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)

$B =\frac{\mu_{0} i }{2 \pi r }$
$\phi=\frac{\mu_{0} i }{2 \pi r } \ell dr$
$\Rightarrow \frac{ d \phi}{ dt }=\frac{\mu_{0} i \ell}{2 \pi r } \cdot \frac{ d r }{ dt }$
$\Rightarrow e =\frac{\mu_{0}}{2 \pi} \cdot \frac{ iv \ell}{ r }$
$i =\frac{ e }{ R }=\frac{\mu_{0}}{2 \pi} \cdot \frac{ iv \ell}{ Rr }$