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  4. A wire of length l, mass m, and resistance R slides without any friction down the parallel conducting rails of negligible resistance (figure). The rails are conneçted to each other at the bottom by a resistanceless rail parallel to the wire so that the wire and the rails form a closed rectangular conducting loop. The plane of the rails makes an angle θ with the horizontal and a uniform vertical magnetic field of induction B exists throughout the region. Find the steadystate velocity of the wire.

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Q. A wire of length $l$, mass $m$, and resistance $R$ slides without any friction down the parallel conducting rails of negligible resistance (figure). The rails are conneçted to each other at the bottom by a resistanceless rail parallel to the wire so that the wire and the rails form a closed rectangular conducting loop. The plane of the rails makes an angle $\theta$ with the horizontal and a uniform vertical magnetic field of induction $B$ exists throughout the region. Find the steadystate velocity of the wire.Physics Question Image

Electromagnetic Induction

Solution:

$B I l \cos \theta=m g \sin \theta$ ...(i)
Here induced emf across slider is $(B \cos \theta) l v$
$\therefore$ Induced current $I=\frac{B l v \cos \theta}{R}$
image
From Eq. (i)
$B l \cos \theta \frac{B l v \cos \theta}{R}=m g \sin \theta$
$\therefore v=\frac{m g R \sin \theta}{B^{2} l^{2} \cos ^{2} \theta}$