6.3.3(a)-(e) - Flux, Faraday's and Lenz's laws and generators

Magnetic flux and the weber

Magnetic flux is a way of measuring how much magnetic field passes through an area. Imagine a loop of wire in a uniform magnetic field. If the field passes straight through the face of the loop, the flux is large. If the loop is turned edge-on to the field, almost no field passes through the area, so the flux is zero.

Here $B$ is magnetic flux density in tesla (T), $A$ is the area in square metres ($\mathrm{m^2}$), and $\theta$ is the angle between the magnetic field direction and a line normal to the area. The unit of magnetic flux is the weber, $\mathrm{Wb}$.

Because $B$ is in $\mathrm{T}$ and $A$ is in $\mathrm{m^2}$,

[DIAGRAM: asset_name: 6.3.3(a)-(e) - Flux, Faraday's and Lenz's laws and generators - diagram 1; asset_slug: 6_03_3a_magnetic_flux_faraday_lenz_and_generators__diagram_01; recommended_method: drawn_physics; caption: Magnetic flux depends on the angle between the field and the normal to the coil plane.; description: Edge-on area geometry shows the normal perpendicular to the plane and angle to the field; cosine graph.]

The angle is the common trap. In $\Phi = BA\cos\theta$, $\theta$ is not the angle between the magnetic field and the plane of the coil. It is the angle between the magnetic field and the normal to the plane. The normal is a line at right angles to the coil face.

If $\theta = 0^\circ$, the field is parallel to the normal, so the field passes straight through the coil face:

If $\theta = 90^\circ$, the field is parallel to the plane of the coil, so no field passes through the area:

Worked example: flux through a coil

A rectangular coil has area $4.0 \times 10^{-3}\ \mathrm{m^2}$. It is placed in a uniform magnetic field of flux density $0.25\ \mathrm{T}$. The normal to the coil makes an angle of $30^\circ$ with the field. Calculate the magnetic flux through one turn.

Use:

Substitute:

The answer is less than $BA = 1.0 \times 10^{-3}\ \mathrm{Wb}$, because the coil is tilted rather than face-on to the field.

Flux linkage and Faraday's law

A coil usually has more than one turn. If each turn links the same magnetic flux, the magnetic flux linkage is the number of turns multiplied by the flux through one turn.

For a coil of $N$ identical turns in a uniform field:

The unit is often written as $\mathrm{Wb}$ turns. The word "turns" is not a new SI base unit; it reminds you that the flux through one turn has been multiplied by the number of turns.

Electromagnetic induction happens when the magnetic flux linkage changes. It can change because the magnetic flux density changes, the coil area linked by the field changes, the angle changes, or the number of turns linked changes. In a generator, the angle changes continuously. In a search-coil experiment, the magnetic field may alternate or the coil may be rotated.

OCR gives the induction equation in the form:

Here $\mathcal{E}$ is the induced e.m.f. in volts (V), $\Delta(N\Phi)$ is the change in magnetic flux linkage in $\mathrm{Wb}$ turns, and $\Delta t$ is the time interval in seconds. The minus sign is not a calculation nuisance. It represents Lenz's law, which you will use for direction in the next part.

For calculation magnitudes, OCR often accepts:

This tells you the average induced e.m.f. over the time interval. If the flux linkage changes non-uniformly, the instantaneous e.m.f. changes too; the largest e.m.f. occurs when the rate of change of flux linkage is greatest.

Worked example: average induced e.m.f.

A coil has $250$ turns. The magnetic flux through each turn changes from $1.8 \times 10^{-4}\ \mathrm{Wb}$ to $0.20 \times 10^{-4}\ \mathrm{Wb}$ in $0.040\ \mathrm{s}$. Calculate the magnitude of the average induced e.m.f.

First calculate the magnitude of the change in flux per turn:

Then calculate the magnitude of the change in flux linkage:

Now use Faraday's law:

The average induced e.m.f. is $1.0\ \mathrm{V}$. A shorter time for the same flux-linkage change would give a larger e.m.f.

Lenz's law and induction direction

Faraday's law gives the size of the induced e.m.f. Lenz's law gives its direction.

The phrase "oppose the change" is precise. It does not mean "oppose the magnetic field" in every situation. It means the induced current, if a complete circuit is available, produces a magnetic effect that tries to reduce the change in magnetic flux linkage.

Use this two-step method for direction questions.

1. Identify the change in magnetic flux linkage.
2. Choose the induced effect that opposes that change.

Example: a north pole moves towards a coil

As the north pole approaches, the magnetic flux linkage through the coil increases. The induced current must produce a magnetic field that opposes this increase. The coil face nearest the magnet therefore behaves like a north pole, repelling the approaching north pole. The induced current direction is the one that gives that pole arrangement, found using the right-hand grip rule for a coil.

Example: a north pole is pulled away from a coil

Now the flux linkage through the coil is decreasing. The induced current must try to maintain it. The near face of the coil behaves like a south pole, attracting the receding north pole. The current direction is reversed compared with the approaching case.

This is an energy-conservation idea. If an induced current helped the original change, the change would grow without an energy input. In real induction, work must be done to move the magnet, turn the generator, or change the field. That input energy is transferred electrically and thermally in the circuit.

Graphs make Faraday and Lenz work together. If flux linkage $N\Phi$ is increasing, $\Delta(N\Phi)/\Delta t$ is positive, so the minus sign gives an e.m.f. in the negative chosen direction. If flux linkage is decreasing, the gradient is negative, so the induced e.m.f. has the opposite sign. If flux linkage is momentarily constant, the induced e.m.f. is zero even if the flux linkage itself is large.

[DIAGRAM: asset_name: 6.3.3(a)-(e) - Flux, Faraday's and Lenz's laws and generators - diagram 2; asset_slug: 6_03_3a_magnetic_flux_faraday_lenz_and_generators__diagram_02; recommended_method: drawn_physics; caption: An induced coil field opposes the approach or retreat of a north magnetic pole.; description: An approaching north pole induces a north coil face that repels it; a retreating north pole induces a south face that attracts it. The opposing effect requires external work.]