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 Fig. Find the electric charge flown through the frame -

• A. $\frac{2 \mu _{0} i a^{2}}{2 \pi R b}$
• B. $\frac{\mu _{0} i}{2 \pi R}log_{e} \frac{b + a}{b - a}$
• C. $\frac{ \, \mu _{0} i a}{2 \pi R}log_{e} \frac{b + a}{b - a}$
• D. None of these

Answer 1

Answer: (C)

$i=\frac{1}{R}\left[\frac{d \phi}{d t} + L \frac{d i}{d t}\right]$
$q=\displaystyle \int i d t=\frac{1}{R}\left[\Delta \phi + 0\right]=\frac{\Delta \phi}{R}=\frac{1}{R}\displaystyle \int _{b - a}^{b + a} B a d x$
$=\frac{1}{R}\displaystyle \int _{b - a}^{b + a} \frac{\mu _{0} i a}{2 \pi x}dx=\frac{\mu _{0} i a}{2 \pi R}log_{e} \frac{b + a}{b - a}$