Potential of a Dielectric sphere and an exterior point charge.

[icelevistus]

Consider a sphere of radius a with dielectric constant epsilon. A point charge q lies a distance d from the center of the sphere. Assume d > a, calculate the potential for all points inside the sphere, and outside the sphere.



The problem is to be solved with an expansion of Legendre Polynomials. Note the azimuthal symmetry of the problem.



This is a boundary condition problem. Coefficients are solved by equating the tangent electric field inside and outside the sphere at its boundary, and equating the normals of the of D also at the boundaries. The main problem is finding a way to express the potential outside the sphere. The point charge of course introduces a singularity that makes it difficult (you can express the potential as a Legendre expansion of the sphere plus the point charge potential, but this creates a mess for applying the boundary conditions. Any ideas?

 

[gabbagabbahey]

I'm assuming you already know the general solution to Laplace's equation?...Well outside the sphere, the potential will obey Laplace's equation everywhere except at the location of the point charge itself; luckily you don't really need to know the potential at that exact location. All you need is to determine suitable boundary conditions for the problem, and then solve for your coefficients as usual...Two of the three necessary BC's are simple: (1)The potential is continuous at the boundary of the sphere (r=a) and (2) The perpendicular components of the displacement field are equal at r=a (since there is no free surface charge on the sphere)...The hard part is finding a suitable 3rd boundary condition...hint: what does the potential look like at points far from the sphere but not too far from the point charge?

 

[icelevistus]

OK, so for the outside potential, all the positive powers of r have zero coefficients.

I went through the other BCs for this, but you're left with two equations relating the two coefficients that can only be solved by setting all the coefficients equal to zero.

 

[icelevistus]

(I deduced that the potential far from the sources should reduce to just the potential of the point charge, since the dipole part of the sphere will die out)

 

[gabbagabbahey]

OK, so for the outside potential, all the positive powers of r have zero coefficients.

I went through the other BCs for this, but you're left with two equations relating the two coefficients that can only be solved by setting all the coefficients equal to zero.

You should be getting an infinite series, with non-zero coefficients...if you post your work (just the part where you apply the BC's), I should be able to help you out on this.

 

[gabbagabbahey]

(I deduced that the potential far from the sources should reduce to just the potential of the point charge, since the dipole part of the sphere will die out)

Yes, exactly the dipole goes like 1/r^2 (and higher moments vanish even faster) while the monopole term of the point charge potential (which should be the only monopole term you get) goes like 1/r.

 

[icelevistus]

I solved the problem, it turned out that including the potential of the point particle in the expansion did not cause as many problems as I had initially anticipated. Thank you for the help.