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 [itex]E_{parallel}[/itex] and [itex]D_{normal}[/itex] must be continuous at the boundary, which he writes as (a is the radius of the dielectric sphere and epsilon is its dielectric constant):
[itex]E_{parallel}[/itex]:
[itex](-1/a)\frac{\partial \phi_{in}}{\partial \theta} |_{r =a}= (-1/a)\frac{\partial \phi_{out}}{\partial \theta}|_{r =a}[/itex]
[itex]D_{normal}[/itex]:
[itex]-\epsilon\frac{\partial \phi_{in}}{\partial r}|_{r =a} = -\epsilon_0\frac{\partial \phi_{out}}{\partial r}|_{r =a}[/itex]
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 [itex]E_{parallel}[/itex] & [itex]D_{normal}[/itex] 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 [itex]D_{normal}[/itex] 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: http://www.physics.utah.edu/~wu/phyc...olutions06.pdf
Thanks!
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 [itex]E_{parallel}[/itex] and [itex]D_{normal}[/itex] must be continuous at the boundary, which he writes as (a is the radius of the dielectric sphere and epsilon is its dielectric constant):
[itex]E_{parallel}[/itex]:
[itex](-1/a)\frac{\partial \phi_{in}}{\partial \theta} |_{r =a}= (-1/a)\frac{\partial \phi_{out}}{\partial \theta}|_{r =a}[/itex]
[itex]D_{normal}[/itex]:
[itex]-\epsilon\frac{\partial \phi_{in}}{\partial r}|_{r =a} = -\epsilon_0\frac{\partial \phi_{out}}{\partial r}|_{r =a}[/itex]
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 [itex]E_{parallel}[/itex] & [itex]D_{normal}[/itex] 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 [itex]D_{normal}[/itex] 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: http://www.physics.utah.edu/~wu/phyc...olutions06.pdf
Thanks!