Question

A circle, having radius "r" has line charge distribution over its circumference having linear charge density $\lambda ={\lambda }_0{\cos }^2\theta$. Calculate the total electric charge residing on the circumference of the circle.
$[\stackrel{2\pi }{\underset{0}{\int }}{\cos }^2\theta d\theta =\pi ]$

Answer 1

Let us take an element of angular width $d\theta$ at angle $\theta$.

Length of this element $=Rd\theta$

Charge on this element is $dq=\lambda Rd\theta ={\lambda }_o{\cos }^2\theta Rd\theta$

Total charge =$Q=\int q={\lambda }_{o}R{\int }_0^{2\pi }{\cos }^2\theta d\theta$

$⟹Q={\lambda }_o\pi R$