Magnetic flux: counting field lines through a loop
Magnetic flux (Φ) tells how much magnetic field passes through a surface. Think of it as the number of field lines going through a loop.
Φ = B A cos θ. Here B is the field, A is the area of the loop and θ is the angle between B and the normal (the line standing straight out of the loop face). SI unit: weber (Wb); 1 Wb = 1 T m².
Flux is largest when the loop faces the field (θ = 0) and zero when the loop is edge-on (θ = 90°).
Faraday's laws of induction
Michael Faraday and Joseph Henry found that a changing flux makes an emf. A steady field, however strong, makes nothing.
First law
Whenever the flux through a circuit changes, an emf is induced. If the circuit is closed, a current flows. This lasts only while the change lasts.
Second law
The size of the induced emf equals the rate of change of flux: |ε| = N dΦ/dt for a coil of N turns.
You can change flux in three ways: change B (move a magnet), change A (stretch or slide part of the loop), or change θ (turn the loop).
Induced emf and induced current
The emf appears even in an open coil. Current flows only when the circuit is closed: I = ε / R.
The charge that flows depends only on the total change in flux, not on how fast: q = N ΔΦ / R. A slow push and a fast push send the same charge; the fast one sends it in a bigger, shorter burst.
Motional emf
A rod of length l sliding with speed v across a field B (all three at right angles) sweeps area l v every second. So ε = B l v. You can also see it as the magnetic force qvB pushing the free electrons to one end of the rod.
Eddy currents
A changing flux through a solid metal plate makes small swirling currents inside it, called eddy currents. They heat the metal and oppose motion. Used in induction cooktops and train brakes; reduced in transformers by using thin laminated sheets.
Lenz's law: which way the current flows
Lenz's law: the induced current flows in the direction that opposes the change in flux that caused it. That is why Faraday's law has a minus sign: ε = −N dΦ/dt.
Magnet N-pole coming in → the near face of the coil becomes N (pushes it back). Magnet going out → the near face becomes S (pulls it back).
Lenz's law is energy conservation. You must do work against this opposing force, and that work becomes electrical energy. If the current helped the change, you would get energy for free, which is impossible.
For a straight moving rod, use Fleming's right-hand rule: thumb = motion, first finger = field, middle finger = induced current.
Self-induction and self-inductance (L)
When the current in a coil changes, its own flux changes, so an emf appears in the same coil. This is self-induction. This emf opposes the change of current, so it is called a back emf.
N Φ = L I and ε = −L dI/dt. L is the self-inductance; unit henry (H). 1 H = 1 V s/A.
For a long solenoid: L = μ₀ n² A l = μ₀ N² A / l. Putting an iron core inside multiplies L by μᵣ.
An inductor acts like inertia for current. Energy stored in its magnetic field: U = ½ L I².
Mutual induction and mutual inductance (M)
When the current in coil 1 changes, the flux through a nearby coil 2 changes, and an emf appears in coil 2. This is mutual induction.
N₂ Φ₂ = M I₁ and ε₂ = −M dI₁/dt. For two long coaxial solenoids (inner radius r₁, length l): M = μ₀ n₁ n₂ π r₁² l. Also M₁₂ = M₂₁.
M is larger when the coils are close, wound on the same iron core and share the same axis. Transformers and wireless chargers use mutual induction.
Try it: a practical
Try it: In the 3D, set Turns N = 4 and drag the magnet in; note the "Last kick". Now set N = 16 and drag at the same speed. The kick is about 4 times bigger, because ε = −N dΦ/dt. At home: a small speaker or a toy motor connected to an LED can light the LED if you spin the motor shaft fast. You are running a generator.
Key formulas and definitions
- Φ = B A cos θ (unit weber, Wb)
- ε = −N dΦ/dt (Faraday + Lenz)
- Motional emf: ε = B l v
- Charge through the circuit: q = N ΔΦ / R
- Rotating rod (length l, angular speed ω): ε = ½ B ω l²
- Self-induction: N Φ = L I, ε = −L dI/dt
- Solenoid: L = μ₀ N² A / l
- Mutual induction: ε₂ = −M dI₁/dt; M = μ₀ n₁ n₂ π r₁² l
- Energy in an inductor: U = ½ L I²
Worked examples
1. A loop of area 0.02 m² lies in a field of 0.5 T. Find the flux when (a) the field is along the normal, (b) at 60° to the normal.
(a) Φ = BA cos 0 = 0.5 × 0.02 × 1 = 0.01 Wb. (b) Φ = 0.5 × 0.02 × cos 60° = 0.01 × 0.5 = 0.005 Wb.
2. The flux through a 200-turn coil drops from 4 mWb to 1 mWb in 0.1 s. Find the average induced emf.
ΔΦ = 1 − 4 = −3 mWb = −3 × 10⁻³ Wb. ε = −N ΔΦ/Δt = −200 × (−3 × 10⁻³)/0.1 = 6 V.
3. A 0.5 m rod moves at 4 m/s at right angles to a 0.3 T field. The rod is part of a circuit of resistance 2 Ω. Find the emf and the current.
ε = B l v = 0.3 × 0.5 × 4 = 0.6 V. I = ε / R = 0.6 / 2 = 0.3 A.
4. The current in a coil of L = 50 mH falls from 5 A to 0 in 10 ms. Find the self-induced emf.
ε = −L dI/dt = −0.05 × (0 − 5)/0.01 = +25 V. The emf tries to keep the current flowing (it opposes the fall).
5. Find L of an air-core solenoid with 500 turns, length 0.5 m and area 4 × 10⁻⁴ m². (μ₀ = 4π × 10⁻⁷)
L = μ₀ N² A / l = 4π × 10⁻⁷ × 250000 × 4 × 10⁻⁴ / 0.5 = 4π × 10⁻⁷ × 200 ≈ 2.51 × 10⁻⁴ H ≈ 0.25 mH.
6. Two coils have M = 1.5 H. The current in the first rises from 0 to 20 A in 0.5 s. Find the emf in the second coil and the flux change through it.
ε₂ = M dI₁/dt = 1.5 × 20 / 0.5 = 60 V. Flux linkage change = M ΔI₁ = 1.5 × 20 = 30 Wb (turns).
7. A 1 m metal rod spins at 20 rad/s about one end in a field of 0.2 T along its axis of spin. Find the emf between its ends.
ε = ½ B ω l² = 0.5 × 0.2 × 20 × 1² = 2 V.
8. A 100-turn coil of resistance 5 Ω has its flux changed by 2 × 10⁻³ Wb. How much charge flows?
q = N ΔΦ / R = 100 × 2 × 10⁻³ / 5 = 0.04 C. It does not depend on how fast the change happens.
Common mistakes
- Thinking a strong steady magnet near a coil makes current. Only a CHANGE in flux induces emf.
- Forgetting to multiply by N (number of turns) in ε = N dΦ/dt.
- Using θ as the angle between B and the plane of the loop. In Φ = BA cos θ, θ is measured from the normal to the loop.
- Saying Lenz's law opposes the flux. It opposes the CHANGE in flux; if flux is falling, the induced current tries to keep it up.