Two identical positive charges are placed at the diagonally opposite corners of a square . Two more identical but negative charges are placed at the other two corners. What is the work done in putting all the charges together at the centre . If q is the magnitude of each charge and 2a is the length of each of the square ?

Options

A. `(sqrt(2)q^(2))/(8piepsilon_(0)a)`
B. `(4+q^(2))/(8piepsilon_(0)a)`
C. `(q^(2)(4-sqrt(2)))/(8piepsilon_(0)a)`
D. `(q^(2)(2-sqrt(2)))/(8piepsilon_(0)a)`

Answer

C

Explanation

`W = U_(2) - U_(1) , U_(2) = 0 ` is P.E at centre
`U_(1) ` = P.E of all in original state

Now
`U_(1) = [ 1/(4piepsilon_(0)) - (q^(2))/(2a) - (q^(2))/(2a)- (q^(2))/(2a) + (q^(2))/(2a)+ (q^(2))/(sqrt(2.2a)) + (q^(2))/(2sqrt(2.a))] `
` = (-q^(2) .(4-sqrt(2)))/(8pi epsilon_(0)a)`
`:. " " W = U_(2) - U_(1) = (q^(2)(4-sqrt(2)))/(8piepsilon_(0)a)`