Question

Figure shows a conducting rod of negligible resistance that can slide on a smooth U-shaped rail made of wire of resistance 1 $\Omega m^{-1}$ . Position of the conducting rod at $t=0$ is shown. A time-dependent magnetic field $B=2t$ (tesla) is switched on at $t=0$.The magnitude of the force required to move the conducting rod at a constant speed of $5cm/s$ at the some intstant of $t=2s$ is equal to 1000$x\times {10}^{-3}N$, where $x$ is (Answer upto 2 decimal places).

Answer 1

At $t=2s,$ length of the wire $=(2\times 40cm)+20cm=1m$
Resistance of the wire $=1\Omega$
Change in flux is given by$d\varphi =B.dA=2t\times vdt\times 0.2=0.4vtdt=0.02tdt$
Current through the rod $=\frac{1}{R}\frac{d\varphi }{dt}=\frac{0.02tdt}{dt}=0.02t=0.04A$
Force on the wire is $=ilB$
$=0.04\times (0.2)\times 4=0.032N$
Same force is applied on the rod in opposite direction to make net force zero.