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Soaring Crane

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## 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**

## Homework Equations

See below.

## The Attempt at a Solution

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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]

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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.

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