Source: Chapters 13 and 14
Tags: Faraday's law, magnetic flux, induced EMF, Lenz's law, mutual inductance, self-inductance, inductor, RL circuit, LC circuit, RLC circuit, magnetic energy
Difficulty: Intermediate to Advanced
Prerequisites: Chapters 11-12 notes (magnetic fields, Ampere's law, solenoids). Understanding of RC circuits from Chapter 10.
This is where electricity and magnetism fully merge. Faraday's law says a changing magnetic flux produces an EMF, which is how generators, transformers, and wireless charging work. Inductance quantifies how effectively a changing current in one circuit (or the same circuit) induces an EMF. The LC and RLC circuits introduced here are the foundation for understanding oscillations, resonance, and ultimately AC circuits and electromagnetic waves.
A changing magnetic flux through a loop induces a voltage (Faraday's law), and the direction of that induced current opposes the change that caused it (Lenz's law). Inductors resist changes in current and store energy in their magnetic fields. When paired with capacitors, they create oscillating circuits (LC) that are the electrical analogue of a mass on a spring.
Magnetic flux (Φ)
The total amount of magnetic field passing through a surface: Φ = ∫ B · dA = BA cos θ for a uniform field through a flat loop. Think of it as: how many field lines thread through the loop.
Faraday's law
The induced EMF in a loop equals the negative rate of change of magnetic flux through the loop: ε = -dΦ/dt. For N turns: ε = -N dΦ/dt. In simple terms, change the magnetic flux, and you get a voltage.
Lenz's law
The direction of the induced current is such that it opposes the change in flux that produced it. This is the physical meaning of the negative sign in Faraday's law.
Mutual inductance (M)
A measure of how effectively a change in current in one coil induces an EMF in a neighbouring coil: ε₂ = -M dI₁/dt. Measured in henrys (H). M = N₂Φ₂₁/I₁.
Self-inductance (L)
A measure of how effectively a coil's own changing current induces an EMF in itself: ε = -L dI/dt. L = NΦ/I. Think of it as: the circuit's resistance to changes in its own current.
Inductor
A circuit element designed to have a specific self-inductance, usually a coil of wire. It opposes changes in current.
RL circuit
A circuit containing a resistor and an inductor. The time constant is τ_L = L/R.
LC circuit
A circuit containing an inductor and a capacitor. Energy oscillates between the electric field of the capacitor and the magnetic field of the inductor at angular frequency ω = 1/√(LC).
RLC circuit
An LC circuit with resistance, which causes the oscillations to decay (damping). The charge oscillates as q(t) = Q₀ e^(-Rt/2L) cos(ω't + φ).
Magnetic flux: Φ = BA cos θ (uniform field, flat surface).
Faraday's law: ε = -N dΦ/dt.
The flux can change because B changes, the area changes, or the angle between B and the surface normal changes. Any of these produces an induced EMF.
The induced current creates a magnetic field that opposes the change in flux.
If the flux through a loop is increasing, the induced current flows in the direction that creates a field opposing the increase (i.e., opposing the external field).
If the flux is decreasing, the induced current flows in the direction that tries to maintain the flux.
When two coils are near each other, a changing current in coil 1 changes the flux through coil 2, inducing an EMF in coil 2.
ε₂ = -M dI₁/dt, and by symmetry ε₁ = -M dI₂/dt. The mutual inductance M is the same regardless of which coil is the source.
A changing current in a coil changes the flux through itself, inducing a back-EMF: ε = -L dI/dt.
Self-inductance of a solenoid: L = μ₀n²Al, where n is turns per unit length, A is cross-sectional area, and l is the length.
Written in terms of the total turns N: L = μ₀N²A/l.
Energy stored in an inductor: U = ½LI².
Magnetic energy density: u_m = B²/(2μ₀).
Power absorbed by an inductor: P = LI dI/dt.
This is the magnetic analogue of U = ½CV² for a capacitor and u_E = ½ε₀E² for the electric field.
When a battery (EMF = ε) is connected to an RL series circuit:
Current: I(t) = (ε/R)(1 - e^(-t/τ_L)), where τ_L = L/R.
The current rises exponentially toward ε/R.
Energy stored in the inductor's field: U = ½LI².
When the battery is removed and the circuit is shorted, the current decays: I(t) = I₀ e^(-t/τ_L).
Energy oscillates between the capacitor (electric field energy ½Q²/C) and the inductor (magnetic field energy ½LI²).
Charge on the capacitor: q(t) = Q₀ cos(ωt + φ), where ω = 1/√(LC).
Current: I(t) = -ωQ₀ sin(ωt + φ).
Total energy is constant: U_total = ½Q₀²/C = ½LI_max².
