# Potential At A Point Due To An Electric Dipole

Let us calculate Potential At A Point Due To An Electric Dipole:-

Let an electric dipole consist of two equal and opposite point charges – q at A and +q at b ,separated by a small distance AB =2a ,with centre at O.

The dipole moment                 p=q*2a

We will calculate potential at any point P,where

OP=r and  angle BOP= θ

Let                                   AP=r1 and BP=r2

Draw AC perpendicular PQ and BD perpendicular PO

In ∆AOC.                                     Cosθ=OC/OA =OC/a

OC= a cos θ

Similarly,                                      OD= a cos θ

Potential at P due to               +q=1/4πε0q/r2

And  Potential at P due to     -q= -1/4πε0 q/r1

Net potential at P due to the dipole

V=q/4πε0r2 –q/4πε0r1

V= q/4πε0[1/r2-1/r1]

Now ,               r1=AP=CP

=OP+OC

=r+a cos θ

And                   r2=BP=DP

=OP –OD

=r- a cos θ

V=q/4πε0[1/r-a cos θ – 1/r+a cos θ]

= q/4πε0[r+a cos θ –r + a cos θ/r2-a2 cos2θ]

V= q/ 4πε0[2a cos θ/r2-a2 cos2 θ]

i.e.                              V=p cosθ/4πε0(r2-a2 cos2 θ)                             (p=q*2a)

Special cases:-

(i)                 When the point P lies on the axial line of the dipole ,θ=00

Cos θ=cos 00 =1

V=p/4πε0(r2-a2)

If                             a<<r.then V=q/4πε0r2

Thus due to an electric dipole ,potential, V∞ 1/r2

(ii)               When the point P lies on the equatorial line of the dipole ,θ=900

Cos θ =cos 900=0

i.e electric potential due to an electric dipole is zero at every point on the equatorial line of the dipole.