# Spherical Capacitor half-filled with a dielectric

1. Jan 8, 2012

### camipol89

1. The problem statement, all variables and given/known data

There is a spherical capacitor half filled with a dielectric of a given costant k.
They're also given the inner radius R1,the outer radius R2 and the 2 potentials on the sphere of radius R1,V1<0,and on the outer spherical shell of radius R2,V2=0.

2. Relevant equations

You need to find the electric field E(r) and the electric potential V(r) everywhere in the capacitor,as well the free and bound charges on the spheres on radii R1,R2.
The problem ask you also to find the total free charge on the inner sphere.

3. The attempt at a solution

To find E(r) and V(r) I used the fact that the electric field is purely radial,so that its tangent component coincides with its tangential component.By using the boundary conditions between the empty region and the one filled with dielectric,I found that sigma1=k*sigma2,where sigma1 is the superficial density charge on the half of the inner sphere facing the region filled with the dielectric,and sigma2 is the density charge on the half of the inner sphere facing the empty region.
After that,I calculated the total charge on the inner sphere,which is Qtot=2*pi*R1^2*(1+k)*sigma2.
By treating the problem as a simple spherical empty capacitor with an effective charge on the inner sphere equal to Qtot,I found out that the electric field in the capacitor is
E(r)=Qtot/(4*pi*epsilon_0*r^2) with R1<r<R2
The potential is then: V(r)=V1 for r<R1 and V(r)=Qtot/(4*pi*epsilon_0*r) + C for R1<r<R2
where C is determinated by the boundary condition for the potential :
C=-Qtot/(4*pi*epsilon_0*R2)
Is what I've done so far right??
I've only one doubt about that: since V1 is negative(and thus Qtot=4*pi*epsilon_0*R1*V1),is the electric field supposed to be radially inward instead of radially outward?

The real problem for me comes when I have to find the charges.I think that the total free charge on the inner sphere is Qtot,and thus the the superficial free charge on the inner sphere will be sigmatot=Qtot/(4*pi*R1^2).
The bound superficial density charge is then calculated for the half of the sphere facing the dielectric(by using the usual formula of scalar product between the polarization vector P and the normal n going outward from the dielectric (which would be parallel to P if the electric field is radially inward)).
Is right what I'm doing,by still treating the problem as a spherical empty capacitor with inner charge Qtot,even when it comes to calculate the free and bound charges?Do I need instead to consider separately the empty region and the one filled with the dielectric?

Here's an image of the spherical capacitor

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