Electromagnetic Induction

Q. Electromagnetic Induction.

Question

A conducting rod of length l is moved at constant velocity Vo on two parallel, conducting, smooth, fixed rails, that are placed in a uniform constant Magnetic field B perpendicular to the plane of rails as shown in the figure. A resistance R is connected between the two ends of the rail, then please explain how the following options are correct?

1) Thermal power dissipated in the resistor is equal to the rate of work done by the external person pulling the rod.

2) If applied external force is doubled, then a part of the external power increases the velocity of the rod.

3) If resistance R is doubled, then power required to maintain the constant velocity v becomes half.

Answer

1)

If "I" is the current flowing through the conducting rod and its length is "L", then the force acting on it due to magnetic field in the loop of intensity B is
F = IBL
=> mechanical power developed ,
P = FVo = IBLVo ... ( 1 )

Thermal power dissipated 
= electric power expended in the wire
= emf developed x current
= BLVo * I
= IBLVo ... ( 2 )

From ( 1 ) and ( 2 ), 
thermal power dissipated in the resistor 
= work done by the external force.

2)
When force F is acting on the conducting rod, velocity is constant at Vo and according to Newton's first law, no net force is working on the rod. This is because, the force applied balances the force developed on the conducting rod in the opposite direction due to rate of change of magnetic flux. Now when F is increased to 2F, initially net force is 2F - F = F which accelerates the rod and increases its velocity due to the acceleration. With increase in velocity, the opposite magnetic force keeps increasing and when the velocity doubles to Vo becomes equal to 2F. At that instant, again the net force becomes zero and the rod continues to move with velocity 2Vo. During this period, net force changes from F to zero, and the varying net force increases the velocity of the rod from Vo to 2Vo.

3)
If R is doubled, current I becomes half and the force required to maintain velocity also becomes half. If a constant velocity Vo is maintained in this condition, then the power required also becomes half as the force is half and velocity is the same since power = force x velocity.