Question

Derive an expression for the electric potential at any point along the axial line of an electric dipole.

Answer 1

Suppose P is a point on the axial position of the dipole.
Length of dipole = 2a  
Suppose point P is at the distance 'r' from the center of the dipole. 

The potential at a point is V = `1/(4piε_0).Q/r`

So, the potential at P due to q is V_q = `1/(4piε_0).q/(a+r)`

Potential at P due to -q is V_-q = `1/(4piε_0).(-q)/(a-r)`

The total potential at P is `V = V_q + V_(-q)`

= `1/(4piε_0).q/((a + r)) + 1/(4piε_0).(-q)/((r - a))`

= `q/(4piε_0)[1/((a + r)) + 1/((a - r))]`

V = `q/(4piε_0).(2a)/((a^2 - r^2))`