#### Motional emf example

Consider a conducting bar placed on a pair of conducting rails that are separated by a distance L. The rails are connected at the left end by a resistor of resistance R - assume the resistance of the bar and rails is negligible compared to R. There is a uniform magnetic field of magnitude B directed perpendicular to the plane of the rails (into the page in the diagram above).

The bar is then subjected to a constant force F directed right. Assuming there is no friction between the bar and the rails, describe the motion of the bar.

1. The bar experiences a constant acceleration, and the speed increases at a constant rate
2. The changing flux gives rise to another force in the same direction as F that accelerates the bar even faster than F would by itself.
3. The changing flux gives rise to another force opposite in direction to F that causes the bar to reach a terminal (constant) velocity
4. The changing flux gives rise to another force opposite in direction to F that causes the bar to come to rest

The third choice is correct. The faster the bar goes the larger the current induced in the loop consisting of bar, rail, resistor, rail. This current gives rise to a force of magnitude ILB opposing the applied force F. When the forces are equal and opposite there is then no net force, so the bar continues to move at a constant velocity.

In what direction is the induced current in the loop?

1. Clockwise
2. Counter-clockwise

The current goes counter-clockwise around the loop.

If R = 2 ohms, L = 50 cm, B = 2 T, and the terminal velocity of the bar is 4 m/s what is the current in the bar when the bar reaches terminal velocity?

The motional emf has a magnitude:

ε = vBL = 4 x 2 x 0.5 = 4 volts

Now bring in Ohm's Law:

ε = IR, so I = ε/R = 4/2 = 2 A.

What is the applied force?

The applied force F is, at terminal velocity, equal and opposite to the force the current feels from the magnetic field.

F = ILB = 2 x 0.5 x 2 = 2 N.