Self-Inductance

Open a switch on a humming electromagnet and the contacts spit fire. The supply was five volts; the arc that jumps the gap needed hundreds. Nothing in electrostatics permits a circuit to out-volt its own battery — but this circuit was not asked. Its current threaded flux through its own windings, and Lenz's Law has already taught us what happens to any circuit whose flux is threatened: it fights. When the switch tried to kill the current in a millisecond, the coil summoned whatever EMF was required to object. This article gives that self-opposition its proper name, its unit, and its geometry. The name is self-inductance. Every circuit — deliberately wound or not — creates a magnetic field proportional to its own current, and therefore links flux proportional to its own current: $\Phi = LI$ . The proportionality constant $L$ is pure geometry, as characteristic of a loop's shape as capacitance is of a conductor's. Change the current and Faraday's law hands back an EMF $-L\,\mathrm{d}I/\mathrm{d}t$ : the circuit resists changes to its own current exactly as it resists changes to any flux. Inductance is Lenz's law institutionalized — electrical inertia, the electromagnetic twin of mass. Mastering $L$ buys three things at once. Computationally: a catalog of the workhorse geometries — solenoid, toroid, coaxial cable — each a two-line flux calculation. Physically: the reason currents ramp instead of jump, why interrupted inductors arc, and where the natural timescale $L/R$ of every magnetic circuit comes from. And structurally: $L$ is the handle by which the next two articles grab the chapter's deeper prizes — the coupling of separate circuits (Mutual Inductance) and the discovery that the work done against the back-EMF is banked in the field itself (Energy Stored in Magnetic Fields).

Key takeaways

Any circuit's own current threads flux through itself in proportion: $\Phi = LI$ . The constant $L$ — the self-inductance — depends only on geometry (and permeable materials), never on $I$ .
Faraday plus $\Phi = LI$ gives the back-EMF $\mathcal{E} = -L\,\mathrm{d}I/\mathrm{d}t$ : circuits oppose changes to their own current. Inductance is electrical inertia; the henry is its unit ( $1\ \mathrm{H} = 1\ \mathrm{V\,s/A}$ ).
The geometry catalog: long solenoid $L = \mu_0 n^2 \ell A$ (the workhorse); toroid $L = \frac{\mu_0 N^2 h}{2\pi}\ln(b/a)$ ; coaxial cable $L/\ell = \frac{\mu_0}{2\pi}\ln(b/a)$ — each from one flux integral.
Because current cannot jump (that would need infinite EMF), inductive circuits evolve on the timescale $\tau = L/R$ : currents ramp in and coast out exponentially.
Interrupt an inductor's current abruptly and $-L\,\mathrm{d}I/\mathrm{d}t$ spikes: the inductive kick that arcs switches and kills transistors — and that flyback diodes exist to tame.
KEY IDEA: self-inductance $L = \Phi/I$ is the geometric constant through which a circuit's own flux resists changes in its own current, with back-EMF $-L\,\mathrm{d}I/\mathrm{d}t$ — electrical inertia, unit henry.

The core insight

Self-inductance $L = \Phi/I$ is the geometry-fixed ratio of a circuit's own flux linkage to its own current; through Faraday's law it produces the back-EMF $\mathcal{E} = -L\,\mathrm{d}I/\mathrm{d}t$ , making every circuit resist changes to its own current like a mass resists acceleration.

Where this lives today

Inductance is the most manufactured magnetic quantity on Earth. Every switching power supply — one per phone charger, laptop brick, LED bulb, and server rail; billions per year — works by deliberately exploiting equation (2): a transistor chops current into an inductor tens of thousands to millions of times per second, and the inductor's insistence on continuous current is what smooths chopped voltage into clean DC. The flyback diode of worked example 2 (or a synchronous transistor playing its role) is soldered next to essentially every one of them. At the other end of the scale, a 1.5-tesla MRI magnet is a self-inductance of order 10–100 henries carrying hundreds of amperes: energized through its $L/R$ physics over tens of minutes, and protected — because $\tfrac12 LI^2$ there is megajoules — by quench-management systems whose entire design brief is this article's warning box at industrial voltage. The frontier entry is quantum. Superconducting qubits are LC oscillators in which the "L" is partly a Josephson junction's nonlinear kinetic inductance — nonlinearity being exactly what makes the oscillator's energy levels unevenly spaced and therefore addressable as a two-level system. Circuit designers at millikelvin temperatures now tune inductances the way this article's catalog tunes $n$ , $A$ , and $\ln(b/a)$ — the henry, defined by a nineteenth-century flux ratio, is a working unit of twenty-first-century quantum hardware.

