Question

The dipole moment of a system of charge + q distributed uniformly on an arc of radius R subtending an angle $\pi /2$ at its centre where another charge $-q$ is placed is $\frac{\sqrt{n}qR}{\pi }$. Then the value of $n$ is (answer upto two decimal places)

Answer 1

$\lambda =\frac{q}{\pi R/2}=\frac{2q}{\pi R}$

$d{p}_2={\int }_0^{\pi /2}dp\cos \theta$
$={\int }_0^{\pi /2}\left(\lambda Rd\theta \right)R\cos \theta$
=$\lambda R^2{\int }_0^{x/2}\cos \theta d\theta$
=$\lambda R^2[\sin \theta {]}_0^{\pi /2}$

$=\lambda R^2\cdot 1=\frac{2q}{\pi R}\cdot R^2=\frac{2qR}{\pi }$

${dp}_1={\int }_0^{\pi /2}dp\sin \theta =\lambda R^2{\int }_0^{\pi /2}\sin \theta d\theta$
dP =$\sqrt{(d{p}_1{)}^2+(d{p}_2{)}^2}$
$p=\frac{2\sqrt{2}qR}{\pi }$
$\therefore \sqrt{n}=2\sqrt{2}$
$⟹n=8$