Potential inside grounded, conducting sphere with dipole at center.

[oddjobmj]

Homework Statement

Suppose a grounded spherical conducting shell of radius R surrounds a pointlike dipole at the center with $\vec{p}=p\vec{k}$ Find the potential $V(r,\theta$) for r <= R. Hint: Use spherical harmonics regular at r=0 to satisfy the boundary condition.

Homework Equations

General solution:
$V(r,\theta)= \sum_{n=0}^\infty A_nr^nP_n(cos\theta) + \sum_{n=0}^\infty B_nr^{-(n+a)}P_n(cos\theta)$

$V_{dip}=\frac{kqdcos\theta}{r^2}$

The Attempt at a Solution

So $V(r,\theta)$ ends up being the sum of the above general solution plus the potential due to the dipole.

I believe we can get rid of the whole $B_n$ term because the potential inside the sphere is finite and at r=0 the summation including $B_n$ would explode so $B_n=0$. Result:

$V(r,\theta)= \frac{kqdcos\theta}{r^2} + \sum_{n=0}^\infty A_nr^nP_n(cos\theta)$

I have solved systems similar to this without the dipole, for example. I'm not sure how to go about solving this with that there. I did see a suggestion somewhere to notice that $P_1(cos\theta)=cos\theta$ but I am not sure how to utilize this fact.

Any suggestions? Thanks!

 

[Orodruin]

The sphere is grounded, what does this tell you?

 

[oddjobmj]

I'm sorry for leaving that out. It was written down but I must have forgotten to type it out:

V(r=R)=0

 

[Orodruin]

Yes, so how can you use this piece of information?

 

[oddjobmj]

$V(R,θ)=0= \frac{kqdcos\theta}{R^2} + \sum_{n=0}^\infty A_nR^nP_n(cos\theta)$

So:
$-\frac{kqdcos\theta}{R^2} = \sum_{n=0}^\infty A_nR^nP_n(cos\theta)$

If n=1 then you can divide out the $cos\theta$ but justifying this (i.e. not considering other n values) is what I am not sure of. Also where to go from there is a little unclear.

 

[Orodruin]

Since you are asking this question, I assume you are taking a course where Legendre polynomials are being discussed? What properties of Legendre polynomials are you aware of? If this does not ring a bell, are you familiar with Fourier series and seeing function spaces as vector spaces?

 

[oddjobmj]

Not sure how to say this accurately but I know they are orthonormal. Perhaps I could multiply by $P_m$ and the only terms that would matter are the ones where m=n. I've seen that done but why and when to do that is unclear. I am familiar with the form of Fourier series but not really seeing function spaces as vector spaces generally.

 

[Orodruin]

So knowing that they are orthonormal is enough in this case (in fact it is enough to know that they are linearly independent). You have already hinted at it but let us make it explicit: Can you express the dipole potential (the term out of the sum) in terms of Legendre polynomials? How does this help? (Note that the sum is a linear combination of Legendre polynomials)

 

[oddjobmj]

The dipole potential is a constant times $P_1$. Is that what you mean?

 

[Orodruin]

Yes. So you have a linear combination of linearly independent functions which should be equal to a constant times one of those functions. What does that tell you about the coefficients in the linear combination? (In terms of a vector space, each Legendre polynomial is a basis vector in a basis spanning the entire space)