Question

The square of side a above contains a positive point charge +Q fixed at the lower left corner and negative point charges –Q fixed at the other three corners of the square. Point P is located at the center of the square.

a. On the diagram, indicate with an arrow the direction of the net electric field at point P.

b. Derive expressions for each of the following in terms of the given quantities and fundamental constants.

i. The magnitude of the electric field at point P

ii. The electric potential at point P

c. A positive charge is placed at point P. It is then moved from point P to point R, which is at the midpoint of the bottom side of the square. As the charge is moved, is the work done on it by the electric field positive, negative, or zero?

____Positive ____Negative ____Zero

Explain your reasoning.

d. i. Describe one way to replace a single charge in this configuration that would make the electric field at the center of the square equal to zero. Justify your answer.

ii. Describe one way to replace a single charge in this configuration such that the electric potential at the center of the square is zero but the electric field is not zero. Justify your answer.

Answer 1

b. i. The fields at point P due to the upper left and lower right negative charges are equal in magnitude and opposite in direction so they sum to zero. The fields at point P due to the other two charges are equal in magnitude and in the same direction so they add.

Using r² = a²/2 we have E = 2 × kQ/r² = 4kQ/a²

ii. V = ΣkQ/r = k(–Q – Q – Q + Q)/r = –2kQ/r with r = a/√2 giving V = –2√2kQ/a

c. Negative. The field is directed generally from R to P and the charge moves in the opposite direction. Thus, the field does negative work on the charge.

d. i. Replace the top right negative charge with a positive charge OR replace the bottom left positive charge with a negative charge. The vector fields/forces all cancel from oppositely located same charge pairs.

ii. Replace the top left negative charge with a positive charge OR replace the bottom right negative charge with a positive charge. The scalar potentials all cancel from equidistant located opposite charge pairs. The field vectors in these cases will not cancel.