# Boundary conditions with dielectrics question

1. Aug 24, 2012

### VortexLattice

Hi all,

I'm doing what should be a pretty simple problem, but some theory is giving me trouble.

Basically, in this problem I have a conducting sphere, surrounded by a thick insulating layer, and then vacuum outside that. I'm attempting to solve for the potential in the insulating layer by using a Legendre Polynomial expansion. To find the coefficients of the expansion in the different regions, I have to match boundary conditions (BC's) at the interfaces.

Now, I just did a different problem in Jackson, a dielectric sphere in a uniform E field. Here, he uses the same method, and the BC's he applies at the dielectric/vacuum interface are that $E_{parallel}$ and $D_{normal}$ must be continuous at the boundary, which he writes as (a is the radius of the dielectric sphere and epsilon is its dielectric constant):

$E_{parallel}$:

$(-1/a)\frac{\partial \phi_{in}}{\partial \theta} |_{r =a}= (-1/a)\frac{\partial \phi_{out}}{\partial \theta}|_{r =a}$

$D_{normal}$:

$-\epsilon\frac{\partial \phi_{in}}{\partial r}|_{r =a} = -\epsilon_0\frac{\partial \phi_{out}}{\partial r}|_{r =a}$

And then he proceeds to solve it like that. So I tried applying that to this problem, but it gave me garbage answers (the potential of the conducting sphere is constant, so $E_{parallel}$ & $D_{normal}$ are zero for it, which then made all the coefficients of the Legendre expansion for the insulator potential 0...which clearly isn't right).

Luckily, I found a solution to this problem (It's actually problem 4.24 in Griffiths), but I still don't understand it. In this problem, the BC's they used were that the potential has to be continuous on the conductor/insulator interface, but nothing about either D or E. They said the same thing about the insulator/vacuum interface, but that one also seems to have the $D_{normal}$ condition.

My question is, why is the continuous potential the only BC on the first interface, while they use both on the second interface? Further, why isn't there the E_parallel BC at all?

If it helps, the solution I found to this problem is here, on page 2: www.physics.utah.edu/~wu/phycs4420/notes/solutions06.pdf

Thanks!

2. Aug 26, 2012

### gabbagabbahey

At each interface, you want to use the continuity condition to "connect" the potential in the two adjacent regions. Conditions (1) just demands that the potential is finite at the origin. Condition and (2) conveys the information that the sphere is placed in an otherwise uniform electric field. Condition (5) conveys the information about the permeability of the dielectric - you could just as well use the same type of boundary condition at the 1st interface instead of at the second if not for the fact that the permealibilty of the conductor is infinite (if you had a hollow core instead of a conductor, you wouldn't have that problem).

Last edited: Aug 26, 2012