Calculating Electric Field and Induced Charge in a Charged Sphere System

[tag16]

Homework Statement

A solid insulating sphere of radius a has a uniform charge density p and a total charge Q. Concentric with this sphere is an uncharged, conducting, hollow sphere whose inner and outer radii are b and c. (a) Find the magnitude of the electric field in the regions r<a, a<r<b, b<r<c, and r<c. (b) Determine the induced charge per unit area on the inner and outer surfaces of the hollow sphere.

Homework Equations

Gauss's Law= S E dA= Q/E_0

The Attempt at a Solution

S E dA= Q/E_0
= E(4pir^2)= Q/E_0

I am doing this right so far? if so not sure what to do next, if not, not sure what to do.

 

[kuruman]

You are doing fine so far. Next you need to apply Gauss' Law by choosing different gaussian surfaces in each of the regions of interest

First at r<a
Second at a<r<b
Third at b<r<c
Fourth at r>c (You say r<c, but I assume you meant r>c)

Apply Gauss' Law as you have stated it. Make sure that you calculate the charge enclosed by each surface correctly and remember that inside a conductor the electric field is zero.

 

[tag16]

ok this is what I have so far...

for r>a

S E dA= Q/E_0
= E(4pir^2)= Q/E_0
E=Q/4piE_0r^2= kQ/r^2

for a<r<b

q(internal)=pV'=p(4/3pir^3)
S E dA = E S dA= E(4pir^2)= q(internal)/E_0
E=qin/4piE_0r^2=p(4/3pir^3)/4piE_0r^2=pr/3E_0
E= (Q/(4/3)pia^3)r/3(1/4pik)= kr(Q/a^3)

for r>c
would it be the same as the 1st one?

for b<r<c
would it be 0?

Did I do any of these right? and for the wrong ones what do I need to do?

 

[kuruman]

ok this is what I have so far...

for r>a

S E dA= Q/E_0
= E(4pir^2)= Q/E_0
E=Q/4piE_0r^2= kQ/r^2

I assume you mean r < a. In this case not all the charge Q is enclosed by the Gaussian surface. Only a fraction of it. Can you figure out what that fraction is?

for a<r<b

q(internal)=pV'=p(4/3pir^3)
S E dA = E S dA= E(4pir^2)= q(internal)/E_0
E=qin/4piE_0r^2=p(4/3pir^3)/4piE_0r^2=pr/3E_0
E= (Q/(4/3)pia^3)r/3(1/4pik)= kr(Q/a^3)

Actually this is the answer to the previous part and the answer to the previous part is the answer to this one. Can you see why?

for r>c
would it be the same as the 1st one?

It would be the same as in the region a<r<b (corrected).

for b<r<c
would it be 0?

Yes it would.

Did I do any of these right? and for the wrong ones what do I need to do?

See above.

 

[tag16]

Thanks...I think I just copied it down wrong in here from what I wrote down.