pt;font-family:"Cambria Math",serif'>)</span></p> | |
| </td> | |
| <td width=23 colspan=2 valign=bottom style='width:17.0pt;border:none; | |
| border-bottom:solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:31.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=4 valign=bottom style='width:3.0pt;padding:0in 0in 0in 0in; | |
| height:31.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=27 colspan=2 rowspan=2 valign=bottom style='width:20.0pt; | |
| padding:0in 0in 0in 0in;height:31.55pt'> | |
| <p class=MsoNormal align=right style='margin-right:5.0pt;text-align:right; | |
| line-height:37.0pt'><span style='font-size:8.5pt;font-family:"Cambria Math",serif'>=</span></p> | |
| </td> | |
| <td width=29 valign=bottom style='width:22.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:31.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:31.55pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:4.55pt'> | |
| <td width=41 valign=bottom style='width:31.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=5 valign=bottom style='width:4.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=27 colspan=3 rowspan=2 valign=bottom style='width:20.0pt; | |
| padding:0in 0in 0in 0in;height:4.55pt'> | |
| <p class=MsoNormal align=right style='margin-right:4.0pt;text-align:right'><i><sup><span | |
| style='font-size:26.0pt'>2</span></sup></i></p> | |
| </td> | |
| <td width=29 rowspan=2 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal align=right style='text-align:right'><i><sup><span | |
| style='font-size:26.0pt'>2</span></sup></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:30.7pt'> | |
| <td width=128 valign=bottom style='width:96.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=21 valign=bottom style='width:16.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=48 valign=bottom style='width:.5in;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=63 colspan=4 valign=bottom style='width:47.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal align=center style='margin-right:12.0pt;text-align:center'><i><span | |
| style='font-size:18.0pt'>4</span></i><i><span style='font-size:25.5pt'> <sub>0</sub></span></i></p> | |
| </td> | |
| <td width=16 valign=bottom style='width:12.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=11 valign=bottom style='width:8.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr height=0> | |
| <td width=128 style='border:none'></td> | |
| <td width=21 style='border:none'></td> | |
| <td width=48 style='border:none'></td> | |
| <td width=41 style='border:none'></td> | |
| <td width=5 style='border:none'></td> | |
| <td width=5 style='border:none'></td> | |
| <td width=11 style='border:none'></td> | |
| <td width=15 style='border:none'></td> | |
| <td width=8 style='border:none'></td> | |
| <td width=4 style='border:none'></td> | |
| <td width=16 style='border:none'></td> | |
| <td width=11 style='border:none'></td> | |
| <td width=29 style='border:none'></td> | |
| <td width=0 style='border:none'></td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:3.6pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>Pretty interesting, huh? Flux | |
| is already showing us new things. Let’s go on.</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection13> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><a name=page12></a><span | |
| style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:17.3pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Electric Potential Energy</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:12.6pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:77%'><span style='font-size:7.5pt; | |
| line-height:77%'>The work done by any force is</span><span style='font-size: | |
| 7.5pt;line-height:77%;font-family:"Cambria Math",serif'> = ∫</span><span | |
| style='font-size:8.5pt;line-height:77%;font-family:"Cambria Math",serif'> ⋅</span><span | |
| style='font-size:7.5pt;line-height:77%'> . Let’s try taking the integral of the | |
| electrostatic force!</span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='border-collapse:collapse'> | |
| <tr style='height:30.95pt'> | |
| <td width=119 valign=bottom style='width:89.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal style='margin-left:67.0pt'><i><sub><span style='font-size: | |
| 21.0pt'>2</span></sub></i></p> | |
| </td> | |
| <td width=37 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=131 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=24 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=69 colspan=5 valign=bottom style='width:52.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal style='line-height:11.6pt'><i><span style='font-size:10.5pt'>2</span></i></p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=32 colspan=2 valign=bottom style='width:24.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=20 rowspan=2 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:30.95pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:8.1pt'> | |
