Concepts regarding Electric Potentials of Spheres

[WWCY]

Homework Statement

My questions are just related to part a of this problem.

Homework Equations

The Attempt at a Solution

I know that potential inside a conductor is equivalent to potential on the surface of the conductor and potential at any point is an algebraic sum of potential contributions from surrounding sources. But this is as far as I got..

I gathered that potential on surface of b is 0, as net charge of the sphere is 0 (after treating both spheres as point charges of +q and -q). By my previous statement, shouldn't potential inside sphere b (for example r_a < r < r_b) 0 as well?

Could anyone point out the gaps in my understanding? Any help is very much appreciated.

 

[BvU]

shouldn't potential inside sphere b (for example r_a < r < r_b) 0 as well

That's not a conducting volume !

 

[WWCY]

That's not a conducting volume !

So...
a) Potential is only constant across solid conducting objects?
b) How do I deduce the Potential inside a hollow sphere? Does the V_ab = ∫ E . dr still come into play?

Thank you

edit:
This source seems to state that there isn't any voltage difference between the surface of the shell and its interior though, what am I missing?
http://www.phys.uri.edu/gerhard/PHY204/tsl93.pdf

 

[WWCY]

Here's another stab at the problem.
I tried to be as detailed as possible in explaining my steps and thought processes... Does this look right? Thank you!

 

[BvU]

Can't read any of that.
How do you determine the potential from the inner sphere if the outer would not be there at all ?

 

[WWCY]

Can't read any of that.
How do you determine the potential from the inner sphere if the outer would not be there at all ?

Oops, apologies.

The potential from the inner sphere is as if it originates from a point charge so... v = kq/r? for r>ra

 

[BvU]

Bingo. But: do you realize you now are ready with the exercise ?

 

[WWCY]

Bingo. But: do you realize you now are ready with the exercise ?

Yeah, i believe i do. Might I clarify a few concepts?

1) My initial assumption that field in a hollow sphere should be 0 is wrong because that was based on the assumption that there was no charge enclosed in it, yes? Since there is now a +q in the hollow space, there exists an E-field, which also means that there is a potential difference between points in the hollow space and the surface of the hollow sphere. i.e. for r where ra < r < rb, therefore V =/= 0 for points in the hollow space.

2) There is no field in the small solid sphere because there is no charge in it. Therefore there is no E-field and no potential difference between points in the sphere "a" and on the surface of that sphere.

So taking V_a to be potential for points in the smaller solid sphere, and using V_a - V_b = ∫ E . dr and then splitting the integral limits into:
1) r<r_a to r_a - where field is 0
2) r_a to _rb - field present
3) r_b to ∞ - field and potential = 0 (this means V_b = 0)

should see me obtain the potential of points in the smaller sphere, is this right?

Again, apologies if I'm not getting my points across as clearly as i should be. Thanks for your patience.

 

[BvU]

1) My initial assumption that field in a hollow sphere should be 0 is wrong because that was based on the assumption that there was no charge enclosed in it, yes?

Correct.

Since there is now a +q in the hollow space, there exists an E-field, which also means that there is a potential difference between points in the hollow space and the surface of the hollow sphere. i.e. for r where ra < r < rb, therefore V =/= 0 for points in the hollow space.

Correct again

2) There is no field in the small solid sphere because there is no charge in it.

In the sense that all the charge sits on the surface, yes.

Therefore there is no E-field

In a conductor there is no E-field because if there were, the charges would move (after all, it's a conductor!) until there is no more E-field

and no potential difference between points in the sphere "a" and on the surface of that sphere.

Right.

should see me obtain the potential of points in the smaller sphere, is this right?

Right again.

Again, apologies if I'm not getting my points across as clearly as i should be. Thanks for your patience.

No need to apologize. And: you're welcome.