Question

A Spherical Capacitor with two concentric shells with radii a and b with a<b. The volume between the shells contains a vacuum and the inner and outer shells hold charges +Q and -Q respectively.

1) Use Gauss law to callculate the displacement field D between the spheres.

2) What is the capacitance of this configuration?

3) If the volume is filled with a dialectric with relaltive permittivity, how does the capacitance change?

Answer 1

1. Given that the geometry of the bodies is very symmetrical we use Gauss' law to calculate the displacement field E:

We took a gaussian sphere or radius r, where a<r<b. Taking this consideration, the dot product between the field E and dA is simply the product of the magnitudes of the vectors:

But with this surface taken, the displacement field E is constant, so we can take it out of the integral:

But integrating dA over all the sphere is just the superficial area of the sphere:

This is the magnitude of the displacement field, but we don't know what's the charge in the sphere, we consider an uniform density charge in order to calculate Q:

Making the integral from a to an arbitrary distance a:

If the density is constant then:

Now we have the expression for the charge in the sphere, so we substitute it in the displacement field:

Taking out the equal terms:

where a<r<b

Now we build the displacement vector, as the field is radial we take the radial vector pointing outside the sphere:

2. The capacitance of the configuration is given by:

We calculate the voltage:

As the electric field is pointing in the same direction of the displacement dL, given the radial symmetry, then the dot product of them is simply the product of their magnitudes:

But as the displacement field we already know it:

Substituing in the voltage:

Solving the integral we find that the voltage is:

We substitute this value in the capacitance:

3. Suppose that the relative permittivity of the diaelectric is k, then we know that the capacitance of a shell in a dialectric increases by this factor k:

So as we see, in a dialectric the capacitance increases by this factor k: the relative permittivity.