| <td width=119 rowspan=2 valign=bottom style='width:89.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal style='margin-left:11.0pt;line-height:13.7pt'><span | |
| style='font-size:3.0pt;font-family:"Cambria Math",serif'>= ∫</span></p> | |
| </td> | |
| <td width=37 rowspan=2 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=131 rowspan=2 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal align=right style='margin-right:3.25pt;text-align:right; | |
| line-height:13.7pt'><span style='font-size:3.0pt;font-family:"Cambria Math",serif'>⋅ | |
| =</span></p> | |
| </td> | |
| <td width=24 rowspan=2 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:13.7pt'><span | |
| style='font-size:3.0pt;font-family:"Cambria Math",serif'>∫</span></p> | |
| </td> | |
| <td width=1 valign=bottom style='width:1.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=27 valign=bottom style='width:20.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=17 valign=bottom style='width:13.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:8.1pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;border:none;border-bottom:solid windowtext 1.0pt; | |
| padding:0in 0in 0in 0in;height:8.1pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=52 rowspan=2 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal style='margin-left:5.0pt;line-height:13.7pt'><span | |
| style='font-size:3.0pt;font-family:"Cambria Math",serif'>= −</span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:8.1pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:7.1pt'><i><span | |
| style='font-size:8.0pt'>1</span></i></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:8.1pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:7.1pt'><i><span | |
| style='font-size:8.0pt'>2</span></i></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal><span style='font-size:7.0pt'> </span></p> | |
| </td> | |
| <td width=88 rowspan=2 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal style='margin-left:3.0pt;line-height:13.7pt'><span | |
| style='font-size:3.0pt;font-family:"Cambria Math",serif'>= − </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:8.1pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:4.55pt'> | |
| <td width=1 valign=bottom style='width:1.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=27 valign=bottom style='width:20.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=17 valign=bottom style='width:13.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:4.6pt'><span | |
| style='font-size:1.0pt;font-family:"Cambria Math",serif'>|</span></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:4.6pt'><i><span | |
| style='font-size:5.0pt'>2</span></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:4.55pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:6.95pt'> | |
| <td width=119 valign=bottom style='width:89.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=37 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=131 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:6.95pt'><i><span | |
| style='font-size:8.0pt'>1 2</span></i></p> | |
| </td> | |
| <td width=24 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=1 valign=bottom style='width:1.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=27 valign=bottom style='width:20.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=27 colspan=2 rowspan=2 valign=bottom style='width:20.0pt; | |
| padding:0in 0in 0in 0in;height:6.95pt'> | |
| <p class=MsoNormal style='margin-left:1.0pt;line-height:19.65pt'><i><sup><span | |
| style='font-size:22.5pt'>2</span></sup></i></p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=36 rowspan=2 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=20 rowspan=2 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:19.65pt'><i><sub><span | |
| style='font-size:21.0pt'>1</span></sub></i></p> | |
| </td> | |
| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:6.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:6.95pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:12.75pt'> | |
| <td width=119 rowspan=2 valign=bottom style='width:89.0pt;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal style='margin-left:61.0pt'><i><sub><span style='font-size: | |
| 21.0pt'>1</span></sub></i></p> | |
| </td> | |
| <td width=37 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal> </p> | |
| </td> | |
| <td width=131 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal> </p> | |
| </td> | |
| <td width=40 colspan=3 rowspan=2 valign=bottom style='width:30.0pt; | |
| padding:0in 0in 0in 0in;height:12.75pt'> | |
| <p class=MsoNormal style='margin-left:11.0pt'><i><sub><span style='font-size: | |
| 21.0pt'>1</span></sub></i></p> | |
| </td> | |
| <td width=27 valign=bottom style='width:20.0pt;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal> </p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal> </p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal> </p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal> </p> | |
