Consider 4 charges (2 electrons and 2 protons) that are placed at the corners of

Question

Consider 4 charges (2 electrons and 2 protons) that are placed at the corners of a square (each side of the square is length a).
(a) Does there exist a configuration such that the electric potential is nonzero at the center? Explain your answer.
(b) For what configuration of charges is both the electric eld and electric potential zero at the center of the square?
(c) Using the configuration of charges in (b), determine how much work it takes to move the charge at the top right corner until it feels no effect from the other three charges

Please help with detailed explanation for each part above.

Explanation / Answer

pat a:

potential due to a charge q at a distance d is given by k*q/d

here all the charges are at same distance from the center of the square no matter what the configuration is.

hence as the sum of charges is zero, total potential at center will always be zero.

part b:

electric field is a vector quantity and direction is away from the charge if the charge is positive and towards the charge if the charge is negative.

hence if we place similar charges in opposite corners, their fields will counterbalance each other and net field will be zero.

in this configuration, total potential will also be zero.

part c:

it wont feel any effect from the other three charges if it removed to an infinite distance.

in the present configuration, let the top right corner charge is +q , then bottom left corner is +q

and other two corners have -q.

so the top right corner is at a distance of d from the negative charges and at a distance of sqrt(2)*d from the positive charge.

then potential at the top right corner=(-k*q/d)+(-k*q/d)+(k*q/(sqrt(2)*d))

=-1.2929*k*q/d

so work required=charge*potential difference

=q*(0-(-1.2929*k*q/d))

=1.2929*k*q^2/d

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