Question

A non conducting ring of radius 0.5 m carries a total charge of $1.11\times {10}^{-10}$ C, distributed non-uniformly on its circumference, producing an electric field in space. What is the value of the line integral - ${\int }_{\infty }^l\overrightarrow{E}.\overrightarrow{dl}$ ( l = 0 being the centre of the ring), in volts.

Answer 1

We know $-{\int }_{\infty }^l\overrightarrow{E}.\overrightarrow{d}l=V_{l=0}-V_{l=\infty }$

$-{\int }_{\infty }^l\overrightarrow{E}.\overrightarrow{d}l=V_{l=0}$ (because as per the usual convention, potential at infinity is 0)

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

$-{\int }_{\infty }^l\overrightarrow{E}.\overrightarrow{d}l=V_{l=0}=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{R}$

$-{\int }_{\infty }^l\overrightarrow{E}.\overrightarrow{d}l=9\times {10}^9\times \frac{1.1\times {10}^{-10}}{0.5}V=1.98V\approx 2V$