Variable dielectric permitivity

[lanwatch]

Homework Statement

The dielectric permitivity need not be a constant. Typically one can find problems where the permitivity varies, but always does so in the direction of the field. I came up with a problem that may (or may not) be solvable without the use of numerical methods.

Consider a spherical capacitor formed by two concentric spheres of radii 'a' and 'b'. The dielectric filling the volume between both spheres has permitivity given by

\varepsilon = \varepsilon_0 (2+\cos(\theta))

Find the capacity of the setup.

The Attempt at a Solution

Still at it, I am trying to decompose the dielectric in sections between two discrete θs and apply the constitutive equations, and then trying to reduce it to a PDE.

 

[tiny-tim]

welcome to pf!

hi lanwatch! welcome to pf!

(have an epsilon: ε and a theta: θ )

can't you treat it as lots of very thin capacitors in parallel, and integrate to find the total capacitance?

 

[lanwatch]

That was my first thought, but you need to prove before that the field is radial. To me that is not obvious...

 

[tiny-tim]

hi lanwatch!

(i'm very sorry i didn't reply earlier … somehow i lost this thread )

That was my first thought, but you need to prove before that the field is radial. To me that is not obvious...

but the electric displacement field (D) is unaffected by the dielectric, so that will be radial, and we are given that the permittivity is scalar, so the E electric field will always be parallel to the D field