| </td> | |
| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal> </p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:12.75pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:12.3pt'> | |
| <td width=37 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=131 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=27 valign=bottom style='width:20.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=17 valign=bottom style='width:13.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:10.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:12.3pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:20.0pt'> | |
| <td width=287 colspan=3 valign=bottom style='width:215.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal align=right style='margin-right:22.25pt;text-align:right'><span | |
| style='font-size:12.0pt'>So, electric potential energy is given by</span></p> | |
| </td> | |
| <td width=24 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=1 valign=bottom style='width:1.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=27 valign=bottom style='width:20.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=17 valign=bottom style='width:13.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:20.0pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:36.7pt'> | |
| <td width=119 valign=bottom style='width:89.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=37 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=131 rowspan=3 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=24 rowspan=3 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal align=right style='text-align:right;line-height:47.85pt'><span | |
| style='font-family:"Cambria Math",serif'>=</span></p> | |
| </td> | |
| <td width=1 valign=bottom style='width:1.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=68 colspan=4 valign=bottom style='width:51.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal align=center style='margin-right:8.0pt;text-align:center'><i><sub><span | |
| style='font-size:26.0pt'>1</span></sub></i><i><sub><span style='font-size: | |
| 25.5pt'> </span></sub></i><i><sub><span style='font-size:26.0pt'>2</span></sub></i></p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:36.7pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:1.7pt'> | |
| <td width=119 valign=bottom style='width:89.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=37 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=1 valign=bottom style='width:1.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=27 valign=bottom style='width:20.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=17 valign=bottom style='width:13.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:1.7pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:16.55pt'> | |
| <td width=119 valign=bottom style='width:89.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=37 valign=bottom style='width:28.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=1 valign=bottom style='width:1.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=53 colspan=3 rowspan=2 valign=bottom style='width:40.0pt; | |
| padding:0in 0in 0in 0in;height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
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| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
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| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
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| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:16.55pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
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| <td width=119 valign=bottom style='width:89.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
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| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
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| <td width=131 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=24 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
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| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=52 valign=bottom style='width:39.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=36 valign=bottom style='width:27.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=23 valign=bottom style='width:17.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=9 valign=bottom style='width:7.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=20 valign=bottom style='width:15.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=88 valign=bottom style='width:66.0pt;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:3.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:3.9pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895824384'><span style='position:absolute;left:183px;top:-102px; | |
| width:259px;height:232px'><img width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image001.jpg"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:13.55pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>Now, let’s consider what | |