Origins · 1 minJoseph Henry and how this was discovered

Joseph Henry

1797–1878 · Albany, Princeton, Smithsonian Institution

Henry is the American mirror of Faraday — and on self-induction, the man who got there first. An Albany schoolteacher who built the most powerful electromagnets of his era (a yard of his insulated winding could lift a ton), he noticed in 1832 the vivid spark that leapt whenever his own coil's circuit was broken: induction of a circuit on itself, published months before anyone else described it, in the same window in which Faraday (unknown to him, and first) found mutual induction. The complementarity is now fossilized in the terminology: Faraday's name on the law, Henry's on the unit — one volt-second per ampere, the henry, ratified in 1893. His later career built American science itself: first Secretary of the Smithsonian, organizer of the weather-telegraph network that became the Weather Service, and the arbiter who backed the telegraph's practicality (Morse leaned directly on Henry's relay — an application of this article's electromagnet physics). The kilovolt kick of worked example 2 is, historically, the first observation anyone made of self-inductance: Henry simply refused to stop staring at the spark.

History

1832

— Henry publishes self-induction: the spark at the breaking of his electromagnet's own circuit

1834

— Lenz's rule fixes the back-EMF's direction; self-induction becomes "electrical inertia"

1853

— Kelvin's circuit analysis puts

L

into the oscillation equation; the LC frequency emerges

1886

— Heaviside coins the very word "inductance" and the symbol-discipline of modern circuit theory

1893

— The Chicago Electrical Congress names the unit: one henry, one volt-second per ampere

1911

— Onnes discovers superconductivity: persistent currents make

\Phi = LI

a conservation law

The physical picture

A current loop is never just a current loop; it is also a magnet, and its own field loops back through its own aperture. Crank the current up and the self-flux rises with it; by Lenz's Law, the loop must oppose that rise — and since the rise is of its own making, the opposition lands on its own current. The loop drags its feet against its own change. Wind the wire into $N$ turns and the effect compounds twice over: each turn's field threads every other turn, so the flux linkage grows as $N \times N$ — the signature $N^2$ that dominates every entry in the inductance catalog.

Figure

A coil linking its own flux: raising the current summons a back-EMF that opposes the raise

The mechanical analogy is exact enough to compute with. Push a mass and it accelerates at a rate set by $m$ ; push a current through a coil and it ramps at a rate set by $L$ : the circuit equation $V = L\,\mathrm{d}I/\mathrm{d}t$ is Newton's second law with $V \leftrightarrow F$ , $I \leftrightarrow v$ , $L \leftrightarrow m$ . A big inductor is a flywheel: slow to spin up, and — the dangerous half of the analogy — slow to stop. Try to halt a flywheel instantly and something breaks; try to halt an inductor's current instantly and the voltage $-L\,\mathrm{d}I/\mathrm{d}t$ grows until something conducts — usually an arc across the opening switch. The analogy will complete itself in the next article, where the flywheel's kinetic energy $\tfrac12 mv^2$ finds its twin $\tfrac12 LI^2$ banked in the field. What kind of quantity is $L$ ? A ratio of flux to current — and since the field of any circuit is strictly proportional to the current producing it (Biot–Savart is linear in $I$ ), the ratio is a constant of the geometry: shape, size, turn count, and whatever permeable material threads the windings, nothing else. In that respect inductance is the perfect twin of capacitance, $C = Q/V$ : one constant says how much flux a shape self-links per ampere, the other how much charge a shape holds per volt. Both are computed by assuming the source, calculating the field, and taking one integral — and both reward the same instinct for symmetry.

