Legendre Expansion solution to sphere of potential V with charge q outside

[phys_student1]

Hi,

It seems that there is no much examples of this particular case.

OK, we all know how to write the general solution to Laplace equation in spherical coordinates in terms of Legendre polynomials (when there is azimuthal symmetry).

There are a lot of cases here but I would like to know how to attack the problem of a sphere held at potential V with a charge 1 outside a distance d.

I know you will have an image charge inside the sphere a distance d*a^2 from sphere's center.

But how to account for this system of image charge and original charge? Can we simply add a term (outside the sum of Legendre Polynomial) ?

Thanks in advance

 

[Meir Achuz]

You can put an image charge, q"=V/a+qa/d, at the center of the sphere of radius a.
You don't need Legendre polynomials for this.
The original image charge should be at d'=a^2/d.

 

[phys_student1]

You can put an image charge, q"=V/a+qa/d, at the center of the sphere of radius a.
You don't need Legendre polynomials for this.
The original image charge should be at d'=a^2/d.

I understand the image charge method.

What I seek is the solution utilizing Legendre Polynomials.

Again, I understand Legendre Polynomial solution as applied to different cases, but not this one.