10.2 Self & Mutual Inductance, Lenz's Law
Key Takeaways
- Mutual induction: changing current in a primary coil induces EMF in a magnetically coupled secondary; e_s ∝ M (di_p/dt)
- Mutual inductance M depends on number of turns, coupling (geometry/core), and permeability of the shared magnetic path
- Lenz’s Law: induced current/EMF polarity opposes the change in flux that produced it — the physical meaning of the minus sign in Faraday’s Law
- Self-induction: a coil induces EMF in itself when its own current (and flux linkage) changes; e = −L (di/dt)
- Self-inductance L = NΦ/I (weber-turns per ampere); unit is the henry (H)
10.2 Self & Mutual Inductance, Lenz's Law
Quick Answer: Mutual induction — changing primary current induces secondary EMF (e_s = −M di_p/dt). Self-induction — a coil opposes its own current change (e = −L di/dt). Lenz’s Law sets polarity so the induced effect opposes the flux change. L and M are in henrys.
Section 10.1 established that changing flux linkage induces EMF. This section names the circuit property that quantifies how much EMF you get for a given rate of current change, splits it into mutual and self inductance, and locks polarity with Lenz’s Law.
Mutual Induction
Two coils share a magnetic path (air, iron core, or transformer-style coupling). Call them primary (energised) and secondary (open or loaded).
When primary current i_p changes, primary flux changes. Part of that flux also links the secondary. Faraday’s Law then induces an EMF in the secondary even though no battery is connected to it.
e_s = −M (di_p / dt)
| Symbol | Meaning |
|---|---|
| e_s | Induced EMF in the secondary |
| M | Mutual inductance (henry, H) |
| di_p/dt | Rate of change of primary current |
Equivalently, if flux linkage of the secondary due to primary current is N_s Φ_21, then M = N_s Φ_21 / i_p (for linear coupling), and e_s = −d(N_s Φ_21)/dt.
Worked example 1. M = 0.05 H; primary current rises at 40 A/s.
|e_s| = M |di/dt| = 0.05 × 40 = 2.0 V.
Worked example 2. Same M, but di/dt = 200 A/s → |e_s| = 10 V. Faster primary current change → larger secondary kick — the transformer and ignition-coil story in miniature.
Rate of change of primary current
Mutual induction needs changing primary current, not merely a steady DC value:
| Primary condition | Secondary induced EMF (ideal) |
|---|---|
| Steady DC after flux has settled | ≈ 0 (dΦ/dt = 0) |
| Rising current (switch-on, AC rising half) | EMF while flux is increasing |
| Falling current (switch-off, AC falling half) | EMF while flux is decreasing (opposite polarity) |
| AC primary | Continuous alternating secondary EMF |
Exam trap. “DC in the primary always induces secondary voltage” is false for steady DC. Induction occurs during transitions and with AC.
Factors Affecting Mutual Inductance
M measures how effectively a change in one coil’s current couples flux into the other.
| Factor | Effect on M | Comment |
|---|---|---|
| Number of turns on either coil | More turns → larger M (roughly ∝ N_p N_s for tight coupling) | Each turn adds flux linkage |
| Coupling / geometry | Closer coils, coaxial alignment, shared core window → higher M | Loose spacing or orthogonal axes → weak coupling |
| Core permeability | Soft iron / ferrite core raises M vs air core | Low-reluctance path increases shared flux |
| Core area / magnetic path | Larger useful core cross-section for shared flux helps | Design/detail beyond Module 3 naming |
| Coefficient of coupling k | M = k √(L_1 L_2) with 0 ≤ k ≤ 1 | k = 1 is perfect coupling; real transformers approach high k |
Aircraft relevance. Transformer secondaries, current transformers (CTs), ignition coils, and some sensor couplings all rely on mutual inductance. A cracked core, open primary, or shorted turn changes effective coupling and induced behaviour — troubleshooting later builds on this vocabulary.
Lenz’s Law and Polarity Rules
Lenz’s Law: the direction of the induced EMF (and of the induced current if the circuit is closed) is such that it opposes the change in magnetic flux that produced it.
