D.4 Induction

Define magnetic flux

Magnetic flux measures the magnetic field passing through an area. Here $\theta$ is the angle between the field and the normal to the surface, not the surface itself. Flux has unit weber, $\mathrm{Wb}=\mathrm{T\,m^2}$.

Worked example — square loop

For side $6.2\,\mathrm{cm}=6.2\times10^{-2}\,\mathrm{m}$, $A=(6.2\times10^{-2})^2\,\mathrm{m^2}$. In $B=4.3\times10^{-4}\,\mathrm{T}$ with $\theta=45^\circ$, $\Phi=BA\cos\theta=1.2\times10^{-6}\,\mathrm{Wb}$. For $N$ identical turns, flux linkage is $N\Phi$.

Use flux linkage

For a coil with $N$ turns, the linked flux is $N\Phi$. A change can result from changing $B$, $A$, $\theta$, or the number of turns. Keep the angle convention explicit: it is not generally the angle between B and the plane itself.

Check the geometry

At $\theta=0$, field and area normal align and $\Phi=BA$. At $\theta=90^\circ$, the field is parallel to the surface and $\Phi=0$. Convert area to square metres before substitution.

Common trap

Do not use the angle between B and the plane when the formula expects the angle to the normal. Do not confuse flux $\Phi$ with flux linkage $N\Phi$.

Use the rate of change

Faraday's law states that a changing magnetic flux linkage induces an emf. For an average emf use the finite change; for an instantaneous emf use a derivative. No change in flux linkage means no induced emf.

Worked example — average induced emf

For $N=1200$ turns, flux per turn increasing from $0$ to $4.8\times10^{-5}\,\mathrm{Wb}$ in $2.7\,\mathrm{ms}=2.7\times10^{-3}\,\mathrm{s}$, $|\varepsilon|=N\Delta\Phi/\Delta t=21\,\mathrm{V}$. The minus sign determines polarity through Lenz's law; it does not make the magnitude negative.

Identify what changes

Flux linkage can change because the field strength, coil area or angle changes. The minus sign gives the direction described by Lenz’s law; the magnitude is determined by how rapidly the flux changes.

Apply it to coils

An alternating current in a primary coil creates an alternating magnetic field and changing flux in a nearby secondary coil, so an emf is induced there. A steady current after switching has no changing flux and does not sustain an induced emf in the secondary.

Common trap

Do not say that a magnetic field alone induces emf. It is the change in flux linkage, not the mere presence of B, that matters.

Use the motional-emf model

A straight conductor of length $L$ moving at speed $v$ perpendicular to a uniform magnetic field $B$ sweeps out area and develops an emf. The stated equation is restricted to the perpendicular geometry.

Worked example — moving conductor

For $B=120\,\mu\mathrm{T}=120\times10^{-6}\,\mathrm{T}$, $v=98.0\,\mathrm{cm\,s^{-1}}=0.980\,\mathrm{m\,s^{-1}}$ and $L=23.0\,\mathrm{cm}=0.230\,\mathrm{m}$, $\varepsilon=BvL=2.70\times10^{-5}\,\mathrm{V}$. Reversing either motion or field reverses polarity.

Check the motion

The conductor must cut across field lines. Motion parallel to the field produces no motional emf; increasing B, v or the length in the field increases the emf. For a complete circuit, the emf can drive current.

Explain charge separation

Moving charge carriers in the conductor experience magnetic force and separate until an internal electric force balances it. The resulting potential difference across the ends is the motional emf.

Common trap

Do not use the wire’s total length if only part is inside the field, and do not expect emf when the motion is parallel to the field lines.

State the direction rule

Lenz’s law says the induced emf and induced current act in a direction that opposes the change in magnetic flux that produces them. It does not oppose the magnetic field itself; it opposes the change.

Use a four-step check

1. Identify whether external flux through the coil increases or decreases.
2. Determine the induced field needed to oppose that change.
3. Use the right-hand rule to find current direction/polarity.
4. Check that the induced current would require external work, consistent with energy conservation.

Interpret emf sign

A positive or negative emf is relative to the chosen loop direction and reference terminal. The sign records polarity/direction; it is not a negative amount of energy or a separate magnitude.

Common trap

Do not make the induced field reinforce an increasing flux. That would violate energy conservation and reverse the predicted current.

Track one rotation

A coil rotating at constant angular speed in a uniform magnetic field has changing flux linkage. The rate of change is zero at the orientations where flux is maximum or minimum, and greatest when the flux passes through zero.

Read the emf waveform

Because the flux varies sinusoidally with rotation angle, the induced emf is sinusoidal. It changes sign every half-turn as the polarity reverses. The emf is zero when the rate of flux change is zero and has maximum magnitude when the rate is greatest.

Connect positions and graph

Match coil orientation to the graph before choosing a phase. A half-turn reverses the emf; a full turn repeats the waveform. The exact phase depends on the stated initial orientation and rotation axis.

Common trap

Do not put the emf maximum where flux is maximum. Faraday’s law depends on the rate of change of flux, not its value.

Read the frequency effect

For a rotating coil with fixed $N$, $B$ and $A$, increasing rotation frequency increases the rate of change of magnetic flux. The induced-emf amplitude therefore increases with frequency, while the emf waveform oscillates more rapidly.

Transform the graph

If the rotation frequency doubles, the emf period halves and the number of cycles per unit time doubles. For the same coil and field, the peak emf also doubles. If frequency is halved, period doubles and peak emf halves.

Hold variables fixed

These comparisons assume the coil area, number of turns and magnetic-field strength stay unchanged. Changing those quantities also changes the emf amplitude, so identify which parameter the question varies.

Common trap

Do not change only the period when frequency changes. For a fixed rotating coil, frequency affects both waveform frequency and peak emf through the rate of flux change.