How Moving a Magnet Through a Coil Creates Electricity

Push a magnet into a coil connected to a meter and the needle kicks; pull it out and the needle kicks the other way. This is electromagnetic induction.

This part covers magnetic flux and the induction experiments, Faraday's laws, Lenz's law, and motional emf with inductance.

**Magnetic flux through a surface is ${\Phi }_B=\vec{B}\cdot \vec{A}=BA\cos \theta$, measured in webers, and the experiments of Faraday and Henry showed that an emf is induced in a coil whenever the magnetic flux through it changes — by moving a magnet, moving a coil, or changing the current in a nearby coil.

Magnetic flux:**

$${\Phi }_B=BA\cos \theta ,1\text{ Wb}=1{\text{ T m}}^2$$

Worked example. A coil of area $0.05$ m${}^2$ has its normal at $60^{\circ }$ to a $0.4$ T field:

$${\Phi }_B=0.4\times 0.05\times \cos 60^{\circ }=0.01\text{ Wb}$$

The experiments:

- Magnet and coil — moving a bar magnet towards a coil connected to a galvanometer deflects it; moving it away deflects it oppositely; holding it still gives no deflection; faster motion gives a larger deflection
- Two coils at rest — switching the current in one coil on or off induces a brief current in the other, while a steady current induces nothing

An everyday example. A bicycle dynamo spins a magnet beside a coil, and the changing flux lights the headlamp — pedalling faster makes it brighter.

The substance. A strong but steady field induces nothing — only a changing flux produces an emf.

**Faraday's laws state that an emf is induced whenever the magnetic flux through a circuit changes, and that its size equals the rate of change of flux linkage, $\epsilon =-N\frac{d{\Phi }_B}{dt}$; in a closed circuit of resistance $R$ the induced current is $\frac{\epsilon }{R}$.

The law:**

$$\epsilon =-N\frac{d{\Phi }_B}{dt}$$

Worked example 1 — a changing field. A coil of $200$ turns and area $0.01$ m${}^2$ lies perpendicular to a field that falls steadily from $0.5$ T to $0.1$ T in $0.2$ s:

$$\epsilon =N\frac{\Delta (BA)}{\Delta t}=200\times \frac{(0.5-0.1)\times 0.01}{0.2}=4\text{ V}$$

With a total resistance of $8\Omega$, the induced current is $\frac{4}{8}=0.5$ A.

Worked example 2 — charge that flows. The charge depends only on the total change in flux:

$$q=\frac{N\Delta \Phi }{R}=\frac{200\times 0.004}{8}=0.1\text{ C}$$

however quickly the change happens.

An everyday example. An induction cooktop passes a rapidly changing current through a coil, and the changing flux induces currents in the steel pan that heat it.

The substance. The emf depends on how fast the flux changes, not on how large the flux is.

Lenz's law states that the induced current flows in the direction that opposes the change in flux producing it — the meaning of the minus sign in Faraday's law — and it must be so, because an induced current that aided the change would create energy from nothing.

Applying the law:

- North pole pushed towards a coil — the near face becomes a north pole, repelling the magnet
- North pole pulled away — the near face becomes a south pole, attracting the magnet back

The energy argument. If the induced current helped the magnet along, the magnet would speed up, raising the current, which would push the magnet faster still — endless energy with no input. Because induction opposes the motion, work must be done to move the magnet, and that work becomes electrical energy and heat.

Worked example — direction. A north pole approaches a coil. To make the near face a north pole, the current in that face, seen from the magnet's side, must flow anticlockwise.

An everyday example. A strong magnet dropped through a thick copper pipe falls surprisingly slowly, because currents induced in the pipe oppose its motion.

The substance. Lenz's law gives only the direction, while Faraday's law gives the size — together they fully describe the induced emf.

**A rod of length $l$ moving at speed $v$ perpendicular to a field $B$ has an induced emf $\epsilon =Blv$, because the area it sweeps changes the flux at that rate; self-inductance $L$ links a coil's own flux linkage to its current, $N\Phi =LI$, and mutual inductance $M$ links the flux linkage of one coil to the current in another, $N_2{\Phi }_2=M{I}_1$.

Motional emf.** A rod slides on rails $l$ apart, enclosing area $lx$ in field $B$:

$${\Phi }_B=Blx\Rightarrow |\epsilon |=\frac{d{\Phi }_B}{dt}=Bl\frac{dx}{dt}=Blv$$

Worked example. A $0.5$ m rod moves at $4$ m s${}^{-1}$ across a $0.3$ T field:

$$\epsilon =0.3\times 0.5\times 4=0.6\text{ V}$$

Self-inductance. A changing current induces $\epsilon =-L\frac{dI}{dt}$ in the same coil. The unit is the henry.

Mutual inductance. A changing current in coil 1 induces ${\epsilon }_2=-M\frac{d{I}_1}{dt}$ in coil 2.

Worked example — self-induced emf. In a $0.2$ H coil, the current falls from $3$ A to $1$ A in $0.05$ s:

$$\epsilon =L\frac{\Delta I}{\Delta t}=0.2\times \frac{2}{0.05}=8\text{ V}$$

An everyday example. The transformer in a phone charger uses mutual inductance — a changing current in one coil induces an emf in a second coil on the same core.

The substance. Inductance acts like inertia for current, opposing changes in current just as mass opposes changes in velocity.