A moving bar

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 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:

Fx = ma = -ILB

m dv/dt = -ILB

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

I = ε/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) = vi e-t/τ

The time constant here is τ = mR/B2L2

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

Why Does the Loop Oppose Change?

Why is there always a tendency to oppose a change in magnetic flux? Why doesn't the loop reinforce the change instead?

Cosider the moving bar. Let's say we gave the bar a push to the left, and that instead of the induced current acting to oppose the change in flux it added to the change in flux. This would require a current in the opposite direction to what we found before, giving rise to a force to the left on the bar. The bar would accelerate to the left, increasing the rate of change of flux, increasing the induced current, increasing the force, etc.

With one tiny push from us to get things moving, the bar would pick up speed all by itself. Clearly this makes no sense from a conservation of energy perspective. The fact that there is always an opposition to a change in flux does make sense from an energy perspective, however. In this situation the initial kinetic energy of the bar is dissipated as heat. To keep the bar moving requires that we continue to apply a force to overcome the opposition of the loop.