Quick answer: Faraday’s law states that a changing magnetic flux through a coil induces an emf equal to −N dΦ/dt, with its direction set by Lenz’s law. Self-inductance L links this emf to changing current: ε = −L dI/dt. In an LR circuit current changes exponentially with time constant τ = L/R.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 13
What does Faraday’s law say?
Magnetic flux through a surface is ΦB = ∫ B · dA, measured in webers (1 Wb = 1 T·m²). For a uniform field it becomes ΦB = BA cos θ, where θ is the angle between the field and the normal to the surface.
When the flux linked with a coil changes, an emf appears in the coil. The size of the effect depends only on how fast the flux changes and on the number of turns, not on what makes the flux change.
Faraday’s law: ε = −N dΦB/dt. The minus sign is Lenz’s law: the induced current flows in the direction whose own magnetic field opposes the change in flux.
The flux can change because B changes, because the area changes, or because the angle changes. A loop of area A in a field B(t) = B₀ sin ωt has flux B₀A sin ωt, so the emf is a cosine: ε = −NB₀Aω cos ωt.
How is motional emf found?
A straight conductor of length ℓ moving at speed v perpendicular to a field B sweeps out area at the rate ℓv. Faraday’s law then gives a motional emf, and the magnetic force on the charges in the rod does the pushing.
ε = Bℓv, with ε in volts (V), B in tesla (T), ℓ in metres (m) and v in metres per second (m/s).
A coil has N = 200 turns and area 4.0 × 10⁻⁴ m². A field perpendicular to the coil falls uniformly from 0.50 T to zero in 0.20 s. The emf magnitude is N A ΔB/Δt = 200 × (4.0 × 10⁻⁴) × 0.50 / 0.20 = 0.20 V.
What is inductance?
A coil that carries a changing current produces a changing flux through itself, so it induces an emf in itself. This self-induction is described by the inductance L, measured in henries (1 H = 1 V·s/A).
L = NΦB/I and εL = −L dI/dt. Energy stored: U = ½LI², in joules. An ideal long solenoid has L = μ₀N²A/ℓ, where N is the number of turns, A the cross-sectional area and ℓ the length.
A solenoid has N = 800, A = 2.0 × 10⁻⁴ m² and ℓ = 0.40 m. Then L = (4π × 10⁻⁷)(800)²(2.0 × 10⁻⁴)/0.40 = 4.0 × 10⁻⁴ H = 0.40 mH.
How does current grow in an LR circuit?
When a switch connects a battery of emf ε to an inductor and resistor in series, Kirchhoff’s loop rule gives ε − IR − L dI/dt = 0. Solving it gives an exponential approach to the final current. At the first instant the inductor carries all the voltage, so dI/dt = ε/L.
Growth: I(t) = (ε/R)(1 − e−t/τ). Decay: I(t) = I₀e−t/τ. The time constant is τ = L/R, in seconds.
| Time | Fraction of final current | Fraction of initial current in decay |
|---|---|---|
| 0 | 0 | 1 |
| τ | 0.632 | 0.368 |
| 2τ | 0.865 | 0.135 |
| 3τ | 0.950 | 0.050 |
| 5τ | 0.993 | 0.007 |
L = 0.50 H, R = 25 Ω, ε = 10 V. Then τ = 0.50/25 = 0.020 s and the final current is 10/25 = 0.40 A. At t = 0.040 s = 2τ, I = 0.40 (1 − e⁻²) = 0.35 A.
Students say an inductor opposes current. It opposes the change in current. At steady state it has no emf across it and behaves like a plain wire.
What happens in an LC circuit?
A charged capacitor connected to an inductor exchanges energy between its electric field and the inductor’s magnetic field. With no resistance the total ½CV² + ½LI² stays constant and the charge oscillates with angular frequency ω = 1/√(LC), so the period is T = 2π√(LC). Resistance adds damping.
To find the sign in Lenz’s law, ask what the loop wants: it fights the change. If flux into the page grows, the loop produces flux out of the page.
- Faraday’s law: ε = −N dΦ/dt, flux in Wb = T·m²
- Motional emf of a rod: ε = Bℓv
- Inductance: L = NΦ/I, solenoid L = μ₀N²A/ℓ, energy U = ½LI²
- LR circuit: τ = L/R, I = (ε/R)(1 − e^(−t/τ))
- LC circuit: ω = 1/√(LC), energy swaps between C and L
Faraday's law, inductors and LR circuits: frequently asked questions
What is Faraday’s law of induction?
The emf induced in a coil equals the negative rate of change of magnetic flux linkage: ε = −N dΦ/dt. It applies whether the flux changes because of field, area or angle.
What does the minus sign in Faraday’s law mean?
It is Lenz’s law. The induced current produces a field that opposes the change in flux, which is how energy is conserved.
What is the time constant of an LR circuit?
It is τ = L/R. After one time constant the current has covered 63 % of its change towards the final value.
How much energy does an inductor store?
An inductor carrying current I stores U = ½LI² in its magnetic field. The energy is released when the current falls.
How do you find the frequency of an LC circuit?
Use ω = 1/√(LC) in rad/s, then f = ω/2π. A larger inductance or capacitance gives a lower frequency.
