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- Homework Statement
- A spherical shell of uniform charge density σ has a circular hole cut out of it as shown below. What is the Electric Field at a radius just outside the sphere, directly over the center of the circular, cut-out hole? HINT: the hole is small enough that you can treat it as flat, and the point at which you are calculating the field is so close to the hole that it can be approximated as an infinite plane.

- Relevant Equations
- $$\oint_{}^{} E \cdot dA = \frac{q_e}{\epsilon_o}$$

First draw a gaussian shape outside of the sphere (a larger sphere) with radius R. The total charge from the (inner) sphere will be:

$$Q = \sigma A$$

$$A = 4\pi r^2$$

$$Q = \sigma 4\pi r^2$$

Use Gauss's Law to derive electric field magnitude

$$\oint_{}^{} E \cdot dA = \frac{q_e}{\epsilon_o}$$

$$E\oint_{}^{} dA = \frac{q_e}{\epsilon_o}$$

$$EA = \frac{q_e}{\epsilon_o}$$

$$E = \frac{q_e}{\epsilon_oA}$$

Substitute for q and A

$$E = \frac{\sigma4\pi r^2}{\epsilon_o4\pi R^2}$$

Cancel

$$E = \frac{\sigma r^2}{\epsilon_o R^2}$$

I'm not surprised that this is wrong, but I feel like I should be allowed to use more variables. Anyways, can anyone help me to see what I did incorrectly here?

Much appreciated

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