Question

4 charges equal to -Q placed at the 4 corners of a square and a charge q at the centre ,if system is in equilibrium q is

Answer 1

Here since the system is in equilibrium hence the net force on each charge q will be 0. We are considering 1 such charge. Here:

Force F = Kq2/a2 +Kq2/a2 + the forces due to diagonal element

First lets find he resultant of the 1st two forces:

R = √2(Kq2/a2)

Now force due to diagonal charge q :

F1 = Kq2/(a√2)2 where a√2 is the length of the diagonal

F1 = Kq2/2a2

Force F2 due to Q :

F2 = KqQ/(a√2/2)2

Net force on q hence = R +F1+F2 = 0

√2Kq2/a2 +Kq2/2a2 +2KqQ/a2 = 0

Cancelling Kq/a2 on both sides:

Q = -(4√2 +2)q/8