Question

Derive a relation for electric WM field of an electric dipole at a point on its equatorial line.

Answer 1

Consider an electric dipole having two charges -q and +q lying at A and B at a distance 2a. Let us determine electric field intensity at point P on the equatorial line at a distance r from the center O of the dipole. Electric field intensity E₊ due to charge +q and E_- due to charge -q at P are given by

\(E_+ = k \frac{q}{(r^2\ - a^2)} \ \ \ [k = \frac{1}{4πɛ_o}]\)

\(E_- = k \frac{q}{(r^2\ - a^2)}\)

Resolving E+ and E- into rectangular components, we have E₊ cos θ + E_- cos θ along PD. E₊ sin θ and E_- sin θ being equal and opposite mutually cancel each other. (Fig.)

∴ Net electric field at P is given by
E_eq = E₊ cos θ + E_- cos θ
= 2E cos θ [∵ E₊ = E_- = E(say)]

-ve sign means that the direction of \(\vec E_{eg}\)  is opposite to \(\vec p\). When point P is very far away from the electric dipole i.e. for a short dipole 2a << r.

\(|\vec E_{eg}| = \frac{|\vec p|}{4πɛ_or^3}\) or \(E_{eg} = \frac{P}{4πɛ_or^3}\)