Question

An infinitely long straight wire carrying current $I$, one side opened rectangular loop and a conductor $C$ with a sliding connector are located in the same plane, as shown in the figure. The connector has length $l$ and resistance $R$. It slides to the right with a velocity $v$. The resistance of the conductor and the self inductance of the loop are negligible. The induced current in the loop, as a function of separation $r$, between the connector and the straight wire is :

• A. $\frac{{\mu }_0}{\pi }\frac{Ivl}{Rr}$
• B. $\frac{{\mu }_0}{2\pi }\frac{Ivl}{Rr}$
• C. $\frac{2{\mu }_0}{\pi }\frac{Ivl}{Rr}$
• D. $\frac{{\mu }_0}{4\pi }\frac{Ivl}{Rr}$

Answer 1

Answer: (B)

$B=\frac{{\mu }_{0}i}{2\pi r}$
$\varphi =\frac{{\mu }_{0}i}{2\pi r}\ell dr$
$\Rightarrow \frac{d\varphi }{dt}=\frac{{\mu }_{0}i\ell }{2\pi r}\cdot \frac{dr}{dt}$
$\Rightarrow e=\frac{{\mu }_0}{2\pi }\cdot \frac{iv\ell }{r}$
$i=\frac{e}{R}=\frac{{\mu }_0}{2\pi }\cdot \frac{iv\ell }{Rr}$