Question

A non-conducting ring of radius $0.5m$, $1.11\times {10}^{-10}$ coulomb charge is non-uniformly distributed over the circumference of ring, produces electric field $E$ around itself. If $\ell =0$ is the centre of ring, then the value of ${\int }_{\ell =-\infty }^0-E.d\ell$ is :-

• A. $2V$
• B. $-2V$
• C. $-1V$
• D. $zero$

Answer 1

Answer: (A)

$2V$

We have,

${\int }_{l=\infty }^{l=0}-\overline{E}.\overline{d}l$

$=V_{i=0}=V_{i=\infty }$

or, ${\int }_{l=\infty }^{l=0}-\overline{E}\overline{dl}=V_{l=0}$ ($\because$ as per the usual conversation, potential at infinity is $0$)

Potential at the centre of the ring, whether the charge distribution is uniform or non-uniform $=\frac{1}{4\pi {ϵ}_0}\frac{Q}{R}$

$\therefore$ ${\int }_{l=\infty }^{i=0}-\overline{E}\overline{dl}=V_{l=0}=\frac{1}{4\pi {ϵ}_0}\frac{Q}{R}$

$\therefore$ ${\int }_{t=\infty }^{l=0}-\overline{E}\overline{dl}=9\times {10}^9\times \frac{1.1\times {10}^{-10}}{0.5}V=2V$