A conducting bar on conducting rails forms a complete loop with resistance R (assume this is constant). Friction between the bar and the rails is negligible. There is a uniform magnetic field directed out of the page. The bar is given an initial velocity v_o to the left.

What does the bar do?

1. Speed up
2. Slow down
3. Move at constant velocity

The flux through the loop is changing, so the loop tries to oppose the change. The flux is changing because the bar is moving, changing the area of the loop. To oppose the change the loop makes the bar slow down.

To see why, let's start with the induced current. What direction is it?

1. clockwise
2. counter-clockwise

Moving the bar to the right decreases the area of the loop, decreasing the flux. The induced current must generate a field out of the page. This requires a counter-clockwise current.

Why does this make the bar slow down? Let's call the +x direction to the left, the direction the bar is moving. The current through the bar is up. A current up in a field out of the page gives a force to the right:

F_x = ma = -ILB

m

dv dt

=

-ILB

The current is given by Ohm's Law, where the emf is the motional emf:

I

=

e R

=

vBL R

Plugging this into our force expression gives:

m

dv dt

=

-vB2L2 R

The solution to this is the function that is basically the negative derivative of itself.

We need a negative exponential:

v(t) = v_o e^-t/t

The time constant here is

t

=

mR B2L2

Thus the velocity (and the induced emf and current) decrease exponentially to zero.