Electromagnetic Induction, Generators and Transformers

• Magnetic flux through a flat loop of area $A$: $\Phi = BA\cos\theta$, where $\theta$ is the angle between the field and the normal to the loop. Unit: the weber ($1\ \text{Wb} = 1\ \text{T m}^2$).
• Faraday's law: the induced e.m.f. equals the rate of change of flux linkage, $\varepsilon = N\,\Delta\Phi/\Delta t$.
• No change, no e.m.f.: a magnet held still inside a coil induces nothing.
• The e.m.f. grows with the number of turns, the field strength, the coil area and the speed of the change. The thickness of the wire does not change the e.m.f.; it only changes the resistance, and so the current.

Lenz's law: the induced current flows in the direction that opposes the change of flux that produced it. (This is the minus sign often written in Faraday's law, $\varepsilon = -N\,\Delta\Phi/\Delta t$.)

• Push a N pole into one end of a coil: that end becomes a N pole, to push the magnet back.
• Pull the N pole out: that end becomes a S pole, to pull it back. The current reverses.
• Swapping the pole (S instead of N) also reverses the current. So pushing a S pole in gives the same current as pulling a N pole out of the same end.
• Moving faster gives a bigger current in the same direction.
• Induced eddy currents in solid metal also oppose motion: this is how magnetic brakes work.

• AC generator: a coil of $N$ turns and area $A$ spinning at angular speed $\omega$ in a field $B$ gives a peak e.m.f. $\varepsilon_0 = NBA\omega$. Faster spinning raises both the peak e.m.f. and the frequency.
• Transformer: $V_s/V_p = N_s/N_p$. More turns on the secondary: step-up; fewer: step-down.
• In an ideal transformer, power in = power out: $V_p I_p = V_s I_s$. Stepping the voltage up steps the current down by the same factor.
• A transformer works only on AC. A steady DC current gives a steady flux, so nothing is induced in the secondary.
• Power lines use very high voltages so that the current, and the $I^2 R$ heating loss in the cables, is small.