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AP Physics C: E&M Study Guide
UNIT 1: ELECTROSTATICS
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.
Firstly, you're going to need a decent grasp of basic calculus. Both Physics C exams are calculus-based, rather than algebra-based, like Physics 1 and 2 are. You will need to know how to do basic 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.
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.
Also, something else of note: I will BOLD 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.
With all that said, let's get started!
Electrostatics is the study of charges at rest. 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 magnetic field 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 very slowly.
Charged Particles Review
Here’s a quick review on charged particles:
● Protons carry a positive charge.
● Electrons carry a negative charge.
● An object with a net charge has an excess of protons or electrons.
● Similar charges repel, opposite charges attract.
Charge is measured in the unit of coulombs. A single proton has a charge of +e, while an electron has a charge of -e. e is the elementary charge, defined as the electric charge a proton carries, and is about 1.602 × 10-19 coulombs. Charged particles exert force on other charged particles, with the direction determined by their charge (same or opposite to each other).
The electrostatic force that a point charge of q1 would exert on another charge q2 is given by
Coulomb’s Law for Electrostatic Force:
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Here’s what the symbols mean:
● F12 is the force that charge q1 exerts on charge q2, in Newtons.
● k is Coulomb’s constant, about 8.99 × 109 N m2/C2.
● q1 and q2 are charges 1 and 2 being considered, in coulombs.
● r12 is the distance vector from q1 to q2.
o |r12|2 is the square of the scalar distance between charges 1 and 2.
o ̂ is the unit vector pointing from charge 1 to 2. This partially determines the
12
direction of the electrostatic force that charge 1 will exert.
If the vector notation scares you, here’s the scalar form that often also works:
1 2
12= 2
Electrostatic Force With Multiple Charges
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 F exerted on charge Q by N charges qi, we simply do a vector sum of all the forces.
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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.”
Electrostatic Force With Objects That Take Up Space
The electrostatic force equations that we just went over only cover interactions between point charges. 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 spatial extent (that takes up space), however, you have to approach it a little differently. Let’s go back to our sum of forces from before.
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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. → ∞? Well, we get:
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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!
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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 Q on q.
Electric Field
What Is An Electric Field?
The electric field E created by a charge q1 is a vector function called a vector field, 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.
Electric Field Near A Point Charge
The electric field a distance r away from point charge q is given by:
( , ) = | |2 ̂
The direction of E is radially outward from a positive point charge and radially inward towards a negative charge.
Once again, if you don’t like or need the vector notation, you can use the scalar form of the equation.
( , )= 2
To find the electrostatic force exerted on a charge due to the electric field it’s in, multiple E and the charge q, just like you would with gravity.
= ⟶ =
See the similarity?
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.
They begin from a charge and end at infinity, and never intersect. They also don’t have any ends:
they extend out to infinity.
They point outwards from positive charges and inwards from negatives charges.
Electric Field With More Than One Charge
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 r.
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Electric Field Of Charges With Spatial Extend
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.
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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 this HyperPhysics page to see an example of this integral in use.
Something To Remember
Electric fields, like gravitational fields, don’t do anything until a charge enters the field. 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.
What Is Flux?
Flux is a concept that is important to many areas of physics. The flux of a vector quantity X is the amount of the quantity flowing through a surface.
The direction of infinitesimal area dA is outward normal to the surface.
Electric Flux
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 X = X(x,y,z). With electric flux, our vector field X is just referring to the electric field E.
= ∫ ⋅
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 ( ) in the diagram above: that one just uses the angle instead of the dot product.)
Gauss’s Law tells us the electric flux if we have a closed surface, like a sphere or cube. The formula is as follows:
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The scary integral symbol with a circle in it is a surface integral: it means you are adding up the infinitesimal bits over the surface you are considering. Qencl is the charge enclosed by the closed surface, while 0 is the permittivity of free space, about 8.85 × 10−12 2/ 2. This is basically how easily an electric field can permeate in a vacuum.
You remember that constant mentioned in the previous 2 chapters, k? Well, it’s quite closely related to the permittivity of free space:
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Now let’s figure out why. Imagine a charge q that is surrounded by a sphere of radius r. Let’s try calculating the electric flux.
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Pretty interesting, huh? Flux is already showing us new things. Let’s go on.
Electric Potential Energy
The work done by any force is = ∫ ⋅ . Let’s try taking the integral of the electrostatic force!
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So, electric potential energy is given by |
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Now, let’s consider what happens when we have positive/negative charges.
- If both charges are positive or negative, Uq will be positive. This means that it takes positive work to bring to 2 charges together from = ∞ 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.
- If 1 charge is positive and the other negative, Uq will be negative. This means that it will take negative work to bring the 2 charges together. This also makes sense, intuitively: the 2 charges will be attracting each other!
Electric Potential (NOT energy!)
Electric potential is electrical potential energy per unit charge and is measured in units of joules per coulomb. For a charge q in a field created by qsource:
= =
Electric potential is measured in volts (joules per coulomb: 1V = 1 J/C). When a charge of q and a charge of 2q 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.
Electric Potential Difference
The change in electric potential is called the electric potential difference, or voltage. It’s also measured in volts (joules per coulomb).
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You can also represent electric potential difference as an integral.
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Equipotential Lines
Equipotential lines are lines perpendicular to the electric field lines, where the electric potential is the same anywhere on the line.
Because the electric potential anywhere on an equipotential line is the same, no work is done when moving between points on an equipotential line.
Electric Field Near An Infinite Plane Of Charge
Let’s talk about the field near an infinitely large plane of charge. It’s got a charge density of . By symmetry, the field E must be perpendicular to the plane. We have a Gaussian surface (in this case, a cylinder) with base area A. The height doesn’t really matter.
Let’s try and calculate flux: nothing is flowing out of the side, only out the ends of the cylinder. The flux is therefore
We can then use Gauss’s Law.
∮ ⋅ == 0 0
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We can see that E is a constant, independent of distance from the plane. Also, both sides of the plane are the same.
Electric Field Between 2 Parallel Charged Plates
*personal photo*
Let’s imagine 2 parallel plates of uniform charge density, with electric field flowing in the same direction between them, separated by distance d with a potential difference of . What is the electric field inside the 3 plates?
Because it’s like 2 charged planes, the electric field between the 2 plates will be uniform.
Imagine the work needed to move a charge q from the positive plate (call it plate A) to the bottom plate (call it plate B).
= =
The potential difference VAB between A and B is
− =−( − )= − = .
Work is also = because of the constant field. = , so = .
= ⟶ =
Again, we have a field that is the same, this time throughout the area between the plates.