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

Answer 1

The correct option is D $\frac{{\mu }_0}{2\pi }\frac{Ivl}{Rr}$

Magnetic field at a distance $r$ from the wire

$B=\frac{{\mu }_{0}I}{2\pi r}$

Magnetic flux for small displacement $dr$,

$\varphi =B.A=Bldr[\because A=ldr\text{and}B.A=BA\cos 0^o]$

$\Rightarrow \varphi =\frac{{\mu }_{0}I}{2\pi r}ldr$

$\text{Emf induced},e=\frac{d\varphi }{dt}=\frac{{\mu }_{0}Il}{2\pi r}.\frac{dr}{dt}\Rightarrow e=\frac{{\mu }_0}{2\pi }.\frac{Ivl}{r}$

Induced current in the loop, $i=\frac{e}{R}=\frac{{\mu }_0}{2\pi }.\frac{Ivl}{Rr}$