An infinitely long straight wire carrying current I, one side ope… - Vidyadip

MCQ

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
$\frac{\mu_{0}}{\pi} \frac{ Iv l}{ Rr }$
✓
$\frac{\mu_{0}}{2 \pi} \frac{ Ivl }{ Rr }$
C
$\frac{2 \mu_{0}}{\pi} \frac{ Ivl }{ Rr }$
D
$\frac{\mu_{0}}{4 \pi} \frac{ Iv l}{ Rr }$

✓

Answer

Correct option: B.

$\frac{\mu_{0}}{2 \pi} \frac{ Ivl }{ Rr }$

b
$B =\frac{\mu_{0} i }{2 \pi r }$

$\phi=\frac{\mu_{0} i }{2 \pi r } \ell dr$

$\Rightarrow \frac{ d \phi}{ dt }=\frac{\mu_{0} i \ell}{2 \pi r } \cdot \frac{ dr }{ dt }$

$\Rightarrow e =\frac{\mu_{0}}{2 \pi} \cdot \frac{ iv \ell}{ r }$

$i =\frac{ e }{ R }=\frac{\mu_{0}}{2 \pi} \cdot \frac{ iv \ell}{ Rr }$

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