Question: Two particles of chargesq₁and q₂ , are separated by distance in Fig. 24-40. The net electric field due to the particles is zero at x = d/4 .With V = 0 at infinity, locate (in terms of ) any point on the x-axis (other than at infinity) at which the electric potential due to the two particles is zero.

The net electric field due the particles q₁ and q₂ is zero at d/4 on the x axis. Thus, the fields due to q₁ and q₂ must be directed opposite to each other at that point otherwise the net electric field will not be zero at that point. This means that the two charges q₁ and q₂ must have the same sign to the charge either negative or positive.

The electric field is defined as the Coulombic force per unit positive charge due to another charge.

$E=\frac{kq}{r^2}$

Here, k is Coulomb’s constant, q charge of the particle, and r is the distance between the charge and the point where the electric field is located.

The electric potential is expressed as follows.

$V=\frac{kq}{r}$

The net electric field due the particles q₁ and q₂ is zero at d/4 on the x-axis/ Thus, the field due to and must be directed opposite to each other at that point otherwise the net electric field will not be zero at that point.

This means that two charges q₁ and q₂ must have the same sign to the charge either negative or positive.

The electric field is a vector quantity. Since the direction of the electric field depends on the sign of the charge. But vector potential is a scalar, so there is no direction for electric potential.

If the two particles have the same charge, then the potentials that they produce will always have the same sign. This means the electric potential at any point on the x axis due to the two charges q₁ and q₂ never cancel out.

Therefore, the net electric potential cannot possibly zero at anywhere (except infinity) on the x-axis.