Definition, back-EMF, and the inertia of current

The definition is the flux-to-current ratio:

\Phi = L\,I \qquad (1)

where: $\Phi$ — the total flux linkage of the circuit's own field through its own contour (for an $N$ -turn coil, the sum over all turns — $N$ times the flux through one turn, when they share it); $I$ — the circuit's current; $L$ — the self-inductance, in henries ( $1\ \mathrm{H} = 1\ \mathrm{Wb/A} = 1\ \mathrm{V\,s/A}$ ). The linearity deserves one honest caveat before we lean on it: it holds because vacuum (and any linear medium, per Linear and Nonlinear Magnetic Media) responds proportionally. Wrap the coil on iron and $L$ becomes a current-dependent effective quantity — the ferromagnetic fine print lives in the E5 chapter, and everything below is the vacuum/linear story. Differentiate equation (1) and hand the result to Faraday:

\mathcal{E} = -\frac{\mathrm{d}\Phi}{\mathrm{d}t} = -L\,\frac{\mathrm{d}I}{\mathrm{d}t} \qquad (2)

where: $\mathcal{E}$ — the self-induced (back-) EMF around the circuit. The minus sign is Lenz's, now pointed inward: rising current meets an EMF that fights the rise, falling current meets one that props it up. Note what the equation forbids — a discontinuous jump in $I$ would require infinite EMF, so the current through an inductor is a continuous function of time, always. That single sentence is the working content of inductance in every circuit ever built: currents ramp, coast, and ring; they never step. In circuit practice the same physics is written from the source's point of view. Driving current into the coil against its objection requires a terminal voltage (passive sign convention)

V_L = +L\,\frac{\mathrm{d}I}{\mathrm{d}t} \qquad (3)

where: $V_L$ — the voltage across the inductor, positive when the current is being forced upward. Pair the inductor with the resistance $R$ that every real circuit carries, and one first-order equation, $L\,\mathrm{d}I/\mathrm{d}t + IR = \mathcal{E}_0$ , governs the switch-on: the current climbs as $I(t) = (\mathcal{E}_0/R)\,(1 - e^{-t/\tau})$ with

\tau = \frac{L}{R} \qquad (4)

where: $\tau$ — the inductive time constant, the natural clock of every magnetic circuit (its decay twin $e^{-Rt/L}$ already appeared in Lenz's Law's stability argument). Big $L$ , small $R$ : stately ramps — a superconducting MRI magnet takes the better part of an hour to energize. The full transient toolkit (switching networks, AC impedance, resonance with capacitors) belongs to circuit theory proper, deferred to this track's circuits chapter; here the one formula earns its place as the physical meaning of $L$ made visible on an oscilloscope. There is also a field-theoretic form of the definition worth recording — flux as the loop integral of the vector potential (Stokes on $\mathbf{B} = \nabla\times\mathbf{A}$ , the machinery of Magnetic Vector Potential):

L = \frac{\Phi}{I} = \frac{1}{I}\oint_C \mathbf{A}\cdot\mathrm{d}\boldsymbol{\ell} \qquad (5)

where: $C$ — the circuit contour. This form quietly flags a subtlety the catalog geometries sidestep: for an idealized filamentary loop, the self-flux diverges logarithmically at the wire (the field grows as $1/s$ right up to a zero-radius filament), so honest single-loop inductances always reintroduce the wire's finite radius. Extended windings and distributed currents — our three catalog cases — are immune, which is one more reason they are the catalog. (The energy-integral route to $L$ , which handles such cases most gracefully, is Energy Stored in Magnetic Fields' machinery; the internal inductance $\mu_0/8\pi$ per unit length of a fat wire falls out there in two lines.)

Derivation AThe geometry catalog — solenoid and toroid by the flux method

The goal is the two bedrock inductances — the long solenoid and the toroidal coil — by the direct method that defines $L$ : assume the current, compute the field (both fields are E4 property, recalled by name), integrate the flux, count the linkages, divide by $I$ . The strategy showcases the $N^2$ anatomy: one factor of $N$ from the field a multi-turn winding produces, one from the turns that flux threads. Watch the toroid case sharpen the solenoid: its field varies across the cross-section, so the flux is an honest integral, and the answer remembers the geometry logarithmically.

