Derive an expression for electric field due to electric dipole along its equatorial axis at a perpendicular distance r from its centre.

In this case we determine the electric field at a point P which is located on the perpendicular bisector of the line joining the two charges. Let the distance of P from this line be y.

Magnitude of electric field at P due to – q:

is directed from P to A. It has vertical component directed in the -y direction, and a horizontal component directed in the – x direction.

Magnitude of E at P due to + q:

is directed from B to P. It has vertical component directed in the +y direction, and a horizontal component directed in the – x direction.

The net electric field at P is calculated by adding vectorially and.

As their vertical components, and, are equal in magnitude but opposite in direction. Hence they cancel out.

And, the horizontal components are both in the –x direction. Hence they add up.

Thus,

From simple trigonometry, where we can substitute cos θ in terms of a and y, we get

For, neglect in the denominator.

Thus, we get

But the dipole moment of an electric dipole is

Hence, for a point P at a very large perpendicular distance from the axis,

(for)

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