Question

A square wire frame of side $a$ is placed a distance $b$ away from a long straight conductor carrying current $I$. The frame has resistance $R$ and self inductance $L$. The frame is rotated by ${180}^o$ about OO' as shown in figure. Find the electric charge flown through the frame.

• A. $\frac{2{\mu }_{0}i{a}^2}{2\pi Rb}$
• B. $\frac{{\mu }_{0}i}{2\pi R}{\log }_e\frac{b+a}{b-a}$
• C. $\frac{{\mu }_{0}ia}{2\pi R}{\log }_e\frac{b+a}{b-a}$
• D. none of these

Answer 1

Answer: (C)

$\frac{{\mu }_{0}ia}{2\pi R}{\log }_e\frac{b+a}{b-a}$

$i=\frac{1}{R}[\frac{d\psi }{dt}+L\frac{di}{dt}]$
$q=\int idt=\frac{1}{R}[\Delta \psi +0]=\frac{\Delta \psi }{R}=\frac{1}{R}{\int }_{b-a}^{b+a}Ba.dx$
$=\frac{1}{R}{\int }_{b-a}^{b+a}\frac{{\mu }_{0}ia}{2\pi x}dx=\frac{{\mu }_{0}ia}{2\pi R}{\log }_e\frac{b+a}{b-a}$