Gauss: Conducting Spherical Shell w/ Point Charge

[calvert11]

Homework Statement

Consider a conducting spherical shell with inner radius 0.8 m and outer radius 1.3 m. There is a net charge 3 μC on the shell. At its center, within the hollow cavity, there is a point charge 3 μC.

Determine the flux through the spherical Gaussian surface S, which has a radius of 0.4 m. Answer in units of N · m2/C.

a= 0.8m
b= 1.3m
q1 = q2 = 3e-6 C

Homework Equations

Flux = E*A
E = (kQ)/r^2
A = 4pir^2

The Attempt at a Solution

I thought I could approach this in a very straight forward way: finding E of the point charge and multiplying it by the spherical shell's surface area (using the outer radius for both r's).

So, E = (8.99e9*3e-6)/1.3^2 = 15958.6
A = 4*pi*1.3^2 = 21.237

EA = 3.389e5 N*m2/C, which is wrong. I'm not sure what I should do.

 

[Doc Al]

Your solution looks good to me. (Not sure why you used the outer radius, but it doesn't matter. For that matter, you could have just used Gauss's law directly.)

 

[nlightner]

Your approach of finding the electric field of the point charge and multiplying it by the surface area of the spherical shell is a good start. However, there are a few things that need to be taken into consideration in order to get the correct answer.

Firstly, since the point charge is located at the center of the spherical shell, the electric field at any point within the hollow cavity will be zero. This is because the electric field inside a conductor is always zero due to the redistribution of charges on the surface. Therefore, the flux through the spherical Gaussian surface will also be zero.

Secondly, in order to find the electric field at the surface of the spherical shell, we need to take into account the electric field due to both the point charge and the net charge on the shell. This can be done by using the superposition principle, which states that the total electric field at any point is the vector sum of the electric fields due to each individual charge.

Therefore, the correct approach would be to find the electric field at the surface of the spherical shell due to the point charge and the net charge separately, and then add them together to get the total electric field at the surface. This total electric field can then be multiplied by the surface area of the spherical shell to get the flux through the Gaussian surface.

Using the equations given in the homework, the correct approach would be:

E1 = (k*q1)/r1^2 = (8.99e9*3e-6)/(0.4)^2 = 168712.5 N/C

E2 = (k*q2)/r2^2 = (8.99e9*3e-6)/(1.3)^2 = 15958.6 N/C

E_total = E1 + E2 = 184671.1 N/C

Flux = E_total * A = 184671.1 * 4*pi*(1.3)^2 = 3.813e6 N*m2/C

Therefore, the flux through the spherical Gaussian surface is 3.813e6 N*m2/C.