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Apr 3, 2020 at 11:51 comment added user224659 Yes. Just some terminology: What you call the "source" I would call the enviroment and what you call the test charge I would call the system. This generalizes the ideas even further and shows that this approximation is not limited to electrodynamics. Glad this was helpful. I just recently read something very similar in Landaus introduction to statistical physics.
Apr 3, 2020 at 11:35 vote accept 13509
Apr 3, 2020 at 11:35 comment added 13509 This is an amazing answer; so essentially we don't write the Hamiltonian for a single test charge, but rather for the whole system. Except in some cases the Hamiltonian can be approximated as only a function of the position of the test charge - but nonetheless, it is still of the entire system!
Apr 3, 2020 at 11:34 history edited user224659 CC BY-SA 4.0
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Apr 3, 2020 at 9:22 comment added user224659 I made some edits to make my point more clear.
Apr 3, 2020 at 9:22 history edited user224659 CC BY-SA 4.0
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Apr 3, 2020 at 9:15 history answered user224659 CC BY-SA 4.0