A charge q is distributed uniformly on a ring of radius R. A sphere of equal radius R is constructed with its centre at the periphery of the ring, find the flux of the electric field through the surface of the sphere

Given Data: A sphere of equal radius R is constructed with its centre at the periphery of the ring as shown below.

Concept Used: Gauss law

Detailed Solution:

The created diagram depicts the given system. 

In $\Delta {\mathrm{OO}}^{\text{'}}\mathrm{A}$

From symmetry, we can say,

$$\begin{gathered} \frac{\theta }{2}=60° \\ \theta =120° \end{gathered}$$

Let the length of ring inside the sphere i.e AOB be $\ell$

So, $\ell =R\theta =R\left(\frac{2\pi }{3}\right)$

Then, the charge on section AOB will be,

$q_{in}$=$\left(\frac{q}{2\pi R}\right)\ell$=$\left(\frac{q}{2\pi R}\right)\left(R\left(\frac{2\pi }{3}\right)\right)=\frac{q}{3}$

Now, the flux through the sphere is given as, by Gauss law,

$\varphi =\frac{q_n}{{ϵ}_0}=\frac{q}{3{ϵ}_0}$