Quick answer: Electromagnetic induction produces an emf whenever the magnetic flux Φ = BA cos θ through a loop changes. Faraday’s law gives its size, ε = −NΔΦ/Δt, and Lenz’s law gives its direction: the induced current opposes the change in flux. A rod moving through a field has ε = BLv.
Level: AP Physics 2 (algebra-based) · Curriculum: AP P2 Unit 12
What is magnetic flux?
Magnetic flux measures how much magnetic field passes through a surface. A field of the same strength gives more flux through a bigger loop, and the most flux when the field is perpendicular to the loop.
Φ = BA cos θ
Φ = magnetic flux (Wb, where 1 Wb = 1 T·m²), B = field strength (T), A = area of the loop (m²), θ = angle between B and the normal to the loop (the line perpendicular to its surface).
A square coil of side 10 cm sits in a 0.40 T field with the normal at 60° to the field. Find the flux through one turn.
A = 0.10 × 0.10 = 0.010 m². Φ = BA cos θ = 0.40 × 0.010 × cos 60° = 2.0 × 10⁻³ Wb.
What does Faraday’s law say?
Electromagnetic induction is the production of an emf in a conductor when the magnetic flux through it changes. The emf exists only while the flux is changing, even if the flux itself is large.
Faraday’s law: ε = −N ΔΦ/Δt. The size of the induced emf equals the number of turns times the rate of change of flux through one turn. The minus sign expresses Lenz’s law.
The flux Φ = BA cos θ can change in three ways, so there are three ways to induce an emf.
| What changes | Example | Effect on emf |
|---|---|---|
| Field strength B | moving a magnet towards a coil | larger ΔB/Δt gives a larger emf |
| Area A | a loop being stretched or a rod sliding on rails | larger ΔA/Δt gives a larger emf |
| Angle θ | a coil rotating in a generator | faster rotation gives a larger emf |
Emma in Leeds winds a coil of 200 turns with area 25 cm². The field through it, perpendicular to the coil, rises from 0 to 0.80 T in 0.50 s. Find the emf.
A = 25 × 10⁻⁴ = 2.5 × 10⁻³ m². ΔΦ = 0.80 × 2.5 × 10⁻³ = 2.0 × 10⁻³ Wb. ε = NΔΦ/Δt = 200 × 2.0 × 10⁻³ / 0.50 = 0.80 V.
How does Lenz’s law give the direction of the current?
Lenz’s law: the induced current flows in the direction whose magnetic field opposes the change in flux that caused it. This is energy conservation: pushing a magnet towards a loop needs work, which becomes electrical energy.
To use it, ask three questions in order. Which way does the original field point? Is the flux through the loop increasing or decreasing? Which field direction would oppose that change? Then use the right-hand grip rule to find the current that makes this field.
A south pole is pulled away from a loop. Find the induced current direction seen from the magnet side.
The field from a south pole points towards the pole, so it points towards the magnet through the loop. The flux is decreasing, so the loop must make a field in the same direction (towards the magnet). That field comes from a counterclockwise current seen from the magnet side.
What is motional emf?
When a conductor of length L moves at speed v at right angles to a field B, the magnetic force pushes charges along the conductor. This produces ε = BLv. On rails, the area of the circuit grows by Lv each second, which is the same result as Faraday’s law.
A rod 0.50 m long slides at 4.0 m/s on rails in a 0.30 T field. The circuit resistance is 2.0 Ω. Find the emf, the current and the force needed to keep it moving.
ε = BLv = 0.30 × 0.50 × 4.0 = 0.60 V. I = ε/R = 0.60 / 2.0 = 0.30 A. F = BIL = 0.30 × 0.30 × 0.50 = 0.045 N. The power 0.045 × 4.0 = 0.18 W equals I²R = 0.18 W.
Students say the induced field “opposes the magnetic field”. It opposes the change in flux. If the flux is decreasing, the induced field points the same way as the original one.
Make a three-box table before every Lenz problem: original field direction, flux increasing or decreasing, induced field direction. Fill it left to right and the current direction follows.
- Magnetic flux: Φ = BA cos θ, in webers (1 Wb = 1 T·m²)
- Faraday’s law: ε = −NΔΦ/Δt; no change in flux means no emf
- Lenz’s law: the induced current opposes the change in flux
- Motional emf: ε = BLv for a rod moving perpendicular to B
- The induced current carries energy taken from the work done on the system
Magnetic flux, Faraday's law and Lenz's law: frequently asked questions
What is the difference between magnetic field and magnetic flux?
Magnetic field B is the strength at a point, measured in tesla. Magnetic flux Φ is the total field passing through a surface, equal to BA cos θ and measured in webers.
Why is there no induced emf in a constant magnetic field?
Faraday’s law depends on the rate of change of flux. If B, the area and the angle all stay the same, ΔΦ/Δt is zero and no emf is induced, however strong the field is.
What does the minus sign in Faraday’s law mean?
It shows that the induced emf drives a current that opposes the change in flux, which is Lenz’s law. For magnitudes only, you can leave it out and find the direction separately.
How do you find the direction of the induced current?
Find whether the flux through the loop is increasing or decreasing, choose the induced field direction that opposes that change, then use the right-hand grip rule to find the current.
How is motional emf related to Faraday’s law?
A rod of length L moving at speed v changes the circuit area by Lv per second, so ΔΦ/Δt = BLv. Faraday’s law then gives ε = BLv, the same as the magnetic force argument.
