Question

Four equal positive point charges, each of charge 8.7?C , are at the corners of a square of side 9.2 cm. What charge should be placed at the center of the square so that all charges are at equilibrium?

Answer 1

Consider the force exerted on the charge q at the top leftcorner of the sqauare

For this charge tobe in equilibrium, the summation F of all the 4 forces must be zero.

The out ward forces due to the 3 charges at a, a and?2 a

?Fout = k(q2/a2 )cos 45+ k(q2/a2 )cos 45 +kq2/2a2

= k(q2/a2 )[0.707 +0.707 +0.5) =1.914 k(q2/a2 )-------(1)

along the diagonal out ward.

So, for equilibrium, the charge Q at the centre must exert anequal force, inward. Then Q < 0.

The distance from centre to the corner is a/?2.

Force exerted by Q on q, F' =kQ.q/(a/?2.)2 = 2kQ.q/a2 inward -----------(2)

From (1) and (2)

2Q. = 1.914 q

Since Q is -ve

Q = - 0.957 q = - 0.957x 8.7 ?C

Q = - 8.32 uC