That is energy conservation in one sentence: induction does not amplify change for free; it fights the change.
| Flux change that causes induction | Induced current’s magnetic effect |
|---|---|
| Flux through coil increasing into the page | Induced current creates flux out of the page (opposes increase) |
| Flux through coil decreasing into the page | Induced current creates flux into the page (tries to maintain flux) |
| Magnet N-pole approaching a coil face | Coil face becomes N (repels approaching magnet) |
| Magnet N-pole receding | Coil face becomes S (attracts, opposing the loss of flux) |
Polarity checklist for Module 3
- Identify what is changing (flux up or down? magnet approaching or leaving?).
- Decide what induced flux would oppose that change.
- Use the right-hand clasp rule on the coil: fingers = conventional induced current direction; thumb = induced north pole / flux direction.
- For generators, Fleming’s right-hand rule agrees with the same opposition principle when motion and field are given.
Worked polarity scenario. A bar magnet’s north pole moves toward the left-hand face of a solenoid. Flux into that face from the magnet’s N is increasing toward the coil. Induced current makes the left-hand face a north pole (repels the approaching N). That induced current direction is the exam answer — not “whichever way looks convenient.”
Self-Induction
A single coil’s own current produces flux that links its own turns. When i changes, self-flux linkage changes, so Faraday induces an EMF in the same coil. That property is self-inductance L (often just “inductance”).
e = −L (di / dt)
Definition (linear inductor):
L = NΦ / I
(flux linkage per ampere). Unit: henry (H). 1 H means 1 V is induced when current changes at 1 A/s.
| Statement | Meaning |
|---|---|
| Large L | Large flux linkage per ampere — strong opposition to rapid di/dt |
| e ∝ di/dt | Fast switching → large self-induced EMF (voltage spikes) |
| Steady DC | di/dt = 0 → self-induced e = 0 after transient |
Worked example 3. L = 0.2 H; current falls at 50 A/s.
|e| = L |di/dt| = 0.2 × 50 = 10 V (polarity tries to keep current flowing — oppose the decrease).
Worked example 4. L = 4 H (large choke); di/dt = 0.5 A/s → |e| = 2 V.
Factors affecting self-inductance L
| Factor | Effect |
|---|---|
| More turns N | L rises (roughly ∝ N² for a given magnetic path) |
| Higher core μ | L rises |
| Larger core cross-section / shorter magnetic path | L tends to rise (lower reluctance) |
| Air gap in core | L falls (reluctance up) but linearity/saturation behaviour improves — design trade-off |
Self-induction is why inductors “resist” AC (later XL = 2πfL) and why switching inductive loads needs suppression (§10.3).
Mutual vs Self — Comparison Table
| Feature | Self-inductance L | Mutual inductance M |
|---|---|---|
| Coils involved | One coil (own flux) | Two (or more) coupled coils |
| Governing rate | di/dt in that coil | di/dt in the other coil (for the EMF of interest) |
| Formula (magnitude) | |e| = L |di/dt| | |e_s| = M |di_p/dt| |
| Unit | henry | henry |
| Transformer view | Leakage and magnetising inductance of each winding | Coupling between primary and secondary |
Section Synthesis
| Topic | One-line takeaway |
|---|---|
| Mutual induction | Changing i_p → e_s via shared flux; e_s = −M di_p/dt |
| di_p/dt | Steady DC → no continuous mutual EMF; change/AC required |
| Factors for M | Turns, coupling geometry, core μ, coefficient k |
| Lenz | Induced effects oppose the causing flux change |
| Self-induction | e = −L di/dt; L = NΦ/I; unit henry |
With Faraday, mutual/self inductance, and Lenz polarity in place, §10.3 applies them to back EMF, saturation, the LR time constant, and real inductor uses on aircraft.
Mutual inductance between two coils is measured in which SI unit?
A secondary linked with M = 0.02 H experiences a primary current change of 150 A/s. What is the magnitude of the induced secondary EMF?
According to Lenz’s Law, if the north pole of a magnet is moved toward a coil, the face of the coil nearest the magnet becomes:
Self-induced EMF in an inductor is zero when: