How to Find Electrostatic Energy in a Spherical Capacitor?

[carllacan]

Homework Statement

The space between two spherical shells kept at potentials V1 and V2, respectively, is filled with a dielectric medium. Find the electrostatic energy on the medium.

Homework Equations

The Attempt at a Solution

I know how to get the energy if I am given the electric field or the charge and potential, but here I am only given two boundary conditions. I can think ot two approaches:
try to obtain the potential field everywhere using the Laplace equation and the boundary conditions, and then obtain the electric field from it, or
try to obtain the charge of the spheres by assuming uniformly distributed charge on the spheres and then asking how much charge would be needed to have potentials V1 and V2.

Which one should I use, if any?

 

[Simon Bridge]

You know how to find the capacitance?

 

[carllacan]

The capacitance is Q/V, but I have neither of those terms.

 

[abbas_majidi]

You can see 'introduction to electrodynamic' by Griffits. It is explain how to calculate with example.

 

[carllacan]

There are a few examples in Griffiths, but I'm not 100% sure how they relate to this problem. Is any of the approaches I mentioned appropriate?

 

[Simon Bridge]

The capacitance is Q/V, but I have neither of those terms.

... the capacitance of a spherical capacitor can be calculated from it's geometry. Google for "spherical capacitor" to see what I mean, or you can derive the relationship in the usual way.
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capsph.html

 

[carllacan]

Oh, I see, I can find the quotient between Q and ΔV before knowing any of them.

The only thing that troubles me is that now I can calculate the energy between the spheres, but what about the energy on the outside? If the charge Q' in the outer sphere is not equal and opposite to the charge Q in the inner sphere there will be an electric field in the surroundings, and therefore there will be energy depending on the charge Q', which I am not given.

 

[Simon Bridge]

You are only asked to find the electrostatic energy stored in the dielectric medium between the shells.

 

[abbas_majidi]

For finding capacitance Q/V, Q is absolute magnitude of the smallest charge on the spherical shells(may be |q1| >,< or = |q2|), And V is difference potential between them.

 

[abbas_majidi]

oh, I forget. V is only due to equal charges(absolute magnitude) on spherical shells. For calculating V we must remove additional charge.

 

[Simon Bridge]

@Abbas: In the above problem, the shells are maintained at a particular potential ... as opposed to the usual case where one shell is charged off the other one.
Thus $\Delta V = |V_2-V_1|$.

 

[abbas_majidi]

C in C=Q/V is constant and it is independent of Q and V. it only depens on the material and shape of the capacitor.