Question

A conducting bar of mass $m$ and length $l$ moves on two frictionless parallel rails in the presence of a constant uniform magnetic field of magnitude $B$ directed into the page as shown in the figure. The bar is given an initial velocity $v_0$ towards the right at $t = 0$. Then, the

• A. induced current in the circuit is in the clockwise direction
• B. velocity of the bar decreases linearly with time
• C. distance the bar travels before it comes to a complete stop is proportional to $R$
• D. power generated across the resistance is proportional to $l$

Answer 1

Answer: (C)

In given arrangement, sliding bar is given an initial speed $v$ towards left.

As, flux linked with loop is increasing induced current must be flowing counter clockwise to oppose the increasing flux.
During induction electrical energy generated is dissipated as heat. Power generated is given by
$P=\frac{B^{2}l^{2}v^{2}}{R}$
or $P\propto l^{2}$
Acceleration of sliding bar is
$ a =\frac{F}{m}$
$a=\frac{B^{2}l^{2}v}{Rm}$
$\Rightarrow \frac{dv}{dt}=\frac{B^{2}l^{2}v}{Rm}$
$\Rightarrow \frac{dv}{dx}.\left(\frac{dx}{dt}\right)=\frac{B^{2}l^{2}v}{Rm}$
$\Rightarrow dv=\frac{B^{2}l^{2}v}{Rm}dx \left[\therefore \frac{dx}{dt}=v\right] $
$\Rightarrow v=\frac{B^{2}l^{2}}{Rm}.x$
or $x=\frac{Rmv}{B^{2}l^{2}}\Rightarrow x\propto R$