| happens when we have positive/negative charges.</span></p> | |
| |
| <p class=MsoNormal style='line-height:16.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-align:justify;text-justify: | |
| inter-ideograph;text-indent:-.25in;line-height:85%'><span style='font-size: | |
| 10.5pt;line-height:85%;font-family:"Verdana",sans-serif'>-<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><span | |
| style='font-size:10.5pt;line-height:85%'>If both charges are positive or | |
| negative, <i>U</i></span><i><span style='font-size:7.0pt;line-height:85%'>q</span></i><span | |
| style='font-size:10.5pt;line-height:85%'> will be <b>positive</b>. This means | |
| that it takes positive work to bring to 2 charges together from</span><span | |
| style='font-size:12.0pt;line-height:85%;font-family:"Cambria Math",serif'> =</span><span | |
| style='font-size:12.0pt;line-height:85%'> <i>∞</i></span><span | |
| style='font-size:10.5pt;line-height:85%'> to . Makes sense, right? When the 2 | |
| charges are both positive or negative, they will repel, which means you will | |
| have to do work to bring them together.</span></p> | |
| |
| <p class=MsoNormal style='line-height:8.75pt'><span style='font-size:10.5pt; | |
| font-family:"Verdana",sans-serif'> </span></p> | |
| |
| <p class=MsoNormal style='margin-left:.5in;text-align:justify;text-justify: | |
| inter-ideograph;text-indent:-.25in;line-height:142%'><span style='font-size: | |
| 12.0pt;line-height:142%;font-family:"Verdana",sans-serif'>-<span | |
| style='font:7.0pt "Times New Roman"'> </span></span><span | |
| style='font-size:12.0pt;line-height:142%'>If 1 charge is positive and the other | |
| negative, <i>U</i></span><i><span style='font-size:8.0pt;line-height:142%'>q</span></i><span | |
| style='font-size:12.0pt;line-height:142%'> will be <b>negative</b>. This means | |
| that it will take <b>negative work</b> to bring the 2 charges together. This | |
| also makes sense, intuitively: the 2 charges will be attracting each other!</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection14> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><a name=page13></a><span | |
| style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:15.95pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Electric Potential (NOT | |
| energy!)</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:19.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:146%'><span style='font-size:12.0pt;line-height:146%'>Electric | |
| potential is electrical potential energy per unit charge and is measured in | |
| units of joules per coulomb. For a charge <i>q</i> in a field created by <i>q</i></span><i><span | |
| style='font-size:8.0pt;line-height:146%'>source</span></i><span | |
| style='font-size:12.0pt;line-height:146%'>:</span></p> | |
| |
| <p class=MsoNormal style='margin-left:181.0pt;line-height:78%'><sub><span | |
| style='font-size:8.0pt;line-height:78%;font-family:"Cambria Math",serif'>=</span></sub><span | |
| style='font-size:10.0pt;line-height:78%'> </span><sub><span | |
| style='font-size:8.0pt;line-height:78%;font-family:"Cambria Math",serif'>=</span></sub></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895823360'><span style='position:absolute;left:288px;top:-5px; | |
| width:29px;height:2px'><img width=29 height=2 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image013.png"></span></span><span | |
| style='position:relative;z-index:-1895822336'><span style='position:absolute; | |
| left:347px;top:-5px;width:38px;height:2px'><img width=38 height=2 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image014.png"></span></span></p> | |
| |
| <p class=MsoNormal style='margin-left:221.0pt'><span style='font-size:18.0pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895821312'><span style='position:absolute;left:183px;top:-100px; | |
| width:259px;height:232px'><img width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image001.jpg"></span></span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:156%'><span style='font-size:11.5pt;line-height:156%'>Electric | |
| potential is measured in volts (joules per coulomb: 1V = 1 J/C). When a charge | |
| of <i>q</i> and a charge of 2<i>q</i> are displaced in the same way from one | |
| point to another in an electric field, in both cases, the ratio of the change | |
| in potential energy to the charge being displaced is equal is to the change in | |
| electric potential (the electric potential difference) from the first point to | |
| the endpoint.</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:12.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Electric Potential | |
| Difference</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:18.45pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>The change in | |
| electric potential is called the <b>electric potential difference</b>, or <b>voltage</b>. | |
| It’s also measured in volts (joules per coulomb).</span></p> | |
