The linear charge density on a dielectric ring of radius R is varying with $\theta$ as $\lambda ={\lambda }_0\cos \frac{\theta }{2}$, where ${\lambda }_0$ is constant. The potential at the centre O of ring [in volt] is_____________

The charge on the infinitesimal elements of arc which subtend an angle d$\theta$ at the centre of the ring

$dQ=\lambda Rd\theta ={\lambda }_0\cos \frac{\theta }{2}Rd\theta$

Potential at the centre of ring due to charge dQ

$\begin{array}{l}dV=\frac{1}{4\pi {ϵ}_0}\frac{dQ}{R}=\frac{{\lambda }_0\cos \frac{\theta }{2}\cdot Rd\theta }{4\pi {\in }_{0}R}\\ V=\int dV\Rightarrow V=\frac{{\lambda }_0}{4\pi {ϵ}_0}{\int }_0^{2\pi } \cos \frac{\theta }{2}d\theta \\ =\frac{{\lambda }_0}{4\pi {ϵ}_0}{\left[\frac{\sin \frac{\theta }{2}}{\frac{1}{2}}\right]}_0^{2\pi }=0\mathrm{V}\end{array}$