Step 1. The long solenoid: $n$ turns per metre, length $\ell$ , cross-section $A$ , current $I$ . Its interior field is uniform and axial, $B = \mu_0 n I$ (Ampère's Law's flagship result), and exterior field negligible. Flux through one turn:

\Phi_1 = BA = \mu_0 n I A.

Step 2. Count the linkages. The winding has $N = n\ell$ turns, each threaded by $\Phi_1$ , so the total linkage is

\Phi = N\Phi_1 = (n\ell)(\mu_0 n I A) = \mu_0 n^2 \ell A\, I \qquad\Longrightarrow\qquad L = \mu_0 n^2 \ell A.

Why $n^2$ : one $n$ built the field, the other collected it. Every winding pays and collects in proportion to its turn density — double the turns in the same length and the inductance quadruples. Per unit volume, $L/(\ell A) = \mu_0 n^2$ : inductance is turn-density squared, times the vacuum's $\mu_0$ .

Step 3. The toroid: $N$ turns on a core of rectangular cross-section, inner radius $a$ , outer radius $b$ , height $h$ . Ampère's law on a circle of radius $s$ inside the core gives $B(s) = \mu_0 N I / 2\pi s$ — no longer uniform, so integrate the flux across the cross-section honestly:

\Phi_1 = \int_a^b \frac{\mu_0 N I}{2\pi s}\, h\,\mathrm{d}s = \frac{\mu_0 N I h}{2\pi}\ln\frac{b}{a}.

Step 4. Link all $N$ turns and divide:

L = \frac{N\Phi_1}{I} = \frac{\mu_0 N^2 h}{2\pi}\ln\frac{b}{a}.

The same $N^2$ , now with the geometry filed under a logarithm — the signature of every $1/s$ field integrated across an annulus, and a shape we are about to meet again in the coax. Limiting check: a thin toroid ( $b - a = w \ll a$ ) has $\ln(b/a) \approx w/a$ , giving $L \approx \mu_0 N^2 (hw)/2\pi a$ — exactly the solenoid formula with length $2\pi a$ and area $hw$ : a skinny toroid is a solenoid bent into a circle ✓. Verified: solenoid matches Griffiths Eq. 7.27 (§7.2.3) and the toroid his Ex. 7.11; the thin-toroid limit collapses onto the solenoid formula as required; worked example 1 recomputes the solenoid with numbers (n = 1000/m, ℓ = 20 cm, r = 1.5 cm: L = 0.178 mH) and the provenance addendum carries the toroid check (N = 500, a = 2 cm, b = 3 cm, h = 1 cm: L = 0.203 mH).

Figure

The catalog geometries: solenoid, toroid, coax — one flux integral each

Go deeper · Derivation B · 2 minThe coaxial cable — inductance per unit length by the flux method

Derivation BThe coaxial cable — inductance per unit length by the flux method

The goal is the third catalog entry, the coaxial cable — inner conductor radius $a$ , outer shell radius $b$ — by the same define-and-integrate route, chosen deliberately because the next article will recompute this exact geometry from stored energy and must agree. The subtlety worth watching: for a coax, "the circuit's aperture" is the radial gap between the conductors — flux is collected on a flat ribbon running the length of the cable, one edge on each conductor. (We compute the field-in-the-gap contribution — the external inductance, which dominates; the small extra piece stored inside the inner conductor's metal is the energy method's showcase, deferred by name to Energy Stored in Magnetic Fields.)

Step 1. The field in the gap. Ampère's law around a circle of radius $s$ , $a < s < b$ , encloses the inner conductor's full current:

B(s) = \frac{\mu_0 I}{2\pi s},

azimuthal, circling the inner conductor — E4's wire field, recalled.

Step 2. Choose the flux-collecting surface: a rectangle of length $\ell$ lying in a plane through the axis, spanning radially from $s = a$ to $s = b$ . The field is everywhere perpendicular to this ribbon (azimuthal field, radial-axial ribbon), so it is pure flux:

\Phi = \int_a^b \frac{\mu_0 I}{2\pi s}\,\ell\,\mathrm{d}s = \frac{\mu_0 I \ell}{2\pi}\ln\frac{b}{a}.