| |
| <p class=MsoNormal align=right style='margin-right:207.0pt;text-align:right; | |
| line-height:75%'><sub><span style='font-size:8.5pt;line-height:75%;font-family: | |
| "Cambria Math",serif'>=</span></sub></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895820288'><span style='position:absolute;left:322px;top:-5px; | |
| width:29px;height:2px'><img width=29 height=2 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image013.png"></span></span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center;line-height:75%'><span | |
| style='font-size:12.0pt;line-height:75%;font-family:"Cambria Math",serif'>↓</span></p> | |
| |
| <p class=MsoNormal style='margin-left:156.0pt;line-height:76%'><span | |
| style='font-size:8.0pt;line-height:76%;font-family:"Cambria Math",serif'>=</span><span | |
| style='font-size:10.0pt;line-height:76%'> </span><span | |
| style='font-size:12.0pt;line-height:76%'>and</span><span style='font-size:10.0pt; | |
| line-height:76%'> </span><span style='font-size:8.0pt;line-height: | |
| 76%;font-family:"Cambria Math",serif'>=</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895819264;margin-left:259px;margin-top:1px;width:32px;height:1px'><img | |
| width=32 height=1 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image015.png"></span><span | |
| style='position:relative;z-index:-1895818240'><span style='position:absolute; | |
| left:384px;top:-2px;width:34px;height:2px'><img width=34 height=2 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image016.png"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:12.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>You can also represent | |
| electric potential difference as an integral.</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection15> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:52.0pt;border-collapse:collapse'> | |
| <tr style='height:92.35pt'> | |
| <td width=231 colspan=2 rowspan=3 valign=bottom style='width:173.0pt; | |
| padding:0in 0in 0in 0in;height:92.35pt'> | |
| <p class=MsoNormal style='line-height:97.55pt'><a name=page14></a><span | |
| style='font-size:18.0pt;font-family:"Cambria Math",serif'>=−</span><span | |
| style='font-size:12.0pt;font-family:"Cambria Math",serif'>∫</span><span | |
| style='font-size:18.0pt;font-family:"Cambria Math",serif'> ⋅ ⟶</span></p> | |
| </td> | |
| <td width=41 colspan=2 valign=bottom style='width:31.0pt;padding:0in 0in 0in 0in; | |
| height:92.35pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=160 rowspan=3 valign=bottom style='width:120.0pt;padding:0in 0in 0in 0in; | |
| height:92.35pt'> | |
| <p class=MsoNormal style='margin-left:3.0pt;line-height:97.55pt'><span | |
| style='font-size:18.0pt;font-family:"Cambria Math",serif'>= =−</span><span | |
| style='font-size:12.0pt;font-family:"Cambria Math",serif'>∫</span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:92.35pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=41 rowspan=3 valign=bottom style='width:31.0pt;padding:0in 0in 0in 0in; | |
| height:92.35pt'> | |
| <p class=MsoNormal style='margin-left:3.0pt;line-height:97.55pt'><span | |
| style='font-size:18.0pt;font-family:"Cambria Math",serif'>⋅ </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:92.35pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:1.4pt'> | |
| <td width=12 valign=bottom style='width:9.0pt;border:none;border-bottom:solid windowtext 1.0pt; | |
| padding:0in 0in 0in 0in;height:1.4pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=29 valign=bottom style='width:22.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:1.4pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:1.4pt'> | |
| <p class=MsoNormal><span style='font-size:1.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:1.4pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:2.85pt'> | |
| <td width=12 valign=bottom style='width:9.0pt;padding:0in 0in 0in 0in; | |
| height:2.85pt'> | |
| <p class=MsoNormal><span style='font-size:2.0pt'> </span></p> | |
| </td> | |
| <td width=29 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:2.85pt'> | |
| <p class=MsoNormal><span style='font-size:2.0pt'> </span></p> | |
| </td> | |
| <td width=15 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:2.85pt'> | |
| <p class=MsoNormal><span style='font-size:2.0pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:2.85pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:8.05pt'> | |
| <td width=100 valign=bottom style='width:75.0pt;padding:0in 0in 0in 0in; | |
| height:8.05pt'> | |
| <p class=MsoNormal><span style='font-size:6.5pt'> </span></p> | |
| </td> | |
| <td width=131 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:8.05pt'> | |
| <p class=MsoNormal><span style='font-size:6.5pt'> </span></p> | |
| </td> | |
| <td width=12 valign=bottom style='width:9.0pt;padding:0in 0in 0in 0in; | |
| height:8.05pt'> | |
| <p class=MsoNormal><span style='font-size:6.5pt'> </span></p> | |
| </td> | |
