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If I have some conductors in space, each at a certain potential (and assume everything is inside a conducting sphere, in order to have some well defined boundary conditions), we can calculate the potential everywhere (inside the sphere) by solving Laplace's equation. Hence a particle placed inside, will have a well defined trajectory. Of course one will need a numerical approach for an arbitrary distribution, but you can get a trajectory as accurately as you want. I am a bit confused about what happens if you place a neutral insulator material inside (assuming that everything else is the same). Will the Laplace's equation be exactly the same outside the insulator, hence the trajectory of the particle will be the same (assuming it doesn't hit the insulator), or do I have to take the insulator material into account somehow? Thank you!

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