Question

Obtain an expression for intensity of electric field at a point in end on position, i.e., axial position of an electric dipole.

Answer 1

Let the the charge of electric dipole be $q$ and length be $2d$.
Electric field at an axial point P which is at a distance $r$ from the centre of diople $E_P=E_{+}-E_{-}$
$\therefore$ $E_P=\frac{Kq}{r_1^2}-\frac{Kq}{r_2^2}$
where $K=\frac{1}{4\pi {ϵ}_o}$
Or $E_P=\frac{Kq}{(r-d{)}^2}-\frac{Kq}{(r+d{)}^2}$
Or $E_P=Kq(\frac{1}{(r-d{)}^2}-\frac{1}{(r+d{)}^2})$
Or $E_P=Kq(\frac{(r+d{)}^2-(r-d{)}^2}{(r-d{)}^2(r+d{)}^2})$
Or $E_P=Kq(\frac{4rd}{(r^2-d^2{)}^2})$
$⟹$ $E_P=\frac{2KPr}{(r^2-d^2{)}^2}$
where $P=2qd$ (Dipole moment)
If $r>>d$, we get $E_P=\frac{2KP}{r^3}$