Question

A charge $Q$ is distributed uniformly on a ring of radius $r$. A sphere of equal radius $r$ is constructed with it's centre at the periphery of the ring. Find the flux of the electric field through the surface of the sphere.

• A. $\frac{Q}{{ϵ}_0}$
• B. $\frac{Q}{3{ϵ}_0}$
• C. $\frac{Q}{2{ϵ}_0}$
• D. $\frac{Q}{4{ϵ}_0}$

Answer 1

The correct option is B $\frac{Q}{3{ϵ}_0}$

From geometry, $OA{D}_1$ is an equilateral triangle as
$OA=O_{1}A=O{O}_1=r$

So, $\angle AO{O}_1={60}^{\circ }$

from symmetry

$\angle AOB=2\times {60}^{\circ }={120}^{\circ }$

Now, length of arc of ring enclosed by sphere,

$l=\frac{\theta }{{360}^{\circ }}\times 2\pi r=\frac{{120}^{\circ }}{{360}^{\circ }}\times 2\pi r=\frac{2\pi r}{3}$

Further, charge enclosed by sphere,

$Q_{\text{enclosed}}=\frac{Q}{2\pi r}\times l=\frac{Q}{2\pi r}\times \frac{2\pi r}{3}=\frac{Q}{3}$

Hence, electric flux through sphere can be calculate by Gauss's law

$\varphi =\frac{Q_{\text{enclosed}}}{{ϵ}_0}=\frac{Q/3}{{ϵ}_0}$

$\Rightarrow \varphi =\frac{Q}{3{ϵ}_0}$

So, $\left(c\right)$ is the correct option.