# Spherical Shells and Gauss' Law

## Homework Statement

A small, insulating, spherical shell with inner radius a and outer radius b is concentric with a larger insulating spherical shell with inner radius c and outer radius d. The inner shell has total charge q distributed uniformly over its volume, and the outer shell has charge -q distributed uniformly over its volume.

http://i131.photobucket.com/albums/p289/SoaringCrane/yf_Figure_22_391.jpg

Calculate the magnitude of the electric field and direction of the field (outward or toward the center) for

i. a < r < b
ii. b < r < c
iii. c < r < d

See below.

## The Attempt at a Solution

--------------------------------------------------------------------------
General workings for shell with inner radius a and outer radius b:

Q = rho*(4/3)*pi[b^3 – a^3], where Q = total uniform charge

indefinite integral[E*dA] = q_enclosed/epsilon_0

E*4*pi*r^2 = [rho*(4/3)*pi[r^3 – a^3]]/[epsilon_0]
--------------------------------------------------------------------------

Note that then rho for inner shell = q/[(4/3)*pi*(b^3-a^3)] and outer shell rho = [-q]/[(4/3)*pi*(d^3-c^3)] in this specific problem.

a. E = [q/(4*pi*epsilon_0)]*[(r^3 – a^3)/(b^3-a^3)]*(1/r^2)

The direction will be away from the center??

b. Total charge is q, so E = (1/4*pi*epsilon_0)*(q/r^2) This will be outward the center, too?

c. This one I am really unsure of—both with the direction and electric field expression since two different charges are featured.
At first, I thought it would be [-q*(r^3 – d^3)]/[4*pi*epsilon*r^2*(c^3 – d^3)], but this alone is wrong???

Would it be a sum:

[-q*(r^3 – d^3)]/[4*pi*epsilon*r^2*(c^3 – d^3)] + [q/(4*pi*epsilon_0)]*[(r^3 – a^3)/(b^3-a^3)]*(1/r^2) + (1/4*pi*epsilon_0)*(q/r^2) ????

As for direction, would it be outward, too? A positive test charge would move away from the positively charged shell and move towards the negative shell?

Any help is appreciated. Thank you.

Last edited:

Thank you again.

Doc Al
Mentor
a. E = [q/(4*pi*epsilon_0)]*[(r^3 – a^3)/(b^3-a^3)]*(1/r^2)

The direction will be away from the center??

b. Total charge is q, so E = (1/4*pi*epsilon_0)*(q/r^2) This will be outward the center, too?
These look OK to me.

c. This one I am really unsure of—both with the direction and electric field expression since two different charges are featured.
At first, I thought it would be [-q*(r^3 – d^3)]/[4*pi*epsilon*r^2*(c^3 – d^3)], but this alone is wrong???
Here it seems like you just considered the outer shell of charge but forgot to include the inner shell.

Would it be a sum:

[-q*(r^3 – d^3)]/[4*pi*epsilon*r^2*(c^3 – d^3)] + [q/(4*pi*epsilon_0)]*[(r^3 – a^3)/(b^3-a^3)]*(1/r^2) + (1/4*pi*epsilon_0)*(q/r^2) ????
It would be the sum of the field from the partial outer shell plus the field from the total inner shell--you've computed them both, just add them up.

As for direction, would it be outward, too? A positive test charge would move away from the positively charged shell and move towards the negative shell?
The field at point r just depends on the total charge within the spherical surface at r. In this problem, that total charge is always positive for all points up to r = d, thus it points outward at all points.

The sum will be:

total inner shell, (1/4*pi*epsilon_0)*(q/r^2), + partial outer shell,
[-q*(r^3 – d^3)]/[4*pi*epsilon*r^2*(c^3 – d^3)] ?

Or did I use/mismatch the wrong expressions?

Doc Al
Mentor
That looks good to me.