Question

Derive an expression for the intensity of the electric field at a point on the axial line of an electric dipole.

Answer 1

Consider the an electric dipole of charges $+q$ and $-q$ separated by distance $2a$ with center at O.

Goal: To find electric field at point P on the axial line of the dipole, $OP=r$.

Let $E_1$ and $E_2$ be electric field on P due to charges $+q$ and $-q$ respectively.

$E_1=\frac{kq}{(r-a{)}^2}$ along AP

$E_2=\frac{kq}{(r+a{)}^2}$ along BP

The resultant electric field at P, $E=E_1-E_2$ (as both $E_1$ and $E_2$ are in opposite direction)

$E=\frac{kq}{(r-a{)}^2}-\frac{kq}{(r+a{)}^2}=kq\frac{4ra}{(r^2-a^2{)}^2}$

Define, $p=2aq$

$E=k\frac{2pr}{(r^2-a^2{)}^2}$

If $r>>a$, then $E=k\frac{2pr}{r^4}=k\frac{2p}{r^3}$.

In vector form, $\overrightarrow{E}=k\frac{2\overrightarrow{p}}{r^3}$.