SELF AND MUTUAL INDUCTANCE

Try to push a current through a coil of wire. A real current, the kind that lights a bulb or spins a motor. You close a switch, connect a battery, and you would think the current jumps instantly from zero to whatever the resistance allows — V/R, the answer Ohm would give.

It doesn't. It rises, gently, over milliseconds or seconds or minutes depending on the coil. And on the way up, the coil pushes back. If you put a voltmeter across its terminals you will see a voltage opposing the battery, shrinking as the current settles. Disconnect the battery and the reverse happens: the current doesn't stop, it coasts downward, and the coil now sources a voltage that keeps it coasting.

This is self-inductance in action. A coil carrying a current I surrounds itself with a magnetic field. Loop through that field and you get a flux — and because every turn of the coil threads through the field of every other turn, the total flux linkage grows with I. When I changes, the flux changes. Faraday's law then produces an EMF inside the coil itself, and Lenz's law points that EMF against the change. The coil fights you whenever you try to alter its state.

The whole story of inductors is packed into that sentence. What follows is its unpacking.

Begin with the flux linkage. For any coil geometry, the total flux through all of its turns is proportional to the current running through it. "Proportional" is the whole claim — double the current, double the field, double the flux through every cross-section, double the linkage:

The constant L is the self-inductance of the coil. It depends only on geometry and the material filling the coil — how many turns, how big a cross-section, how long, whether there is iron in the core. It has nothing to do with I itself.

Now feed this into Faraday's law. The EMF induced around a closed loop equals the negative rate of change of flux through it. In plain words: the symbol dI/dt means "how fast the current is changing per second," and the integral sign that appears when we compute flux is just an area sum — both are spelled out in detail in the Faraday topic. Applied to the coil's own flux, Faraday gives:

That minus sign is Lenz's law writ small. If I is rising, dI/dt > 0, so the EMF is negative — a back-EMF that pushes against the rise. If I is falling, dI/dt < 0, the EMF flips positive and tries to keep the current going. The coil is an electrical flywheel.

The SI unit for L is the henry, abbreviated H. From EQ.02, one henry is one volt per ampere-per-second — a coil that produces one volt of back-EMF when the current in it changes at one ampere per second.

Joseph Henry was a mathematics teacher at the Albany Academy in upstate New York when, in 1830, he noticed the same phenomenon Faraday would publish the following year: changing magnetic flux produces an EMF. Henry was first to observe self-induction specifically — the spark that jumps the gap when you open the switch on a big coil. He was also first to wind insulated wire on an iron bar and so build a practical electromagnet powerful enough to lift a ton.

He published late. Teaching load, no scientific society nearby, a reluctance to dash into print. Faraday's 1831 paper won the priority race, and Henry's claim survived only because his colleagues at Princeton — where he became chair of natural philosophy in 1832 — vouched for the chronology. When the Smithsonian Institution was chartered in 1846 Henry became its first Secretary, a post he held for three decades and used to build American science from nearly nothing.

The SI unit of inductance bears his name. Every time you see the letter H on a component datasheet you are reading Joseph Henry's initial, tucked into the engineering language a century after he missed being first in print.

For the long solenoid geometry from §03, L comes out in two lines. Inside the solenoid the field is uniform at B = μ₀ · (N/ℓ) · I, where N is the total turn count and ℓ is the length. The flux through one turn is B · A, so the total linkage through all N turns is:

Notice the N². Wind twice as many turns on the same form and the inductance quadruples — one factor of N comes from a stronger interior field, the other from more turns catching that field. This is why every inductor you have ever seen is a coil, and why MRI gradient coils, transformer windings, and wireless-charging primaries are all tightly packed helices.

Close the switch and watch. The current climbs on an (1 − e^(−t/τ)) curve toward its Ohm's-law limit V/R. Green-cyan across the inductor is the back-EMF — huge at t = 0, fading as I stabilises. Slide L upward and the whole rise stretches out: bigger coil, slower climb. Press reset to replay.

If one coil produces flux, a second coil sitting nearby sees that flux too. Let I_1 flow in coil 1 and ask how much of its flux links coil 2. The answer is again proportional:

M is the mutual inductance of the pair. Like L, it depends only on geometry — how the two coils are oriented, how far apart, whether they share a core. Applying Faraday to coil 2:

A changing current in one coil produces a voltage in the other — without any wires between them. This is the principle of every transformer, every induction motor, every RFID card, every wireless phone charger. You drive coil 1 with AC; coil 2 picks up an AC voltage; no direct electrical contact is needed.

The reciprocity theorem says M_{12} = M_{21} — the two directions give the same coupling constant — and a coupling coefficient k = M / √(L_1 L_2) lives in the interval [0, 1], with k = 1 meaning every field line from coil 1 threads coil 2 (an idealised transformer) and k = 0 meaning the coils are geometrically decoupled.

A sinusoidal primary current drives flux through a smaller secondary sitting on the axis. The secondary picks up an EMF −M · dI_1/dt, shown in green-cyan, whose phase leads the primary current by a quarter cycle (because it tracks the derivative, not the current itself). Slide the secondary out of the primary's bore and the coupling k collapses — field lines miss the secondary, linkage drops, induced EMF shrinks.

Close the loop with Ohm. In a series RL circuit — battery V, resistor R, inductor L — Kirchhoff's voltage law says at every instant:

Solve with I(0) = 0:

The RL time constant τ = L/R sets the clock. After one τ the current has reached (1 − 1/e) ≈ 63.2 % of its asymptote; after 2τ, 86 %; after 3τ, 95 %; it never arrives at V/R in finite time.

Three curves, same voltage, same resistor, three different inductances — three different rhythms. The guide lines at 63 / 86 / 95 % mark the universal milestones of any first-order rise. "Faster" means smaller L or bigger R; "slower" means bigger L or smaller R. Every engineer who has ever spec'd a snubber circuit, an ignition coil, or a DC-DC converter has done arithmetic around this one equation.

Self-inductance is the reason every switched-mode power supply works. A MOSFET chops the input DC into high-frequency pulses; those pulses drive a small coil; the coil's L smooths the output through the same RL-rise mechanism FIG.23c shows. Your phone charger, your laptop brick, the regulator inside your GPU — all of them lean on τ = L/R and the back-EMF.

Ignition coils in petrol engines do it backwards. Current builds up through a coil, then a fast switch opens; dI/dt spikes hugely negative, EMF = −L · dI/dt spikes hugely positive, and a 20 kV pulse arcs across the spark plug gap. All of that voltage comes from the coil's refusal to let the current stop suddenly.

Mutual inductance is the transformer. Step up, step down, isolate — three functions of one pair of coupled coils. Your doorbell runs on a 120 V → 12 V step-down; the substation transformer down the street runs 35 kV → 240 V; the Tesla coil in your physics department runs a step-up of a couple thousand.

Mutual inductance is also the NFC in your phone's contactless-payment module: the reader's coil drives an AC field, your phone's coil picks up a few volts of induced EMF, and that is enough to power the chip and return a bit stream. Nothing touches. Magnetic-card readers, wireless earbuds' charging cases, implantable medical devices — every one of them is Φ_{21} = M · I_1 in industrial disguise.

Faraday's law was the first time ∂B/∂t entered the physics. Self- and mutual inductance are its working forms. In FIG.24 we ask where the work done against back-EMF goes — and find that the magnetic field itself is the reservoir.