Electromagnetic induction

What you'll learn

Understand magnetic flux: how much magnetic field passes through a loop.
Use Faraday’s law: how quickly changing flux produces an induced voltage.
Use Lenz’s law: the direction rule for induced currents.
Apply induction to moving conductors, generators and transformers.

The basic idea

Electromagnetic induction is about producing a voltage using a magnetic field. You do not need a battery: if the magnetic field through a coil changes, a voltage can be induced.

Definition

Electromagnetic induction and emf

Electromagnetic induction is the production of an electromotive force (emf) when the magnetic field through a circuit changes. An emf is energy transferred per unit charge, measured in volts (V). Despite the name, emf is not a force.

A current is only induced if there is a complete conducting circuit. If the circuit is open, there can still be an induced emf, but no continuous current.

Common Mistake

Emf is not always current

A changing magnetic field induces an emf. It only induces a current if the circuit is closed.

Magnetic flux

Before using the induction equations, you need the idea of magnetic flux.

A normal is a line drawn at right angles to a surface. For a coil in a magnetic field, the angle in the flux equation is measured between the magnetic field and the normal to the coil, not the plane of the coil.

Magnetic flux through a rectangular coil showing the normal and angle theta

Definition

Magnetic flux and flux linkage

The magnetic flux through one turn of a coil in a uniform magnetic field is

Φ=BAcos⁡θ\Phi = BA\cos\theta

where $B$ is magnetic flux density in tesla (T), $A$ is area in square metres, and $θ\theta$ is the angle between the field and the normal. Magnetic flux is measured in webers (Wb). For a coil with $N$ turns, the flux linkage is $NΦN\Phi$ , measured in weber turns.

Flux is maximum when the normal is parallel to the field, so $θ=0∘\theta = 0^\circ$ . Flux is zero when the normal is perpendicular to the field, so $\theta = 90^\circ`.

Example

Calculating flux linkage

A 100-turn coil of area $2.0 \times 10^{-3}\ \text{m}^2$ is in a uniform magnetic field of flux density $T0.35\ \text{T}$ . The normal to the coil is at $60∘60^\circ$ to the field. Calculate the flux linkage.

Use the angle between the field and the normal: $θ=60∘\theta = 60^\circ$ , so $cos⁡60∘=0.50\cos 60^\circ = 0.50$ .
Calculate the flux through one turn: $Wb\Phi = BA\cos\theta = 0.35 \times 2.0 \times 10^{-3} \times 0.50 = 3.5 \times 10^{-4}\ \text{Wb}$ .
Multiply by the number of turns: $turnsN\Phi = 100 \times 3.5 \times 10^{-4} = 3.5 \times 10^{-2}\ \text{Wb turns}$ .

Faraday’s law

Faraday’s law tells you the size of the induced emf.

Key Idea

Faraday’s law

The magnitude of the induced emf is equal to the rate of change of flux linkage.

For a coil:

ε=−Δ(NΦ)Δt\varepsilon = -\frac{\Delta(N\Phi)}{\Delta t}

If the number of turns is constant:

ε=−NΔΦΔt\varepsilon = -N\frac{\Delta\Phi}{\Delta t}

The minus sign is not for calculating the size. It represents Lenz’s law, which gives the direction of the induced emf.

Example

Calculating an induced emf

A 200-turn coil is placed with its normal parallel to a magnetic field. The coil area is $4.0 \times 10^{-3}\ \text{m}^2$ . The magnetic flux density increases from $T0.20\ \text{T}$ to $T0.65\ \text{T}$ in $s0.080\ \text{s}$ . Calculate the magnitude of the average induced emf.

The normal is parallel to the field, so $cos⁡θ=1\cos\theta = 1$ . The change in flux through one turn is $Wb\Delta\Phi = A\Delta B = 4.0 \times 10^{-3} \times (0.65 - 0.20) = 1.8 \times 10^{-3}\ \text{Wb}$ .
Calculate the change in flux linkage: $turns\Delta(N\Phi) = 200 \times 1.8 \times 10^{-3} = 0.36\ \text{Wb turns}$ .
Divide by the time taken: $V|\varepsilon| = \frac{0.36}{0.080} = 4.5\ \text{V}$ .

Lenz’s law

Lenz’s law is the direction rule for induction.

Definition

Lenz’s law

The direction of the induced emf, and any induced current, is such that it opposes the change that produced it.

For example, if a north pole moves towards a coil, the coil produces a magnetic field that opposes the increase in flux. The near face of the coil becomes a north pole, repelling the approaching north pole.

North pole moving towards a coil showing induced current and Lenz law opposition

Tip

Lenz’s law is about change

The induced effect opposes the change in flux, not simply the magnetic field itself. If flux is increasing, the coil tries to reduce it. If flux is decreasing, the coil tries to maintain it.

