Flux and flux linkage
Magnetic flux — The product of magnetic flux density and the area perpendicular to it, Φ = BA cos θ. Measured in webers.
Induction is about change, so the first job is a quantity that can change. That quantity is magnetic flux — loosely, how many field lines pass through a loop.
It depends on three things: how strong the field is, how large the loop is, and how the loop is oriented. A loop face-on to the field has the most flux through it; turned edge-on, it has none, because no lines pass through.
For a coil of N turns the same flux threads every turn, so the total is NΦ — the flux linkage. This is why coils in generators and transformers have many turns: it multiplies the effect.
The weber is a large unit. Typical school-laboratory flux values are in milliwebers or less.
- Φ
- magnetic fluxWb
- B
- flux densityT
- A
- aream²
- N
- number of turns
The angle is measured from the normal
In Φ = BA cos θ, θ is the angle between the field and the normal to the loop, not between the field and the loop itself. Face-on gives θ = 0 and maximum flux; edge-on gives θ = 90° and zero. Measuring from the plane of the loop swaps sine for cosine and every answer with it.
Faraday's law
Move a magnet into a coil connected to a sensitive meter and the needle deflects. Hold the magnet still inside the coil and it reads zero, however strong the magnet. Pull it out and the needle deflects the other way.
That experiment contains the whole law. An e.m.f. is induced only while the flux is changing, and its size depends on how fast the change happens — not on how much flux there is.
Faraday's law: the induced e.m.f. is proportional to the rate of change of flux linkage. Move the magnet faster, use a stronger magnet, or add more turns, and the e.m.f. rises.
The same law explains why induction works when nothing visibly moves. In a transformer the coils are stationary; it is the alternating current that makes the field change, and a changing field through a stationary coil induces an e.m.f. just as well as a moving magnet does.
- N
- turns
- ΔΦ/Δt
- rate of change of fluxWb s⁻¹
- v
- speed of the conductorm s⁻¹
Two different quantities, drawn as two different things. The green fill inside the coil is the flux through it. The needle is how fast that flux is changing — which is the e.m.f. Watch what happens when you stop the magnet inside the coil: the fill stays bright and the needle drops to nothing. Faraday’s law is about the rate, not the amount, and this is the case that separates them. Notice too that the needle passes through zero as the magnet crosses the middle of the coil — most flux, no change, no e.m.f.
Lenz's law and where the energy comes from
Lenz's law — The direction of an induced current is always such that it opposes the change producing it.
Push the north pole of a magnet towards a coil and the induced current flows so that the near face of the coil becomes a north pole — pushing back. Pull the magnet away and the near face becomes south, trying to hold it.
The coil always resists what you are doing. That is not perversity; it is conservation of energy. If the induced current helped the motion instead of opposing it, the magnet would accelerate on its own and produce electrical energy from nothing.
So you have to do work against that opposition, and the work you do is exactly what becomes electrical energy. This is why a bicycle dynamo makes pedalling harder, and why a power station burns more fuel when demand rises — the generators literally become harder to turn.
The minus sign in Faraday's law is Lenz's law written into the equation.
A coil of 250 turns and area 4.0 × 10⁻³ m² sits in a field of 0.30 T, face-on. The field is reduced to zero in 0.020 s. Find the average induced e.m.f.
- Initial flux
Φ = BA = 0.30 × 4.0 × 10⁻³ = 1.2 × 10⁻³ Wb.Face-on, so cos θ = 1. - Final flux is zero, so
ΔΦ = 1.2 × 10⁻³ Wb. - Flux linkage change
= NΔΦ = 250 × 1.2 × 10⁻³ = 0.30 Wb.The turns multiply the effect. - Uses
e.m.f. = NΔΦ/Δt = 0.30 / 0.020. e.m.f. = 15 V.Taking the magnitude; the sign only tells you the direction.
15 V
The a.c. generator
A generator is a motor run backwards. A coil is rotated in a magnetic field, the flux through it changes continuously, and an e.m.f. is induced.
The output is sinusoidal, and the shape follows from the geometry. When the coil is face-on to the field the flux is greatest — but it is changing most slowly at that instant, so the e.m.f. is zero. When the coil is edge-on the flux is zero, but it is changing fastest, so the e.m.f. peaks. Maximum flux and maximum e.m.f. occur a quarter of a turn apart.
Every half turn the coil passes through the position where the induced e.m.f. reverses, which is what makes the output alternating.
The output is taken through slip rings — two continuous rings with brushes. They keep contact without reversing the connections, so the alternating output is preserved. A d.c. generator uses a split-ring commutator instead, which flips the connections every half turn and folds the negative halves upwards.
A bigger output comes from spinning faster, a stronger field, more turns, or a larger coil area.
Slip rings or commutator?
Slip rings preserve the alternating output — a.c. generator. A split-ring commutator reverses the connections every half turn, giving d.c. The same distinction appears on the motor side of the syllabus, and questions often ask you to name the part rather than describe it.
Key points
Φ = BA cos θ, with θ measured from the normal to the loop.- An e.m.f. appears only while the flux is changing — a stationary magnet induces nothing.
- Faraday: e.m.f. is proportional to the RATE of change of flux linkage.
- Lenz: the induced effect opposes the change, because energy must be conserved.
- In a generator, e.m.f. peaks when the coil is edge-on, where the flux is changing fastest.