Magnetic flux and flux linkage for A-Level Physics
Magnetic flux is the total magnetic field passing through a given area, written as Φ = BA cos θ, where B is the magnetic flux density, A is the area, and θ is the angle between the field and the normal to the surface. Flux linkage is flux multiplied by the number of turns of the coil, written as NΦ. Both ideas sit at the heart of the AQA A-Level electromagnetic induction topic.
This guide walks through the definitions, units, formulas, and the way examiners test them. It covers the geometry that trips students up, a worked example, and the common mistakes that lose marks on Paper 2.
Flux is measured in webers
One weber (Wb) is one tesla metre squared. Magnetic flux is a measure of how much field passes through a loop, not the strength of the field itself.
Two formulas to know
Magnetic flux Φ = BA cos θ and flux linkage NΦ = BAN cos θ. The cos θ term catches almost everyone the first time.
Flux linkage drives induction
Faraday's law states that induced EMF equals the rate of change of flux linkage. If NΦ is not changing, no EMF is induced.
Defining magnetic flux properly
Magnetic flux through a surface is the product of the magnetic flux density perpendicular to the surface and the area of the surface. The mark-scheme wording is: The magnetic flux through an area A is Φ = BA cos θ, where θ is the angle between the magnetic field and the normal to the area.
The key idea is the word perpendicular. Only the component of the field at right angles to the surface counts. If the field lies flat in the plane of the loop, no flux passes through it and Φ = 0.
What the angle θ actually means θ is the angle between the magnetic field B and the normal (the line at right angles to the surface), not the angle between B and the surface itself. When B is perpendicular to the surface, θ = 0 and cos θ = 1, giving maximum flux. When B lies in the plane of the surface, θ = 90° and cos θ = 0.
Magnetic flux linkage
Flux linkage is the magnetic flux multiplied by the number of turns of the coil. For a coil with N turns, flux linkage is NΦ = BAN cos θ. The unit is the weber-turn (Wb-turn), although the SI unit is just the weber.
The reason flux linkage matters is that each turn of a coil experiences the same flux Φ, so the total EMF induced across the whole coil is N times the EMF induced in a single turn. That is why Faraday's law uses NΦ rather than just Φ.
| Quantity | Formula | Unit | Symbol |
|---|---|---|---|
| Magnetic flux density | B | tesla (T) | B |
| Magnetic flux | Φ = BA cos θ | weber (Wb) | Φ |
| Flux linkage | NΦ = BAN cos θ | weber-turn (Wb) | NΦ |
| Induced EMF | ε = –dNΦ/dt | volt (V) | ε |
Faraday's law and Lenz's law
Faraday's law of electromagnetic induction states that the induced EMF is equal to the rate of change of flux linkage. In symbols: ε = –d(NΦ)/dt. The negative sign comes from Lenz's law, which says the induced current flows in a direction that opposes the change in flux that caused it.
In exam questions you will usually be asked to find the magnitude of the induced EMF, so the negative sign is dropped. But if a question asks for the direction of the induced current, Lenz's law is the tool you reach for.
Lenz's law in plain English If you push a north pole into a coil, the coil pushes back. The induced current sets up a magnetic field that opposes the incoming flux, which means the end of the coil facing the magnet acts as a north pole. Pulling the magnet away reverses everything, and the induced current flips direction.
How flux changes in a rotating coil
When a coil rotates in a uniform magnetic field at angular speed ω, the angle between the field and the normal changes with time: θ = ωt. Substituting into the flux linkage equation gives NΦ = BAN cos(ωt). Differentiating with respect to time gives the induced EMF: ε = BANω sin(ωt).
This is why a generator produces an alternating EMF that varies as a sine wave. The peak EMF is ε₀ = BANω, reached when the coil is parallel to the field (θ = 90°, so the rate of change of flux is greatest). When the coil is perpendicular to the field, flux is at a maximum but the EMF is zero, because the rate of change is zero at that instant.
Worked example: Flux through a rotating coil
A square coil of side 0.10 m with 200 turns sits in a uniform magnetic field of flux density 0.50 T. The coil rotates at 50 revolutions per second. Find the peak EMF induced.
Step 1: Calculate the area. A = 0.10 × 0.10 = 0.010 m².
Step 2: Convert frequency to angular speed. ω = 2πf = 2π × 50 = 314 rad/s (to 3 s.f.).
Step 3: Apply the peak EMF formula. ε₀ = BANω = 0.50 × 0.010 × 200 × 314 = 314 V.
The peak EMF is 314 V. Notice how each factor matters: Doubling the number of turns doubles the EMF, doubling the rotation speed doubles the EMF, and doubling the field strength doubles the EMF.
Where students lose marks on flux questions
Examiner reports for the AQA Year 13 Physics paper flag the same handful of mistakes year after year. Almost all of them come down to careless geometry or confusing flux with flux linkage.
Common mistakes that cost easy marks Forgetting the cos θ term when the field is not perpendicular to the coil. Confusing the angle to the surface with the angle to the normal. Writing flux instead of flux linkage in a question about a multi-turn coil. Forgetting the negative sign when asked about the direction of induced current. Using degrees instead of radians when differentiating cos(ωt). Mixing up B (flux density) with Φ (flux) because both are called magnetic field in everyday language.
Key facts to memorise for the exam
- Magnetic flux: Φ = BA cos θ, measured in webers (Wb)
- Flux linkage: NΦ = BAN cos θ, also measured in webers
- θ is the angle between B and the normal to the surface, not the surface itself
- Faraday's law: Induced EMF equals the rate of change of flux linkage, ε = –d(NΦ)/dt
- Lenz's law: The induced current opposes the change in flux that caused it
- Peak EMF in a rotating coil: ε₀ = BANω, reached when the coil is parallel to the field
- 1 weber = 1 tesla metre squared = 1 V·s
- If the flux is constant in time, the induced EMF is zero
