Four charges of +q, +q, +q and +q are placed at the corners A, B, C and D of a square of side a. Find the resultant force on the charge at D. 

Explanation

Let side = `a, BD= sqrt(2)a`
Along AD force `F_1  = 1/(4 pi epsilon_0)cdot (q^2)/(a^2)`
Along CD force `F_2 = 1/(4 pi epsilon_0) cdot (a^2) :. F_1 = F_2`
Along AD force `F_1 = 1/(4 pi epsilon_0) cdot (q^2)/(a^2) "   " :. F_1 = F_2`
Resultant of `F_1` and `F_2 = sqrt((F_1^2) + F_2^2)) = sqrt(2) .F_1`
Force along BD
`F_3 = 1/(4 pi epsilon_0) cdot (q^2)/((sqrt(2)a)^2) = 1/2 1/(4 pi epsilon_0) (q^2)/(a^2)`
Resultant force at `D = sqrt(2) F_1 + F_3`
`= sqrt(2) 1/(4 pi epsilon_0) cdot (q^2)/(a^2) + 1/2 1/(4 pi epsilon_0) cdot (q^2)/(a^2)`
`= 1/(4 pi epsilon_0) (q^2)/(a^2) [ sqrt(2) + ½] = (q^2)/(8 pi epsilon_0 a^2) cdot [1 + 2sqrt(2)]`