Why this ribbon is "the loop": the go current runs up the core, the return runs down the shell, and the circuit they form is closed by the caps; the ribbon is a surface bounded by that go-and-return contour. Any other spanning surface gives the same flux — $\nabla\cdot\mathbf{B} = 0$ guarantees it.

Step 3. Divide by $I$ and quote per unit length:

\frac{L}{\ell} = \frac{\mu_0}{2\pi}\ln\frac{b}{a}.

For the classic lab ratio $b/a = 7$ : $L/\ell = (2\times10^{-7})\ln 7 = 3.9\times10^{-7}\ \mathrm{H/m} \approx 0.4\ \mathrm{\mu H/m}$ — the number printed, in effect, on every metre of RG-58 on every bench.

Step 4. Read the structure. No $N^2$ — a coax is a one-turn device — but the same logarithm as the toroid, for the same reason: a $1/s$ field integrated across an annular gap. And note the pairing with the coax's capacitance per length, $C/\ell = 2\pi\varepsilon_0/\ln(b/a)$ (E1's result): the product is $LC/\ell^2 = \mu_0\varepsilon_0$ — the geometry cancels entirely, a first quiet hint of the constant $1/\sqrt{\mu_0\varepsilon_0}$ whose identity Maxwell's Equations in Vacuum will reveal with full ceremony. Verified: matches Griffiths Ex. 7.13 route (his via energy — the cross-check the next article completes) and Jackson §5.17 register; the b/a = 7 evaluation recomputed: (μ0/2π)ln 7 = 2×10⁻⁷ × 1.9459 = 3.89×10⁻⁷ H/m; the LC product check: (μ0/2π)ln(b/a) × 2πε0/ln(b/a) = μ0ε0 identically.

Limiting cases

Case	Result	Meaning
solenoid, turns doubled at fixed length	$L \to 4L$	the $N^2$ anatomy: build the field, then collect it
thin toroid, $b-a \ll a$	$L \to \mu_0 N^2 A_{\rm cs}/2\pi a$	a solenoid bent into a circle — catalog self-consistency
coax, $b \to a$	$L/\ell \to 0$	no gap, no flux ribbon: go and return currents cancel
filamentary loop, wire radius $\to 0$	$L \to \infty$ (log)	self-flux diverges at the wire; real inductances remember the wire's thickness
DC steady state	$V_L = 0$	an ideal inductor is a plain wire once nothing changes
switch opened in time $\Delta t \to 0$	$\|\mathcal{E}\| = L I/\Delta t \to \infty$	the inductive kick: something will conduct — design the path or the arc picks one
$R \to 0$ with source removed	$I$ persists, $\Phi$ frozen	the superconducting loop of Lenz's Law, now with its constant named: $\Phi = LI$ conserved

Common trapAn inductor's current cannot jump — and it will out-volt any switch that pretends otherwise

The misconception: "the supply is 5 V, so nothing in this circuit can exceed 5 V." False the moment an inductor's current is interrupted. The back-EMF is $-L\,\mathrm{d}I/\mathrm{d}t$ , and $\mathrm{d}I/\mathrm{d}t$ at an opening switch is set by how fast the contacts separate, not by the supply: break 2 A in a millisecond through half a henry and the coil generates a kilovolt (worked example 2), enough to arc the gap and burn the contacts — or destroy the transistor doing the switching. The current will keep flowing somewhere for at least an instant; your only choice is where. Hence the flyback (freewheeling) diode across every relay coil and motor winding: it offers the current a polite exit ramp, converting the would-be kilovolt spike into a gentle decay at one diode-drop. The design rule in one line: never open an inductive circuit without answering, in advance, the question where does the current go next?

Worked examples

1 · A bench solenoid, from geometry to back-EMF (confidence builder)

A solenoid has $n = 1000$ turns per metre, length $\ell = 20\ \mathrm{cm}$ , and radius $r = 1.5\ \mathrm{cm}$ . Find its inductance, and the back-EMF when its current is ramped from 0 to 2.0 A in 5.0 ms.

