Spherical Shell using Gauss' Law

[kgleeso]

Homework Statement

An insulator is in the shape of a spherical shell. The insulator is defined by an inner radius a = 4 cm and an outer radius b = 6 cm and carries a total charge of Q = + 9

C (1

C = 10-6 C). You may assume that the charge is distributed uniformly throughout the volume of the insulator.

What is Ey, the y-component of the electric field at point P which is located at (x,y) = (0, -5 cm)?

Homework Equations

Φ_net = Q_enclosed / ∈₀ = E × ∫_surface dA

The Attempt at a Solution

I found the charge density ρ:
Total charge 9e-6 C / ( 4/3 π (0.06² - 0.04²)) = ρ

Then, with my Gaussian surface a sphere with radius 5 cm, I found Q_enclosed
Q_enclosed = ρV = ρ * 4/3 π (0.05² - 0.04²)

I know that
Φ_net = Q_enclosed / ∈₀ = E × ∫_surface dA
which I can rearrange such that
E (this is what I need to find - the electric field on the surface) = Q_enclosed / (ε₀ * ∫_surface dA)
And since ∫_surface dA = 4 π r²
I can find E:
E = 1/4πε₀ * Q_enclosed / r² = 1/4πε₀ * 4.05e-6 / 0.05² = 14563800

This isn't correct, though. Where did I go wrong? I think it is in those last few equations - do I have the wrong area?

The units do work out: k has units N*m²/C², Q_enc has units C, and r² is in m², which works out to N/C

 

[TSny]

Hello! Welcome to PF!

Your expression for the volume of a sphere is not correct. The power of r is not right.

Otherwise, your method looks good to me

 

[kgleeso]

Oops! Thank you.