Question

Drive an expression for Electric field due to an Electric dipole at a point on the axial line.

Answer 1

Let the electric field at an axial point P at an distance $r$ from the center of the dipole be $E_p$.

Electric field at P, $E_p=E_q-E_{-q}$

Or $E_p=\frac{kq}{(r-a{)}^2}-\frac{kq}{(r+a{)}^2}$ where $k=\frac{1}{4\pi {ϵ}_o}$

Or $E_p=kq[\frac{1}{(r-a{)}^2}-\frac{1}{(r+a{)}^2}]$

Or $E_p=kq\left[\frac{(r+a{)}^2-(r-a{)}^2}{(r+a{)}^2(r-a{)}^2}\right]$

Or $E_p=kq\left[\frac{4ar}{(r^2-a^2{)}^2}\right]$

Or $E_p=\frac{2kPr}{(r^2-a^2{)}^2}$ where $P=2qa$

For $r>>a$, we can neglect $a^2$ compared to $r^2$.

We get $E_p=\frac{2kPr}{r^4}$

$⟹$ $E_p=\frac{2kP}{r^3}$