Question

Question:

An elecric dipole of two charged particles of magnitude q but of opposite sign, oriented along z-axis separated by distance d as shown in fig.1. A point P is located on z-axis at distance z from the center of the electric dipole. Find out the expression of the electric field E of dipole at point P when z>>d.

Answer 1

$E=E_+-E_-=\frac{1}{4\pi \epsilon_0}\frac{q}{r^2_+}-\frac{1}{4\pi \epsilon_0}\frac{q}{r^2_-}=$

$=\frac{1}{4\pi \epsilon_0}\frac{q}{(z-\frac{1}{2}d)^2}-\frac{1}{4\pi \epsilon_0}\frac{q}{(z+\frac{1}{2}d)^2}=$

$=\frac{q}{4\pi \epsilon_0}(\frac{1}{(1-\frac{d}{2z})^2}-\frac{1}{(1+\frac{d}{2z})^2})=$

$=\frac{q}{4\pi \epsilon_0z^2}\cdot \frac{2d/z}{(1-(\frac{d}{2z})^2)^2}=\frac{q}{2\pi \epsilon_0z^3}\cdot \frac{d}{(1-(\frac{d}{2z})^2)^2}.$

$z>>d$

So, we have

$E=\frac{qd}{2\pi \epsilon_0z^3}=\frac{p}{2\pi \epsilon_0z^3}$.