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Cryo

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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.

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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Cryo

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Griffiths "Introduction to Electrodynamics", Example 4.8 (Ch4)

Jackson "Classical Electrodynamics", Section 4.4

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