Revise Induction for IB Physics (D.4)

To describe induction we first need to measure how much magnetic field passes through a surface. This quantity is the magnetic flux, written with the Greek letter $\Phi$ (phi). It depends on the field strength $B$, the area $A$ of the surface, and how the surface is tilted relative to the field.

$\Phi = BA\cos\theta$

Here $\theta$ is the angle between the magnetic field lines and the normal — the line drawn perpendicular to the surface. When the field is perpendicular to the surface (so it is parallel to the normal), $\theta = 0$ and the flux is a maximum, $\Phi = BA$. When the field lies in the plane of the surface, $\theta = 90^\circ$ and the flux is zero. The unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T m². For a coil of $N$ turns, each turn is threaded by the same flux $\Phi$, so the total flux linkage is $N\Phi$. It is the flux linkage, not the flux, that matters for induction.

Faraday's law states that an electromotive force (emf), $\varepsilon$, is induced in a circuit whenever the magnetic flux linkage through it changes, and that the magnitude of the induced emf equals the rate of change of flux linkage.

$\varepsilon = -N\frac{\Delta\Phi}{\Delta t}$

The key word is rate. A large flux sitting unchanged induces nothing; it is the change per unit time that generates an emf. Double the number of turns, or halve the time over which the same change happens, and you double the induced emf. Because the flux is $\Phi = BA\cos\theta$, you can produce that change in three ways — by varying the field $B$, the area $A$, or the angle $\theta$ — and every generator and transformer is just an engineered way of changing one of them.

The minus sign in Faraday's law is not decoration — it carries the physics of direction, and it is a statement in its own right known as Lenz's law: the induced current flows in whatever direction opposes the change in flux that produced it. Push a magnet north-pole-first towards a coil and the coil responds by inducing a current that makes the near face into a north pole, repelling the incoming magnet. Pull the magnet away and the induced current reverses, making the near face a south pole to attract it back. In both cases the coil opposes the change.

This opposition is required by conservation of energy. If the induced current instead reinforced the change, the growing flux would drive an ever-larger current with no energy source — a perpetual-motion machine. Because the current opposes the change, you must do work against a resisting force to keep the flux changing, and that mechanical work is precisely the electrical energy that induction delivers. Lenz's law is conservation of energy written into the direction of the current.

A particularly clean case of induction happens when a straight conductor slides through a constant magnetic field. The free charge carriers inside the conductor move with it, so each experiences a magnetic force $F = qvB$ that pushes it along the conductor. Charge piles up at one end, leaving the other end oppositely charged, and this separation is an emf across the conductor's ends — a motional emf.

\varepsilon = BvL

This holds when the field $B$, the conductor's length $L$ and its velocity $v$ are all mutually perpendicular. You can see it agrees with Faraday's law: as a rod of length $L$ slides at speed $v$ along rails, it sweeps out area at a rate $Lv$, so the flux changes at a rate $B\,Lv$, giving exactly $\varepsilon = BvL$. If the velocity makes an angle to the field, use only the component of velocity perpendicular to the field.

A generator turns motion into electricity by rotating a coil in a magnetic field. As the coil of $N$ turns and area $A$ spins at a steady rate in a uniform field $B$, the angle $\theta$ between the field and the coil's normal changes continuously, so the flux $\Phi = BA\cos\theta$ rises and falls smoothly. By Faraday's law the induced emf follows the rate of change of this flux linkage, which is itself sinusoidal — so the output is an alternating emf that swings between a positive and a negative peak, reversing every half turn.

The timing is the part students most often get wrong. The emf is greatest when the coil's plane lies along the field (the flux is momentarily zero but changing fastest as the field lines sweep across the turns) and the emf is zero when the coil's plane is perpendicular to the field (the flux is at its maximum but, for that instant, not changing). Peak emf and peak flux are always a quarter-cycle apart. Spinning the coil faster raises both the frequency and the peak emf, because the flux then changes more rapidly.

A transformer changes the voltage of an alternating supply. It has two coils wound on a shared iron core: a primary of $N_p$ turns fed by the input AC, and a secondary of $N_s$ turns delivering the output. The alternating primary current produces a continuously changing flux in the core; the core guides that same changing flux through the secondary, inducing an emf in it. Because both coils link the same changing flux per turn, the voltages are in the same ratio as the turns.

$\frac{V_s}{V_p} = \frac{N_s}{N_p}$

If the secondary has more turns than the primary the voltage is stepped up; if it has fewer, the voltage is stepped down. A transformer works only on alternating current, because a steady direct current gives a constant flux and hence no induced secondary emf.

An ideal transformer transfers energy without loss, so the power delivered to the primary equals the power leaving the secondary. This links the currents to the voltages:

V_p I_p = V_s I_s

Stepping the voltage up therefore steps the current down in the same ratio, and vice versa. This is why electricity is transmitted across the grid at very high voltage and low current: the power carried is the same, but the low current means far smaller heating losses ($P_{\text{loss}} = I^2R$) in the transmission cables. Real transformers lose a little energy to resistance in the windings, to eddy currents and to repeatedly magnetising the core, but the ideal relations are an excellent first model.

Induction ties the fields you have already met into working technology. The magnetic flux defined here reappears wherever a circuit meets a changing field, and Faraday's and Lenz's laws govern inductors, eddy-current braking and the behaviour of AC circuits. The generator and transformer you have analysed are the two ends of the power grid — one making the alternating emf, the other reshaping its voltage for efficient transmission. Master the habit — identify what is changing, find $\Delta\Phi$, apply Faraday's law, then use Lenz's law and energy conservation to fix the direction — and the electromagnetism that follows becomes variations on a method you already own.