I Resultant of Different Permittivity substances

[Yajee]

Say the space is divided into two parts: one has permittivity e1 and other e2. A charge q1 is placed at a distance d1 from the interface of two media in region in permittivity e1 and another charge q2 is placed at a distance d2 in region of permittivity e2. What is the force experienced by a charge due to the field of the other charge? How to calculate it?

 

[Cryo]

Solve Maxwell's Equations, to find electric field due to charges. Finding the force is then trivial.

Electrostatics is fine for this problem, so ($\vec{E}$ - electric field, $\phi$ - scalar potential, $\rho$ - charge density)

$\vec{\nabla}.\vec{E}=\rho/\epsilon$
$\vec{\nabla}\times\vec{E}=\vec{0}$

Which is satisfied if $\vec{E}=-\vec{\nabla}\phi$ and $\epsilon\nabla^2\phi=-\rho$. Since you have point-charges, the charge density is zero almost everywhere, so begin by solving $\nabla^2\phi=0$ in both domains and then determine how to stitch the solution at the boundary.

 

[Cryo]

Problem (or something very similar) is treated in

Griffiths "Introduction to Electrodynamics", Example 4.8 (Ch4)
Jackson "Classical Electrodynamics", Section 4.4