Question

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

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

Answer 1

Answer: (A)

$+2$

The electric field due to ring at distance $l$ from the center is $E=\frac{Ql}{4\pi {ϵ}_0(l^2+R^2{)}^{3/2}}$

now, ${\int }_0^{\infty }\overrightarrow{E}.\overrightarrow{dl}={\int }_0^{\infty }\frac{Ql}{4\pi {ϵ}_0(l^2+R^2{)}^{3/2}}dl$

$=\frac{Q}{4\pi {ϵ}_0}{\int }_R^{\infty }\frac{y}{y^3}dy$ let,$l^2+R^2=y^2,2ldl=2ydy$

$=\frac{Q}{4\pi {ϵ}_0}[-\frac{1}{y}{]}_R^{\infty }$

$=\frac{Q}{4\pi {ϵ}_{0}R}$

$=\frac{1.11\times {10}^{-10}}{4\pi \times 8.854\times {10}^{-12}\times 0.5}=2$ volt