Question

Consider an electron between two grounded conducting parallel plates perpendicular the x axis, and intersecting x at ±a. A particle with charge q is placed at x = a/2 between the plates. Derive an expression for the magnitude and direction of the force exerted on the particle. Write your answer in terms of the parameters q and a of the problem, and use a computer if necessary to obtain a numerical value for your answer. Computed numerical values should be provided with at least four significant figures of accuracy.

Answer 1

Since the plates are infinite, we won't be able to get a Gauss's surface around them, but, to find the answer to this question, we don't have to. If the distance between the plates is not infinite, we could make the Gauss's surface big enough to achieve any level of accuracy.

Since the potential of the plates is zero, there won't be any electric field on their external surfaces and no electric field around or inside the grounding wires (presumably, one wire for each plate). The field between the plates won't be zero, but, if the Gauss's surface is big enough, it could be made as small as we choose.

Therefore, the flux through the Gauss's surface could be made as close to zero as we choose. If so, according to the Gauss's law, the charge enclosed inside the surface should be zero as well and, since the known charge between the plates is Q

, the charge induced into the plates must be −Q

.

Obviously, these charge will be located on the inside surfaces of the plates