The big idea: Hold a wire hoop out in the rain and count the drops falling through it — tilt it edge-on and almost none get through. Magnetic flux Φ is the same idea for a magnetic field: how much field passes through a loop of area A.
Flux is biggest when the loop faces the field square-on, and zero when the loop lies edge-on (the lines just skim past).
The angle is measured to the normal: θ in the formula is the angle between B and the normal (a line sticking straight out of the loop), not the angle to the surface. Loop facing the field square-on ⇒ θ = 0 and cos θ = 1 (maximum flux). Loop edge-on ⇒ θ = 90° and cos θ = 0 (zero flux).
- magnetic flux (weber, Wb)
- magnetic flux density / field strength (T)
- area of the loop (m²)
- angle between B and the normal to the area
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Change the flux through a coil and you induce an emf (a voltage). It is the rate of change that matters: a fast change makes a big emf, a slow change a small one. More turns N multiplies the effect, because the same flux change is felt by every turn.
What induces an emf?: Any way of changing the flux works: moving a magnet in or out, growing or shrinking the loop, or rotating it. No change of flux ⇒ no emf, even in the strongest steady field.
- induced emf (V)
- number of turns on the coil
- change in magnetic flux (Wb)
- time over which the flux changes (s)
The magnetic flux through a 50-turn coil falls steadily from 0.20 Wb to 0 in 0.10 s. Find the magnitude of the induced emf.
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Which way does the current flow?: Lenz's law: the induced current always flows in the direction that opposes the change producing it. Push a magnet's north pole toward a coil and the coil's near face becomes a north pole too, pushing back.
The minus sign = energy conservation: The minus sign in Faraday's law is Lenz's law written down. The induced effect must oppose the change — if it helped the change instead, you would get energy from nothing (a perpetual-motion machine). You must do work against this opposing force, and that work becomes the electrical energy. The minus sign is conservation of energy in action.
Faraday's law — how big?
- Gives the size of the emf
- ε ∝ rate of change of flux
- More turns N ⇒ bigger emf
- Faster change ⇒ bigger emf
Lenz's law — which way?
- Gives the direction of the emf
- Current opposes the change
- Source of the minus sign
- It is energy conservation
A moving rod is a battery: Slide a conducting rod of length L through a magnetic field and the free charges in it feel a magnetic force that pushes them along the rod. One end goes positive, the other negative — the rod acts like a small battery with an emf ε = BvL.
This is just Faraday's law again: as the rod moves it sweeps out new area, so the flux through the circuit changes.
- induced (motional) emf (V)
- magnetic flux density (T)
- speed of the rod, perpendicular to B (m s⁻¹)
- length of the rod in the field (m)
A metal rod of length L = 0.40 m slides at v = 3.0 m s⁻¹, perpendicular to a uniform magnetic field of flux density B = 0.50 T. Find the emf induced across the rod.
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Where it shows up: Electromagnetic induction is HL only (D.4):
- Paper 1A — a one-step 'what is the flux?', 'what is the emf?', or 'which way does the current flow?' (Lenz). - Paper 2 — determine an emf from a flux–time graph (gradient × N) or from a moving rod, and state the direction by Lenz's law.
Three easy marks: (1) Use the angle to the normal in Φ = BA cos θ. (2) Faraday needs the rate of change ΔΦ/Δt, not just ΔΦ. (3) For direction, quote Lenz's law — the current opposes the change.
A single square loop of side 0.20 m lies flat in a uniform field of flux density 0.30 T, with the field perpendicular to the loop. The loop is collapsed to zero area in 0.050 s. (a) Determine the magnitude of the emf induced. (b) State and explain the direction of the induced current.
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