Question

A conducting rod length $l$ is moved at constant velocity $v_0$ on two parallel, conducting, smooth, fixed rails, which are placed in a uniform constant magnetic field $B$ perpendicular to the plane of the rails as shown in figure. A resistance $R$ is connected between the two ends of the rail. Then which of the following is/are correct?

• A. The thermal power dissipated in the resistor is equal to the rate of work done by an external person pulling the rod
• B. If applied external force is doubled, then a part of the external power increases the velocity of the rod
• C. Lenz's law is not satisfied if the rod is accelerated by an external force.
• D. If resistance $R$ is doubled, then power required to maintain the constant velocity $v_0$ becomes half.

Answer 1

Answer: (A), (B), (D)

The thermal power dissipated in the resistor is equal to the rate of work done by an external person pulling the rod
If applied external force is doubled, then a part of the external power increases the velocity of the rod
If resistance $R$ is doubled, then power required to maintain the constant velocity $v_0$ becomes half.

EMF induced in the rod due to changing flux through the loop=$\frac{d\varphi }{dt}=Bvl$.

Thus current in the loop$=\frac{Emf}{R}=\frac{Bvl}{R}=i$

Thus the force exerted by magnetic field on the conducting rod inside the loop$=Bil=\frac{B^{2}v{l}^2}{R}$.

Hence an external force equal to this magnitude is required to be applied for the rod to move with constant velocity.

$⟹F_{ext}=\frac{B^{2}v{l}^2}{R}$

Statements B and D are easily derivable.

Since magnetic field does not do any work on the conducting rod, from conservation of energy, the thermal power dissipated in the resistor is equal to the rate of work done by an external person pulling the rod.