| <td width=29 rowspan=2 valign=bottom style='width:22.0pt;padding:0in 0in 0in 0in; | |
| height:8.05pt'> | |
| <p class=MsoNormal><span style='font-size:6.5pt'> </span></p> | |
| </td> | |
| <td width=160 valign=bottom style='width:120.0pt;padding:0in 0in 0in 0in; | |
| height:8.05pt'> | |
| <p class=MsoNormal><span style='font-size:6.5pt'> </span></p> | |
| </td> | |
| <td width=15 rowspan=2 valign=bottom style='width:11.0pt;padding:0in 0in 0in 0in; | |
| height:8.05pt'> | |
| <p class=MsoNormal><span style='font-size:6.5pt'> </span></p> | |
| </td> | |
| <td width=41 valign=bottom style='width:31.0pt;padding:0in 0in 0in 0in; | |
| height:8.05pt'> | |
| <p class=MsoNormal><span style='font-size:6.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:8.05pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
| </tr> | |
| <tr style='height:9.9pt'> | |
| <td width=100 valign=bottom style='width:75.0pt;padding:0in 0in 0in 0in; | |
| height:9.9pt'> | |
| <p class=MsoNormal><span style='font-size:8.5pt'> </span></p> | |
| </td> | |
| <td width=131 valign=bottom style='width:98.0pt;padding:0in 0in 0in 0in; | |
| height:9.9pt'> | |
| <p class=MsoNormal><span style='font-size:8.5pt'> </span></p> | |
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| <p class=MsoNormal style='margin-left:48.0pt;line-height:35.6pt'><span | |
| style='font-size:8.5pt;font-family:"Cambria Math",serif'>= −</span><span | |
| style='font-size:7.0pt;font-family:"Cambria Math",serif'> ∫</span></p> | |
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| style='font-size:8.5pt;font-family:"Cambria Math",serif'>⋅</span></p> | |
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| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Equipotential Lines</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895817216;margin-left:72px;margin-top:12px;width:480px;height:457px'><img | |
| width=480 height=457 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image017.jpg"></span></p> | |
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| |
| <p class=MsoNormal align=center style='text-align:center'><a | |
| href="http://hyperphysics.phy-astr.gsu.edu/hbase/electric/equipot.html"><span | |
| style='font-size:12.0pt;color:#1155CC'>Picture Credit</span></a></p> | |
| |
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| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:145%'><span style='font-size:12.0pt; | |
| line-height:145%'>Equipotential lines are lines perpendicular to the electric | |
| field lines, where the electric potential is the same anywhere on the line.</span></p> | |
| |
| <p class=MsoNormal style='line-height:9.3pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:145%'><span style='font-size:12.0pt; | |
| line-height:145%'>Because the electric potential anywhere on an equipotential | |
| line is the same, no work is done when moving between points on an | |
| equipotential line.</span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection16> | |
| |
| <p class=MsoNormal><a name=page15></a><span style='font-size:12.0pt'>Weird | |
| questions sometimes come up involving this: remember it.</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895816192;margin-left:183px;margin-top:298px;width:259px;height:232px'><img | |
| width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image001.jpg"></span></p> | |
| |
| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection17> | |
| |
| <div> | |
| |
| <table cellspacing=0 cellpadding=0 hspace=0 vspace=0 width=81 height=16> | |
| <tr> | |
| <td valign=top align=left height=16 style='padding-top:0in;padding-right: | |
| 0in;padding-bottom:0in;padding-left:0in'> | |
| <p class=MsoNormal style='line-height:85%'><a name=page16></a><span | |
| style='font-size:2.5pt;line-height:85%;font-family:"Cambria Math",serif'>= | |
| (</span><i><span style='font-size:2.5pt;line-height:85%'>2</span></i><span | |
| style='font-size:2.5pt;line-height:85%;font-family:"Cambria Math",serif'> )</span><span | |
| style='font-size:2.5pt;line-height:85%'>.</span></p> | |
| </td> | |
| </tr> | |
| </table> | |
| |
| </div> | |
| |
| <p class=MsoNormal><b><span style='font-size:18.0pt'>Electric Field Near An Infinite | |
| Plane Of Charge</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895815168;margin-left:151px;margin-top:51px;width:321px;height:470px'><img | |
| width=321 height=470 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image018.jpg"></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| <p class=MsoNormal style='line-height:18.1pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center'><a | |
| href="http://www.physics.louisville.edu/cldavis/phys299/notes/elec_gauss_apps.html"><span | |
| style='font-size:12.0pt;color:#1155CC'>Picture Credit</span></a></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
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| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:14.55pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:147%'><span style='font-size:12.0pt;line-height:147%'>Let’s talk | |
