# Permittivity of a capacitor

1. Nov 7, 2007

### parsifal

1. The problem statement, all variables and given/known data
A spherical capacitor (shell radiuses a and b, a<b) has the space between the shells filled with a dielectric, and the permittivity changes as a function of the radius so that the energy density stays constant (when radius R=a, then er=e1). Find the permittivity as a function of R.

2. Relevant equations
$$E=\frac{Q}{4 \pi \epsilon R^2}$$

$$\epsilon = \epsilon _0 \epsilon _r$$

Energy density:
$$u=\frac{1}{2} \epsilon E^2= \frac{1}{2} \epsilon _0 \epsilon _r \frac{Q^2}{16 \pi ^2 \epsilon _0^2 \epsilon _r^2 R^4}$$

3. The attempt at a solution
I don't have a clue how this should be done. Now, u reduces to:
$$u=\frac{1}{2} \epsilon E^2= \frac{1}{2} \epsilon _0 \epsilon _r \frac{Q^2}{16 \pi ^2 \epsilon _0^2 \epsilon _r^2 R^4} = \frac{Q^2}{32 \pi ^2 \epsilon _0 \epsilon _r R^4}$$

So I guess that leaves me with only er to play with, in order to get rid of the R^4, which in turn is required to be taken out if u is needed to be constant. But as it was initally required that when R=a then er=e1 , and I don't know how I could get rid of the R.