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

Answer 1

Answer: (C)

$i=\frac{1}{R}\left[\frac{d\varphi }{dt}+L\frac{di}{dt}\right]$
$q=\int idt=\frac{1}{R}\left[\Delta \varphi +0\right]=\frac{\Delta \varphi }{R}=\frac{1}{R}{\int }_{b-a}^{b+a}Badx$
$=\frac{1}{R}{\int }_{b-a}^{b+a}\frac{{\mu }_{0}ia}{2\pi x}dx=\frac{{\mu }_{0}ia}{2\pi R}lo{g}_e\frac{b+a}{b-a}$