| about the field near an infinitely large plane of charge. It’s got a charge | |
| density of . By symmetry, the field <b>E</b> must be perpendicular to the | |
| plane. We have a Gaussian surface (in this case, a cylinder) with base area <i>A</i>. | |
| The height doesn’t really matter.</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:13.5pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:75%'><span style='font-size:8.0pt; | |
| line-height:75%'>Let’s try and calculate flux: nothing is flowing out of the | |
| side, only out the ends of the cylinder. The flux is therefore</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:13.75pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>We can then use Gauss’s Law.</span></p> | |
| |
| <p class=MsoNormal style='line-height:4.55pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal align=right style='margin-top:0in;margin-right:147.0pt; | |
| margin-bottom:0in;margin-left:261.0pt;margin-bottom:.0001pt;text-align:right; | |
| text-indent:-113.35pt;line-height:77%'><sub><span style='font-size:24.0pt; | |
| line-height:77%;font-family:"Cambria Math",serif'>∮<span style='font:7.0pt "Times New Roman"'> | |
| </span></span></sub><sub><span style='font-size:36.0pt;line-height:77%; | |
| font-family:"Cambria Math",serif'> ⋅ =</span></sub><sub><span | |
| style='font-size:11.5pt;line-height:77%;font-family:"Cambria Math",serif'>= </span></sub><i><sub><span | |
| style='font-size:10.0pt;line-height:77%'>0 0</span></sub></i></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895814144'><span style='position:absolute;left:322px;top:-25px; | |
| width:50px;height:1px'><img width=50 height=1 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image019.png"></span></span></p> | |
| |
| <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 | |
| style='margin-left:2.0in;border-collapse:collapse'> | |
| <tr style='height:31.25pt'> | |
| <td width=24 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:31.25pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
| </td> | |
| <td width=184 rowspan=3 valign=bottom style='width:138.0pt;padding:0in 0in 0in 0in; | |
| height:31.25pt'> | |
| <p class=MsoNormal style='margin-left:5.0pt;line-height:44.7pt'><span | |
| style='font-size:10.5pt;font-family:"Cambria Math",serif'>= (</span><i><span | |
| style='font-size:10.5pt'>2</span></i><span style='font-size:10.5pt; | |
| font-family:"Cambria Math",serif'>)⟶ =</span></p> | |
| </td> | |
| <td width=33 valign=bottom style='width:25.0pt;padding:0in 0in 0in 0in; | |
| height:31.25pt'> | |
| <p class=MsoNormal><span style='font-size:12.0pt'> </span></p> | |
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| height:31.25pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
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| <td width=24 valign=bottom style='width:.25in;border:none;border-bottom:solid windowtext 1.0pt; | |
| padding:0in 0in 0in 0in;height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=33 valign=bottom style='width:25.0pt;border:none;border-bottom: | |
| solid windowtext 1.0pt;padding:0in 0in 0in 0in;height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:3.5pt'> </span></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:4.2pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
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| <td width=24 valign=bottom style='width:.25in;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal style='margin-left:1.0pt'><i><sub><span style='font-size: | |
| 26.0pt'>0</span></sub></i></p> | |
| </td> | |
| <td width=33 valign=bottom style='width:25.0pt;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal align=center style='text-align:center'><i><span | |
| style='font-size:18.0pt'>2</span></i><i><span style='font-size:25.5pt'> <sub>0</sub></span></i></p> | |
| </td> | |
| <td width=0 valign=bottom style='width:0in;padding:0in 0in 0in 0in; | |
| height:30.7pt'> | |
| <p class=MsoNormal><span style='font-size:.5pt'> </span></p> | |
| </td> | |
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| </table> | |
| |
| <p class=MsoNormal style='line-height:8.75pt'><span style='font-size:2.5pt; | |
| font-family:"Cambria Math",serif'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:145%'><span style='font-size:12.0pt; | |
| line-height:145%'>We can see that <i>E</i> is a constant, independent of | |
| distance from the plane. Also, both sides of the plane are the same.</span></p> | |
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| </div> | |
| |
| <span style='font-size:11.0pt;font-family:"Times New Roman",serif'><br | |
| clear=all style='page-break-before:always'> | |
| </span> | |
| |
| <div class=WordSection18> | |
| |
| <p class=MsoNormal><a name=page17></a><b><span style='font-size:18.0pt'>Electric | |
| Field Between 2 Parallel Charged Plates</span></b></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895813120;margin-left:14px;margin-top:50px;width:597px;height:200px'><img | |
