Question

A square loop of side $a$ and a straight long wire are placed in the same plane as shown in figure. The loop has a resistance $R$ and inductance $L$. The frame is turned through ${18}^o$ about the axis $O{O}^{\prime }$. What is the electric charge that flows through the loop?

• A. $\frac{{\mu }_{0}Ia}{2\pi R}\ln \left(\frac{2a+b}{b}\right)$
• B. $\frac{{\mu }_{0}Ia}{2\pi R}\ln \left(\frac{b}{b^2-a^2}\right)$
• C. $\frac{{\mu }_{0}Ia}{2\pi R}\ln \left(\frac{a+2b}{b}\right)$
• D. None of these

Answer 1

Answer: (D)

None of these

${\varphi }_i={\int }_b^{b+a}\frac{{\mu }_0}{2\pi }\frac{i}{x}\left(adx\right)$

$=\frac{{\mu }_{0}ia}{2\pi }\ln \left(\frac{b+a}{a}\right)$
Similarly

${\varphi }_f=\frac{{\mu }_{0}ia}{2\pi }\ln \left(\frac{b-a}{a}\right)$

$\Delta \varphi =|Q_i-Q_f|=\frac{{\mu }_{0}ia}{2\pi }\ln \left(\frac{b+a}{b-a}\right)$

$\Delta q=\frac{\Delta \varphi }{R}=\frac{{\mu }_{0}ia}{2\pi R}\ln \left(\frac{b+a}{b-a}\right)$