A Green's function for problems involving linear isotropic media

[spin_100]

TL;DR Summary

Is there a way to tackle problems involving linear isotropic dielectric media with permittivity separated by a boundary directly using an appropriate green's function? I am studying electrodynamics from Jackson's electrodynamics and after learning about the power of using green's function to solve boundary value problems, I was wondering if there is something similar to this for dielectric media. I have shown my approach here but I am stuck at a point.

I am considering a simple problem of a sphere of isotropic dielectric media (permittivity $\epsilon$ and Radius $R$) placed in a uniform electric field $E_0$ (z-direction). The problem is from Griffiths Chapter 4, example 7.
Since, it is a linear dielectric material, $D = \epsilon E$ Since there is a discontinuity in $\epsilon$
We can model $\epsilon (r) = \epsilon \theta (R-r) + \epsilon_0 \theta (r-R)$
Taking the divergence of D and since there are no free charges. (external charges) we get $$ 0 = \epsilon(r) \nabla \cdot E + \delta (R-r) ( \epsilon_0 - \epsilon ) E \cdot r$$

Then, we get the possion's equation $$ \nabla \cdot E = - \delta (R-r) ( \epsilon_0 - \epsilon ) E \cdot r $$ After that I don't know how to proceed further. I solved for the potential using an appropriate green's function but the result I am getting is wrong.

 

[vanhees71]

You can get the Green's function using the image-charge concept. The Green's $G(\vec{x},\vec{x}')$ function is given by the solution of the problem having a "unit charge" at $\vec{x}'$. Then you need an image charge (distinguishing the cases when $\vec{x}'$ is inside or outside the sphere). The case of a homogeneous electric field is given by taking two charges $\pm Q$ to infinity and moving them to $\pm \infty \vec{e}_z$ (for the asymptocally constant electric field in the $z$ direction).

You can also solve the problem with the asymptotically constant electric field by making a multipole-expansion ansatz. From symmetry it's a priori clear that you only need to go to the dipole order.

 

[spin_100]

You can get the Green's function using the image-charge concept. The Green's $G(\vec{x},\vec{x}')$ function is given by the solution of the problem having a "unit charge" at $\vec{x}'$. Then you need an image charge (distinguishing the cases when $\vec{x}'$ is inside or outside the sphere). The case of a homogeneous electric field is given by taking two charges $\pm Q$ to infinity and moving them to $\pm \infty \vec{e}_z$ (for the asymptocally constant electric field in the $z$ direction).

You can also solve the problem with the asymptotically constant electric field by making a multipole-expansion ansatz. From symmetry it's a priori clear that you only need to go to the dipole order.

Please provide a reference where such a method is discussed.

 

[vanhees71]

See Sect. 1.8.2 of my E&M manuscript. It's however in German :-):

https://itp.uni-frankfurt.de/~hees/publ/theo2-l3.pdf