Question

Four charges are placed at the corners of a square, where q is positive (q > 0). Initially there is no charge in the center of the square.

a) Find the work W required to bring the charge Qf = +q from infinity and place it at the center of the square.

b) What is the magnitude of the electric force on the charge q when it is placed at the center of the square?

c) What is the magnitude of the total electrostatic energy of the system?

d) What is the sign of the total energy associated with the five point charges?

Answer 1

A) Based on the superposition principle, the potential at the center due to the charges at the corners is

V = V1 +V2 +V3 +V4 = k (q/r) (−1−1+1+1) = 0

so r = 1/√2 is the common distance from the center to the corners. The work required is then W = q V = 0.

B) Due to the symmetry of the charge distri- bution, the electric field is directed along the negative x-axis. Furthermore, the field gen- erated by each charge contributes an equal amount, so

F = 4 F1 cos 45° =4{(k_eq²) / (a/√2)²} (√2/2)

F = 4√ 2 k_e (q²/a²)

C) Total electrostatic energy, U = U12 +U13 +U14 +U23 +U24 +U34

(k_eq²/a)[ 1- (1/√ 2) -1 -1 - (1/√2) +1]

U = - √2 k_eq²/a

D) Negative sign

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