Exercise practice: Electromagnetic Induction

4.1 Flux and Faraday's law numericals

The standard exercise pattern gives a flux that changes with time and asks for the emf. If the flux is phi = 5t squared + 3t + 16 webers, its rate of change is 10t + 3, so at t = 4 s the emf has magnitude 43 V and the sign is negative by Lenz's law. If instead the field changes from 0.5 T to zero through a coil of 100 turns and area 0.02 square metre in 0.1 s, the change in flux is 0.01 Wb, the rate is 0.1 Wb per second and the emf is 10 V.

Another common numerical gives the current induced in a coil of known resistance and asks for the change in flux. Since the emf is I R and also equals the flux change divided by the time, the flux change is I R times the time. For a current pulse of 2 A lasting 0.2 s in a coil of 100 ohm, the flux change is 2 x 100 x 0.2 = 40 Wb. The area under the current time graph gives the same result when the current is not constant.

4.2 Motional emf numericals

A rod of length 0.5 m slides at 2 m per second on rails in a field of 0.3 T, with its length perpendicular to both the field and the velocity. The emf is B l v = 0.3 x 0.5 x 2 = 0.3 V. If the circuit resistance is 1.5 ohm, the current is 0.2 A, the force needed to keep the rod moving at constant speed is B I l = 0.3 x 0.2 x 0.5 = 0.03 N, and the power is F v = 0.06 W, which equals I squared R.

For a rod of length 0.5 m rotating at 60 rpm in a field of 4 mT, the angular speed is 2 pi radians per second, and the emf is half B omega l squared = 0.5 x 0.004 x 6.28 x 0.25 = 3.14 x 10 to the power -3 V, about 3 mV. The same formula answers the question about a wheel or a rod pivoted at one end.

4.3 Inductance numericals

A solenoid has 2000 turns, length 0.5 m and area 4 x 10 to the power -4 square metre. Its self inductance is mu0 N squared A / l = 4 pi x 10^-7 x 4 x 10^6 x 4 x 10^-4 / 0.5 = 4.0 x 10 to the power -3 H, or 4 mH. If the current in it changes at 100 A per second, the emf induced is L di/dt = 0.4 V.

Two coaxial solenoids of the same length and area have 500 and 1000 turns. Their mutual inductance is mu0 N1 N2 A / l; with A = 2 x 10^-3 square metre and l = 1 m it comes out about 1.26 mH. If the current in the first changes at 10 A per second, the emf in the second is 0.0126 V. The mutual inductance is the same whichever coil carries the changing current, provided the geometry is the same.

The energy stored in an inductor of 2 H carrying 3 A is half L I squared = 9 J. That energy came from the source while the current was building up, and it is returned when the current falls, which is why an inductor can produce a large spark when a circuit is broken quickly.

4.4 Direction questions

For the direction, apply Lenz's law to the change in flux. If a magnet is brought towards a coil with its north pole facing the coil, the flux through the coil grows, so the coil's face towards the magnet must be a north pole to oppose the approach; the current direction then follows from the right hand rule. If the magnet is pulled away, the same face must be a south pole to hold on, so the current reverses. When a loop is pulled out of a field pointing into the page, the flux into the page decreases, so the induced current produces a field into the page, which means a clockwise current. Entering a field gives the opposite direction.