Suppose we place two charges and in the cartesian plane in the points and and we want to find the locus where the potential of the field is zero (Notice that in order to simplify the problem we put the two charges lying in the x axis).
Without any loss of generality we can assume that (1), where λ is a costant. In order for the net potential of the field to give zero, the two charges need to be of opposite signs and thus we conclude that λ in equation (1) is a negative constant. Consequently, it is true that (2).
Now, lets assume there is a point in 2D space which satisfies the condition that its potential is zero. From the superposition principle it is true that:
which expands to:
.From the assumption (1) that we introduced, the equation above transforms to:
and are non-zero constants so equation (3) becomes:
It is true that:
and (4) transforms to:
Now both sides of the equation above are positive and thus we can square both sides:
If (5) becomes:
Equation (6) describes a circle (in the form ) with its centre moving in the x' axis.
Its centre K is
and its radius ():
Now we need to check the case of :
From the equation (5) we get:
So the zero potential locus is either a circle when or the perpendicular bisector of (AB) when and thus .