Four charges are arranged at the corners of a square ABCD pf side d, as shown in Fig. Find the work required to put together this arrangement (b) A charge `q_(0)` brought to the center E of the square, the four charges being held fixed at the corners . How much extra work in needed to do this ?

Explanation

(a) Here, AB = BC = CD = DA = d
`AC = BD = sqrt(d^(2) + d^(2)) = d sqrt(2)`
Total work done = P.E. of system fo 4 charges
`= (1)/(4pi in_(0)) sum_("all pairs") (q_(j) q_(k))/(r_(jk))`
`= (1)/(4pi in_(0)d) [q(-q) + (-q) (+q)`
`+q(-q) + (-q) (+q)] + (1(q^(2) + q^(2)))/(4pi in_(0) d sqrt(2))`
`= (-4 q^(2))/(4pi in_(0) d) + (2q^(2))/(4pi in_(0) d sqrt(2))`
`= (-q^(2))/(4pi in_(0) d) (4 - sqrt(2))`
(b) Extra work needed = charge `xx` potential at E due to charges at four corners.
`W = q_(0) xx ((q-q+q-q)/(4pi in_(0) (d sqrt(2)/(2)))) = zero`