- #1

Raen

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**Charged Sphere with a Hole -- Check my work?**

## Homework Statement

You have a spherical shell of radius a and charge Q. Your sphere is uniformly charged except for the region where θ<= 1° (which has σ = 0).

Imagine that your field point is somewhere on the positive z-axis (so z could be larger or smaller than a). Determine E as a function of z.

I believe I can represent this as a uniformly charged sphere without a hole and a thin disk with a charge density of -σ. Then the law of superposition let's me add the two together. I think I did it right, but before I go on to the computer program portion of the assignment, I'd love if somebody would double-check my logic and work. If you see an error, please let me know. If you think it's correct, let me know that, too.

## Homework Equations

sin(1°) = r/a Where r is the radius of the disk. r = sin(1°)a = 0.017a

E field of a sphere: E(r) = Q/(4∏r

^{2}ε

_{0}) = σa

^{2}/(r

^{2}ε

_{0})

E field of a disk: E(z) = q/(2∏ε

_{0}(0.017a

^{2})*(1-z

^{2}/(√z

^{2}+(0.017a)

^{2}))

## The Attempt at a Solution

Along the z-axis, the E field of the sphere can be written as E(z) = σa

^{2}/(z

^{2}ε

_{0})

Therefore, Etotal = σa

^{2}/(z

^{2}ε

_{0}) + q/(2∏ε

_{0}(0.017a

^{2})*(1-z

^{2}/(√z

^{2}+(0.017a)

^{2}))

or

Etotal = σ/ε

_{0}*(a

^{2}/z

^{2}- 1/2 + z/√(z

^{2}+ (0.017a)

^{2})

Is this correct? I'm sorry if it's messy and thank you, thank you in advance.