Question

A semi circular ring of radius $R$ as shown in figure has charge per unit length $\lambda ={\lambda }_o\cos \theta$. Electric dipole moment of this ring is

• A. $\lambda R^2$
• B. $2\lambda R^2$
• C. $\frac{\lambda R^2}{2}$
• D. $\frac{3}{2}\lambda R^2$

Answer 1

The correct option is B $2\lambda R^2$


The dipole moment by a small elemental length $dl$ having charge $dq$ making an angle $\theta$ with the x-axis at origin as shown in the figure, is given by :

$dp=dq2R\cos \theta$
(considering the symmetry along y-axis)

Here,
$dq=\lambda Rd\theta$

Therefore,
$dp=\lambda Rd\theta \times 2R\cos \theta =\lambda 2R^2\cos \theta d\theta$

$\Rightarrow p=\int dp=\lambda 2R^2{\int }_0^{\frac{\pi }{2}}\cos \theta d\theta$
$(\because$ Dipole moment due to each elemental length lies in the same direction$)$

$\Rightarrow p=2\lambda R^2$