| width=597 height=200 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image020.jpg"></span></p> | |
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| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:11.75pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center'><span | |
| style='font-size:12.0pt'>*personal photo*</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:absolute; | |
| z-index:-1895812096;margin-left:183px;margin-top:0px;width:259px;height:232px'><img | |
| width=259 height=232 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image021.jpg"></span></p> | |
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| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
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| <p class=MsoNormal style='line-height:13.3pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:148%'><span style='font-size:12.0pt;line-height:148%'>Let’s imagine | |
| 2 parallel plates of uniform charge density, with electric field flowing in the | |
| same direction between them, separated by distance <i>d</i> with a potential | |
| difference of . What is the electric field inside the 3 plates?</span></p> | |
| |
| <p class=MsoNormal style='line-height:8.25pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>Because it’s like 2 charged | |
| planes, the electric field between the 2 plates will be uniform.</span></p> | |
| |
| <p class=MsoNormal style='line-height:15.5pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='text-align:justify;text-justify:inter-ideograph; | |
| line-height:145%'><span style='font-size:12.0pt;line-height:145%'>Imagine the | |
| work needed to move a charge <i>q</i> from the positive plate (call it plate A) | |
| to the bottom plate (call it plate B).</span></p> | |
| |
| <p class=MsoNormal style='margin-left:170.0pt;line-height:79%'><span | |
| style='font-size:5.0pt;line-height:79%;font-family:"Cambria Math",serif'>=</span><span | |
| style='font-size:10.0pt;line-height:79%'> </span><span | |
| style='font-size:5.0pt;line-height:79%;font-family:"Cambria Math",serif'>=</span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:18.3pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>The potential difference V</span><span | |
| style='font-size:8.0pt'>AB</span><span style='font-size:12.0pt'> between A and | |
| B is</span></p> | |
| |
| <p class=MsoNormal align=center style='text-align:center'><span | |
| style='font-size:6.0pt;font-family:"Cambria Math",serif'>− =−( −</span><span | |
| style='font-size:10.0pt'> </span><span style='font-size:6.0pt;font-family: | |
| "Cambria Math",serif'>)=</span><span style='font-size:10.0pt'> </span><span | |
| style='font-size:6.0pt;font-family:"Cambria Math",serif'>−</span><span | |
| style='font-size:10.0pt'> </span><span style='font-size:6.0pt; | |
| font-family:"Cambria Math",serif'>=</span><span style='font-size:10.0pt'> </span><span | |
| style='font-size:12.0pt'>.</span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>Work is <i>also</i></span><span | |
| style='font-size:10.0pt'> </span><span style='font-size:9.0pt; | |
| font-family:"Cambria Math",serif'>=</span><span style='font-size:10.0pt'> </span><span | |
| style='font-size:12.0pt'>because of the constant field.</span><span | |
| style='font-size:10.0pt'> </span><span style='font-size:9.0pt; | |
| font-family:"Cambria Math",serif'>=</span><span style='font-size:10.0pt'> </span><span | |
| style='font-size:12.0pt'>, so</span><span style='font-size:10.0pt'> </span><span | |
| style='font-size:9.0pt;font-family:"Cambria Math",serif'>=</span><span | |
| style='font-size:10.0pt'> </span><span style='font-size:12.0pt'>.</span></p> | |
| |
| <p class=MsoNormal align=center style='margin-right:9.0pt;text-align:center; | |
| line-height:75%'><span style='font-size:11.5pt;line-height:75%;font-family: | |
| "Cambria Math",serif'>=</span><span style='font-size:10.0pt;line-height:75%'> </span><span | |
| style='font-size:11.5pt;line-height:75%;font-family:"Cambria Math",serif'>⟶</span><span | |
| style='font-size:10.0pt;line-height:75%'> </span><span style='font-size: | |
| 11.5pt;line-height:75%;font-family:"Cambria Math",serif'>=</span></p> | |
| |
| <p class=MsoNormal style='line-height:1.0pt'><span style='position:relative; | |
| z-index:-1895811072'><span style='position:absolute;left:400px;top:-13px; | |
| width:37px;height:2px'><img width=37 height=2 | |
| src="60a033ada7191c04f9eadf39_AP-Physics-C-EM-Electrostatics_files/image022.png"></span></span></p> | |
| |
| <p class=MsoNormal style='line-height:10.0pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal style='line-height:15.9pt'><span style='font-size:10.0pt'> </span></p> | |
| |
| <p class=MsoNormal><span style='font-size:12.0pt'>Again, we have a field that | |
| is the same, this time throughout the area between the plates.</span></p> | |
| |
| </div> | |
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| </body> | |
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