Question

Derive an expression for the electric field E due to a dipole of length ${}^{\prime }2a^{\prime }$ at a point distant r from the centre of the dipole on the axial line.

Answer 1

Electric field at point P at a distance r from centre of dipole due to +q
$E_B=\frac{1}{4\pi {ϵ}_0}\frac{q}{B{P}^2}$ along BP =$\frac{1}{4\pi {ϵ}_0}\frac{q}{(r-a{)}^2}$
Electric field at point P due to -q is
$E_A=\frac{1}{4\pi {ϵ}_0}\frac{q}{A{P}^2}$ along PA =$\frac{1}{4\pi {ϵ}_0}\frac{q}{(r+a{)}^2}$
Net field at P is given by
$E_P$=$E_B-E_A$=$\frac{1}{4\pi {ϵ}_0}(\frac{q}{(r-a{)}^2}-\frac{q}{(r+a{)}^2})=\frac{q}{4\pi {ϵ}_0}\frac{4ar}{(r^2-a^2{)}^2}=\frac{1}{4\pi {ϵ}_0}\frac{2pr}{(r^2-a^2{)}^2}$ along BP .
Where p is the dipole=2qa