Question

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 connected 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 horizontal and a uniform vertical magnetic field of induction $B$ exists throughout the region. Find the steady-state velocity of the wire.

• A. $\frac{mg}{R}\frac{\sin \theta }{B^2{l}^2{\cos }^2\theta }$
• B. $\frac{mg}{R}\frac{{\sin }^2\theta }{B^2{l}^2{\cos }^2\theta }$
• C. $\frac{mgR\sin \theta }{B^2{l}^2{\cos }^2\theta }$
• D. $mgR\frac{{\sin }^2\theta }{B^2{l}^2{\cos }^2\theta }$

Answer 1

The correct option is C $\frac{mgR\sin \theta }{B^2{l}^2{\cos }^2\theta }$

$Bilcos\theta =mgsin\theta$
$\Rightarrow B\left(\frac{Blvcos\theta }{R}\right)lcos\theta =mgsin\theta$
$\Rightarrow v=\frac{mgRsin\theta }{B^2{l}^2{cos}^2\theta }$