This is the electrical analogue of simple harmonic motion (mass-spring system).
The resistance causes damping. Charge: q(t) = Q₀ e^(-Rt/2L) cos(ω't + φ).
Damped angular frequency: ω' = √(1/LC - R²/4L²).
If R²/4L² < 1/LC, the circuit oscillates with decreasing amplitude (underdamped).
If R²/4L² > 1/LC, no oscillation occurs (overdamped).
If R²/4L² = 1/LC, the system is critically damped.
Quantity | Formula |
|---|---|
Magnetic flux | Φ = BA cos θ |
Faraday's law | ε = -N dΦ/dt |
Mutual inductance EMF | ε = -M dI/dt |
Self-inductance EMF | ε = -L dI/dt |
Solenoid inductance | L = μ₀n²Al |
Energy in inductor | U = ½LI² |
Magnetic energy density | u_m = B²/(2μ₀) |
RL time constant | τ = L/R |
RL current (charging) | I(t) = (ε/R)(1 - e^(-Rt/L)) |
LC angular frequency | ω = 1/√(LC) |
LC charge | q(t) = Q₀ cos(ωt + φ) |
RLC damped charge | q(t) = Q₀ e^(-Rt/2L) cos(ω't + φ) |
Faraday's law is the operating principle of every electrical generator and transformer. Wireless charging pads for phones use mutual inductance: a changing current in the pad's coil induces a current in the phone's coil. The LC oscillator is the basic building block of radio tuning circuits, where you select a station by adjusting C or L to match the station's broadcast frequency.
Students often forget that it is the change in flux, not the flux itself, that induces an EMF. A constant, uniform magnetic field through a stationary loop produces zero EMF.
Lenz's law does not say the induced field cancels the external field. It opposes the change. If the external flux is decreasing, the induced field is in the same direction as the external field (trying to maintain the flux).
Students sometimes confuse the RL time constant (τ = L/R) with the RC time constant (τ = RC). In an RL circuit, larger R means faster current rise (shorter time constant), which is the opposite of RC behaviour.
In LC circuits, the frequency depends only on L and C, not on the initial charge or energy. Students sometimes think a larger initial charge changes the frequency; it only changes the amplitude.
⚠️ Faraday's law: be ready to identify which quantity is changing (B, A, or θ) and compute dΦ/dt.
⚠️ Know the LC frequency formula ω = 1/√(LC) and how to use it to find the period, frequency, or the component values.
⚠️ RL circuits follow the same exponential pattern as RC circuits but with τ = L/R instead of τ = RC. Know the charging and decaying current expressions.
⚠️ Energy conservation in LC circuits: ½Q²/C + ½LI² = constant. Exam problems often ask you to find the current when the charge is at some fraction of its maximum, or vice versa.
Fill in the blank: Faraday's law states ε = ________.
-N dΦ/dt.
True or false: An LC circuit oscillates at a frequency that depends on the initial charge stored on the capacitor.
False. The frequency depends only on L and C.
True or false: The energy stored in an inductor is proportional to the square of the current.
True (U = ½LI²).
Fill in the blank: The time constant of an RL circuit is τ = ________.
L/R.
Q: A coil with 200 turns has a magnetic flux of 0.05 Wb through each turn. If the flux drops to zero in 0.1 s, what is the magnitude of the induced EMF?
A: ε = N ΔΦ/Δt = 200 × 0.05 / 0.1 = 100 V.
Q: An LC circuit has L = 10 mH and C = 100 μF. What is the oscillation frequency?
A: ω = 1/√(LC) = 1/√(0.01 × 10⁻⁴) = 1/√(10⁻⁶) = 1000 rad/s. f = ω/(2π) ≈ 159 Hz.
Q: An RL circuit has R = 50 Ω and L = 0.2 H, connected to a 10 V battery. What is the final (steady-state) current and the time constant?
A: I_final = ε/R = 10/50 = 0.2 A. τ = L/R = 0.2/50 = 0.004 s = 4 ms.
Faraday's law is one of Maxwell's four equations and is the foundation for AC circuits (Ch. 15) and electromagnetic waves (Ch. 16). The LC oscillator's frequency formula reappears as the resonant frequency in AC circuits. The energy density formula u_m = B²/(2μ₀) pairs with u_E = ½ε₀E² when computing the energy carried by electromagnetic waves.
Faraday's law, electromagnetic induction, magnetic flux, induced EMF, Lenz's law, mutual inductance, self-inductance, inductor, henry, back-EMF, RL circuit, LC circuit, RLC circuit, oscillation, damping, angular frequency, resonance, energy storage, magnetic energy density, generator, transformer, PHYS 212