# Resultant of Different Permittivity substances

• I
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
Gold Member
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.

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• Yajee
Cryo
Gold Member
Problem (or something very similar) is treated in

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

• Yajee