Question

4.2 One evening, you are at home playing with point charges. During a commercial break for Star Trek Discovery (official sponsor of PHYS321), you assemble 8 point charges Q to be fixed onto the corners of a cube of edge length b. Your room-mate asks, "Huh, if I take another charge Q and stick it at the center, won't that be in static equilibrium?" After a moment of thought, you loudly scoff at the idea, knock over all the furniture, and tell them that Klingon house leader L'Rell would insist that their idea "has no honor", A. Using Gauss's law, prove that the apparent equilibrium position at the center of cube must be a point of unstable equilibrium B. Using the self-averaging property of solutions to Laplace's equation, prove that the apparent equilibrium position at the center of cube must be a point of unstable equilibrium. (Hint: Your answers should not depend on the exact charge distribution, simply that is static and that there is some spatial separation between the test charge and the rest of the charges, so don't try to do any nasty superposition of field or potential from the charges at the corners.)

Can someone carefully answer and explain questions A and B in detail, please?

Answer 1

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