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AP Physics C: E & M

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Unit 1

# 1.5 Other Charge Distributions - Fields & Potentials

Peter Apps

## 1.5: Other Charge Distributions - Fields & Potentials

This topic is a catch-all for more advanced applications of Charge Distributions, Gauss' Law, and calculating Electic Potentials. All of the concepts are covered in the previous sections, so if you need to, re-read those first.

### Extended Charge Distributions

Up until now, we've dealt with charged objects as points, but sometimes we can't approximate the charge to be in a single location. These extended charge distributions are cases where the charge is in a ring, or line, or sheet.
Ring of Charge Example:

Image from dev.physicslab.org

If we assume that x >>> a, then we get the same result as for a point charge we found in Section 1. 2! This is a more general case, but we can see that it simplifies to the basic case!
Line of Charge Example:

Image created by author

### Gauss' Law for Various Shapes

You should be able to use Gauss' Law to derive these if you need to. Check out hyperphysics for more info.
Line of Charge:
Point, Hoop, or Sphere (fully enclosed):
Sphere (not fully enclosed):
Note for Spheres: The electric field inside a uniformly charged sphere is proportional to r until r = R, then it behaves as an inverse square.

Image Courtesy of phys.libretexts.org (CC BY 4.0)

Insulating Sheet of Charge:

### Potential Difference for a Variety of Shapes

This is more straightforward. For all the shapes, simply apply Gauss' Law to find an expression for E, then plug that into
Line of Charge Example:
Conducting Sheets Example (moving from the sheet to a distance d away):

### Practice Questions

1.
a)
b)

Image from collegeboard.org

2.

Image from collegeboard.org

3.

Image from collegeboard.org

D is correct. Outside the spheres E = 0 since the total charge enclosed is 0. Inside the negative shell, the potential looks like a positively charged sphere (constant when r < a, and proportional to 1/r when r > a)

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