Question

(a) Define electric dipole moment. Is it a scalar or a vector? Derive the expression for the electric field of a dipole at a point on the equatorial plane of the dipole.

(b) Draw the equipotential surfaces due to an electric dipole. Locate the points where the potential due to the dipole is zero.

Answer 1

It is defined as the product of either charge and the distance between the two equal and opposite charges

Derivation

At a point of equatorial plane : Consider a point P on broad side on the position of dipole formed of charges + q and - q at separation 2l. The distance of point P from mid point (O ) of electric dipole is r. Let E1 and E2 be the electric field strengths due to charges + q and - q of electric dipole.

To find the resultant of vector E1 and vector E2, we resolve them along and perpendicular to AB.

Component of vector E1 along AB = E1cosθ, parallel to B vector A

Component of vector E1perpendicular to AB = E1sinθ along O to P

Component of vector E2 along AB = E2cosθ, parallel to B vector A

Component of vector E2 perpendicular to AB = E2sinθ along P to O

Clearly components of vector E1 and vector E2 perpendicular to AB : E1sinθ and E2sinθ being equal and opposite cancel each other, while the components of vector E1 and vector E2 along AB : E1cosθ and E2cosθ, being in the same direction add up and give the resultant electric field whose direction is parallel to B vector A.

If dipole is infinitesimal and point P is far away, we have l < <r , so l2 may be neglected as compared to r2 and so equation (3) gives.

i.e. electric field strength due to a short dipole at broadside on position

Electric potential is zero at all points in the plane passing through the dipole equator.