Four electric charges `+q, +q, -q` and `-q` are placed at the corners of a square of side 2L (see figure). The electric potential at point `A`, mid-way between the two charges `+q` and `+q`, is

Options

A. `(1)/(4pi in_(0)) (2q)/(L) (1 + sqrt(5))`
B. `(1)/(4pi in_(0)) (2q)/(L) (1 + (1)/(sqrt(5)))`
C. `(1)/(4pi in_(0)) (2q)/(L) (1 - (1)/(sqrt(5)))`
D. Zero

Answer

C

Explanation

Refer to Fig

`AB = AE = L`
`AC = AD = sqrt((2L)^(2) + L^(2)) = L sqrt(5)`
Electric potential at `A` due to given charge configuration is
`V_(A) = (1)/(4pi in_(0)) [(q)/(AB) + (q)/(AE) + ((-q))/(AC) + ((-q))/(AD)]`
`= (1)/(4pi in_(0)) [(q)/(L) + (q)/(L) - (q)/(L sqrt(5)) - (q)/(L sqrt(5))]`
`= (1)/(4pi in_(0)) [(2q)/(L) - (2q)/(L sqrt(5) )- (q)/(L sqrt(5))]`
`= (1)/(4pi in_(0)) [(2q)/(L) - (2q)/(L sqrt(5))] = (1)/(4pi in_(0)), (2q)/(L) [1- (1)/(sqrt(5))]`