Derive an expression for electric field …

Electric field at point on equatorial line of an electric dipole.

Consider on electric dipole consisting of two point-charges +q and –q separated by distance 2a.

Let P be a point at a distance r from the center of the dipole on the equatorial line, where the electric field is to be calculated.

Let E₁ be electric field intensity at P due to charge –q. Therefore,

From right-angled triangle AOP, we have AP = √r² + a² . Therefore, 

The direction of Vector E₁ is along PA.

Now, let E₂ be electric field intensity at P due to charge +q.

From right-angled triangle BOP, we have BP = √ r² + a².There fore,

The direction of E₂ is along BP.

The resultant electric field Vector E is the difference of the two fields:

Thus, the electric potential due to electric dipole is zero at every point on the equatorial line of the dipole.