Motional emf

A rod moving in a field

Narrated video. Transcript:

Here is a U-shaped conductor lying in a uniform magnetic field B, pointing into the page. A straight rod of length l rests across its two arms and can slide along them.

The rod and the U-shape make a closed rectangle. If the rod is a distance x from the closed end, the area of the rectangle is l x, and the flux through it is Φ_B equals B l x.

Now slide the rod with a steady speed v. The distance x changes at the rate v, so the flux changes at the rate B l v.

By Faraday’s law, the induced emf is ε equals B l v. This emf, produced by moving a conductor through a magnetic field, is called motional emf.

For example, a rod 0.5 metre long moving at 4 metres per second across a field of 0.2 tesla gives ε equals 0.2 × 0.5 × 4, which is 0.4 volt.

We can also see where this emf comes from. Every free charge q in the rod moves with the rod, at speed v, through the field. So it feels a Lorentz force, q v B, along the length of the rod.

Here this force pushes positive charges towards the top end of the rod. One end becomes positive, the other negative. The rod acts like a battery.

The work done by this force in carrying a charge q along the rod is q v B times l. Work per unit charge is the emf: ε equals B l v. The same answer as from Faraday’s law.

If the rod is still and the magnet moves instead, the charges are not moving through the field. Then the emf is due to an electric field, produced by the changing magnetic field.

The force on the rod, and the energy

Narrated video. Transcript:

Join the two rails with a resistor R. Now the moving rod drives a current I equals ε by R, which is B l v over R, around the circuit.

Which way? The flux into the page is increasing, so by Lenz’s law the current flows anticlockwise. In the rod, it flows upward.

But a current-carrying rod in a magnetic field feels a force, I l B. With the current upward and B into the page, this force points backward, against the velocity. Lenz’s law again: it opposes the motion.

Its size is I l B. Put in I equals B l v over R, and the force is B squared l squared v over R.

So to keep the rod moving at a steady speed, someone must keep pulling it with this force. The power they supply is force times velocity: P equals B squared l squared v squared over R.

Now the heat produced in the resistor is I squared R. Put in I equals B l v over R, and we get B squared l squared v squared over R. Exactly the same!

Mechanical energy becomes electrical energy, which becomes heat. Not a joule is created or lost.

Numbers: B equals 0.5 tesla, l equals 0.4 metre, v equals 2 metres per second, R equals 1 ohm. ε equals 0.4 volt, I equals 0.4 ampere, and the force is 0.4 × 0.4 × 0.5, which is 0.08 newton.

The power is 0.08 × 2, which is 0.16 watt, and the heat is 0.4 squared × 1, also 0.16 watt. Stop pulling, and the opposing force slows the rod down until it stops, its kinetic energy all turned into heat.

A rotating rod

Narrated video. Transcript:

A metal rod of length 1 metre turns with angular frequency 400 radians per second about an axis through one end, perpendicular to the rod. Its other end slides on a circular metal ring.

A uniform field of 0.5 tesla points along the axis, everywhere. What emf develops between the centre and the ring?

Different parts of the rod move at different speeds. A small piece of length dr, at distance r from the centre, moves at speed v equals ω r. Its emf is B v dr, which is B ω r dr.

Add up all the pieces from the centre to the end, R. The sum of r dr is R squared over 2. So ε equals ½ B ω R squared.

Another way to see it: the average speed of the rod is half the speed of its tip, ω R over 2. So ε equals B times R times ω R over 2.

Put in the numbers: ε equals ½ × 0.5 × 400 × 1 squared, which is 100 volts.

Now a bicycle wheel with 10 metal spokes, each 0.5 metre long, turns at 120 revolutions per minute, in a plane normal to the horizontal component of the Earth’s field, 0.4 gauss, that is 0.4 × 10⁻⁴ tesla.

120 revolutions per minute is 2 revolutions per second, so ω equals 4π radians per second. ε equals ½ × 0.4 × 10⁻⁴ × 4π × 0.5 squared, about 6.3 × 10⁻⁵ volt.

Does it matter that there are 10 spokes? No. The spokes are joined side by side between the axle and the rim, in parallel, so the emf is the same as for one spoke.

Try it: rod on rails

Interactive simulation. Things to try:

• Press Start. These are the numbers from the second video: ε = 0.4 V, I = 0.4 A, F = 0.08 N, and both powers 0.16 W.
• In the power graph the blue line (F v) and the dashed amber line (I²R) lie on top of each other: energy is conserved.
• Double v. The emf and current double, but the power goes up four times.
• Choose “Push and let go”. The opposing force slows the rod, and the heat produced equals the kinetic energy it started with.
• Drag the rod by hand, left and right. The current and the force reverse, but the force always opposes the motion.

Keeping it moving

A rod of length 0.5 m slides at a steady 2 m/s on rails joined by a 2 Ω resistor, in a field of 0.4 T perpendicular to the rails. What force must be applied to keep it moving at this speed?

1. No force: it moves at constant speed
2. 0.04 N
3. 0.08 N
4. 0.4 N

A spinning rod

A metal rod 0.5 m long rotates at 200 rad/s about one end, in a plane perpendicular to a uniform field of 0.2 T. What emf develops between its two ends?

1. 2.5 V
2. 5 V
3. 10 V
4. 20 V

Summary

• Motional emf (rod ⟂ B, v ⟂ rod): ε = Blv
• Retarding force with resistance R: F = IlB = B²l²v/R
• Power = Joule heat: P = Fv = B²l²v²/R = I²R
• Rod rotating about one end: ε = ½ BωR²

• A conductor moving through a magnetic field has a motional emf ε = Blv, from Faraday’s law (flux Blx) or from the Lorentz force qvB on its charges.
• The induced current makes a force IlB that opposes the motion (Lenz’s law).
• The power needed to keep the rod moving equals the Joule heat I²R: mechanical → electrical → heat.
• For a rod rotating about one end, use the average speed ωR/2: ε = ½BωR². Spokes of a wheel are in parallel, so their number does not matter.