Question

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

• A. $\frac{q}{{ϵ}_0}$
• B. $\frac{q}{2{ϵ}_0}$
• C. $\frac{q}{3{ϵ}_0}$
• D. Zero.

Answer 1

The correct option is C $\frac{q}{3{ϵ}_0}$

Let point of intersection of ring & sphere are $P.Q.........$ let center of ring is $O$ & center of sphere is $O^{\prime }$........

$O{O}^{\prime }$ bisects $PQ$ in two equal ,if $S$ is the point of intersection of $O{O}^{\prime }$ & $PQ$ then

In triangle right angles $POS$

$PO=R,OS=R/2$ so angle $POS=\theta$

Using trigonometry, $cos\theta =OS/PS=1/2$

$\theta =\pi /3$

total angle subtended by arc is $2\pi /\mathrm{3.....}$ this is the arc of ring which lies inside sphere & substends

$2\pi /3$ angle at the center of ring.....

for $2\pi$ radian charge =q

for unit radian $=\frac{q}{2\pi }$

for $\frac{q}{2\pi }$ radian $q_1=\left(\frac{q}{2\pi }\right)\left(\frac{2\pi }{3}\right)=\frac{q}{3}$

total flux through sphere= total charge enclosed $\frac{q}{3{\epsilon }_0}$

$\therefore$ Option $C$ is correct.