Two charges `q_(1) and q_(2)` are placed at (0,0,d) and (0,0,-d) respectively. Find locus of points where the potential is zero.

Explanation

In fig, we have shown two charges `q_(1) (0,0,d) and q_(2) (0,0,d)`. For potential to be zero at (x,y,z) we should have `(q_(1))/(4pi in_(0) sqrt(x^(2) + y^(2) + (z - d)^(2))) + (q_(2))/(4pi in_(0) sqrt(x^(2) + y^(2) + (z + d^(2))) = 0`
`(q_(1))/(sqrt(x^(2) + y^(2) + (z - d^(2)))) = (-q_(2))/(sqrt(x^(2) + y^(2) + (z + d^(2))))` ...(i)
Clearly, total potential can be zero when `q_(1), q_(2)` have opposite signs.
Sqaring both sides of (i), we get `q_(1)^(2) [x^(2) + y^(2) + (z + d)^(2)] = q_(2)^(2) [x^(2) + y^(2) + (z - d)^(2)]`
on simlifying we get `x^(2) + y^(2) + z^(2) + [((q_(1)//q_(2))^(2) + 1)/((q_(1)// q_(2))^(2) - 1)] (2zd) + d^(2) = 0`.
This is the equaction of a sphere will center at `[0,0, -2d ((q_(1)^(2) + q_(2)^(2))/(q_(1)^(2) - q_(2)^(2)))]`. If `q_(1) = -q_(2)`, then `z = 0`.
Therefore, locus of points where potential is zero is the plane through mid point of the two charges.