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 $$E$$ everywhere in space. The value of the integral $$\int\limits_{\ell \, = \,\infty }^{\ell \, = \,0} { - E.d\ell } $$ ($$\ell \, = \,0$$ being center of the ring) in volts is
A.
$$+ 2$$
B.
$$- 1$$
C.
$$- 2$$
D.
zero
Answer :
$$+ 2$$
Solution :
$$\eqalign{ & \int_{\ell \, = \,\infty }^{\ell \, = \,0} { - \overrightarrow E .\overrightarrow {d\ell } } = {V_0} - {V_\infty } \cr & = \frac{{kq}}{r} - 0 \cr & = \frac{{9 \times {{10}^9} \times 1.11 \times {{10}^{ - 10}}}}{{0.5}} \cr & \approx 2\,V \cr} $$
$$\eqalign{ & \int_{\ell \, = \,\infty }^{\ell \, = \,0} { - \overrightarrow E .\overrightarrow {d\ell } } = {V_0} - {V_\infty } \cr & = \frac{{kq}}{r} - 0 \cr & = \frac{{9 \times {{10}^9} \times 1.11 \times {{10}^{ - 10}}}}{{0.5}} \cr & \approx 2\,V \cr} $$