To find the current direction in a coil, use the right-hand grip rule: curl your fingers in the current direction and your thumb points towards the coil’s north pole.

Example

Finding the induced pole of a coil

A north pole is pulled away from the left face of a coil. Decide what magnetic pole the left face of the coil becomes.

The magnetic flux through the coil due to the nearby north pole is decreasing as the magnet moves away.
To oppose this decrease, the coil must try to keep the magnet nearby by attracting the north pole.
A face that attracts a north pole must be a south pole, so the left face of the coil becomes south.

Moving conductors: cutting field lines

A straight conductor can have an induced emf if it moves through a magnetic field. Free charges in the conductor experience a magnetic force, so charge separates along the conductor and a voltage is produced.

For a conductor of length $ℓ\ell$ moving at speed $v$ at right angles to a uniform magnetic field:

ε=Bℓv\varepsilon = B\ell v

Common Mistake

Perpendicular formula

The equation $ε=Bℓv\varepsilon = B\ell v$ applies directly only when the conductor, its velocity, and the magnetic field are mutually perpendicular. Otherwise you need the perpendicular component.

Example

Calculating motional emf

A wire of length $m0.30\ \text{m}$ moves at $s−12.5\ \text{m s}^{-1}$ at right angles to a magnetic field of flux density $T0.80\ \text{T}$ . It is part of a circuit with total resistance $Ω4.0\ \Omega$ . Calculate the induced current.

The motion is perpendicular to the field, so use $ε=Bℓv\varepsilon = B\ell v$ .
Calculate the emf: $V\varepsilon = 0.80 \times 0.30 \times 2.5 = 0.60\ \text{V}$ .
Use $\frac{V}{R}$ for the circuit current: $\frac{0.60}{4.0} = 0.15\ \text{A}$ .

Generators and transformers

AC generators

In an AC generator, a coil rotates in a magnetic field. The flux linkage changes continuously, so an alternating emf is induced.

When the flux linkage is maximum, the rate of change of flux linkage is zero, so the induced emf is zero. When the flux linkage is changing fastest, the induced emf is maximum.

AC generator with rotating coil and sinusoidal induced emf graph

If a coil rotates steadily, the induced emf is sinusoidal. Increasing the magnetic flux density, coil area, number of turns, or rotation speed increases the maximum emf.

Transformers

A transformer uses induction between two coils. An alternating current in the primary coil produces a changing magnetic flux in the core, inducing an emf in the secondary coil.

For an ideal transformer:

VsVp=NsNp\frac{V_s}{V_p} = \frac{N_s}{N_p}

If it is 100% efficient:

V_p I_p = V_s I_s

A step-up transformer has more secondary turns than primary turns, so it increases voltage. A step-down transformer has fewer secondary turns, so it decreases voltage.

Common Mistake

Transformers need changing flux

A transformer needs an alternating current or changing current. A steady direct current gives no continuous induced emf in the secondary coil after the switching moment.

Example

Using the transformer equations

An ideal transformer has $N_p = 600$ turns and $N_s = 3000$ turns. The primary voltage is $V230\ \text{V}$ and the secondary current is $A0.20\ \text{A}$ . Calculate the secondary voltage and primary current.

Use the turns ratio: $Vs230=3000600=5\frac{V_s}{230} = \frac{3000}{600} = 5$ , so $VV_s = 1150\ \text{V}$ .
Calculate the output power: $WP_s = V_s I_s = 1150 \times 0.20 = 230\ \text{W}$ .
For an ideal transformer, input power equals output power, so $AI_p = \frac{230}{230} = 1.0\ \text{A}$ .

Practical note

You can observe induction using a coil connected to a centre-zero galvanometer or data logger. Moving a magnet faster gives a larger deflection because the rate of change of flux linkage is greater. Reversing the magnet’s motion reverses the deflection, showing that the induced emf has changed direction.

Exam technique

In the exam

Draw the coil normal, then decide whether flux linkage is increasing or decreasing.
Use Faraday’s law for the magnitude of the emf, and Lenz’s law for the direction.
Convert areas to square metres and times to seconds before substituting.
For transformers, check that the question assumes an ideal transformer before using equal input and output power.

Self review

Check yourself

Why is the induced emf zero when the flux linkage is maximum in a rotating coil?
A south pole moves towards a coil. What pole does the near face of the coil become?
What happens to the secondary current in an ideal step-up transformer if the voltage increases?

Electromagnetic induction

What you'll learn

The basic idea

Magnetic flux

Faraday’s law

Lenz’s law

Moving conductors: cutting field lines

Generators and transformers

AC generators

Transformers

Practical note

You've reached the end

Electromagnetic induction Revision Guide