Step 1. Cross-section: $A = \pi r^2 = \pi(0.015)^2 = 7.1\times10^{-4}\ \mathrm{m^2}$ .

Step 2. Catalog formula: $L = \mu_0 n^2 \ell A = (1.26\times10^{-6})(10^6)(0.20)(7.1\times10^{-4}) = \mathbf{0.18\ mH}$ . (Sanity: $N = n\ell = 200$ turns — a modest lab coil, and a modest fraction of a millihenry to match.)

Step 3. The ramp: $\mathrm{d}I/\mathrm{d}t = 2.0/5.0\times10^{-3} = 400\ \mathrm{A/s}$ , so $|\mathcal{E}| = L\,\mathrm{d}I/\mathrm{d}t = 1.78\times10^{-4} \times 400 = \mathbf{71\ mV}$ , polarity opposing the rise. Gentle — because the ramp is gentle. Hold that thought for example 2.

Step 4. Scale awareness: wind the same wire density on the same form but double the turns per metre ( $n = 2000$ ): $L$ quadruples to 0.71 mH. The $N^2$ is the fastest lever in magnetics — and the reason "more turns" is the first answer to almost every inductance problem.

2 · The inductive kick, and the diode that tames it (exam level)

A relay coil has $L = 0.50\ \mathrm{H}$ and winding resistance $R_{\rm coil} = 100\ \Omega$ , carrying $I_0 = 2.0\ \mathrm{A}$ . The switch opens, separating its contacts in about $\Delta t = 1.0\ \mathrm{ms}$ . (a) Estimate the voltage the coil generates with no protection. (b) A flyback diode is fitted across the coil; describe the current decay and find the energy the coil dissipates.

Step 1. Unprotected: the switch demands $\mathrm{d}I/\mathrm{d}t \approx -I_0/\Delta t = -2.0\times10^{3}\ \mathrm{A/s}$ , so the coil answers with

|\mathcal{E}| = L\left|\frac{\mathrm{d}I}{\mathrm{d}t}\right| = 0.50 \times 2.0\times10^{3} = \mathbf{1.0\ kV}

— from a coil that may have been running on 12 V. In practice the voltage climbs only until the contact gap ionizes; the "1 kV" becomes an arc, the arc sustains the current (Lenz satisfied), and the contacts erode. Faster switches make it worse: halve $\Delta t$ , double the spike.

Step 2. Protected: the instant the switch opens, the coil's collapsing-flux EMF forward-biases the flyback diode, which closes the loop across the coil. The current transfers to the diode path without discontinuity — $I$ through the inductor is continuous, as equation (2) demands — and then decays through the coil's own resistance:

I(t) = I_0\,e^{-t/\tau}, \qquad \tau = \frac{L}{R_{\rm coil}} = \frac{0.50}{100} = \mathbf{5.0\ ms}.

Step 3. The voltage across the coil during decay is one diode drop ( $\approx 0.7$ V) — three orders of magnitude below the unprotected spike. The cost is a slower release: the relay holds for a few $\tau$ (~15–25 ms) after switch-off, a real design trade in fast actuators.

Step 4. The energy audit. The coil entered the transient holding magnetic energy $W = \tfrac12 L I_0^2 = 0.5 \times 0.50 \times 4.0 = \mathbf{1.0\ J}$ (the formula the next article derives from exactly this kind of ramp-down bookkeeping — quoted here on credit from Energy Stored in Magnetic Fields). Every joule of it ends as heat in $R_{\rm coil}$ (and a trickle in the diode): $\int_0^\infty I^2 R\,\mathrm{d}t = \tfrac12 L I_0^2$ ✓. Where that joule was, microseconds earlier — in the metal? in the space around it? — is precisely the next article's question.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.2.3 · Jackson, Classical Electrodynamics, 3rd ed., §5.17 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 7 · Zangwill, Modern Electrodynamics, §14.5.

Provenance. Equations cross-checked against Griffiths 4th ed. §7.2.3 (solenoid $L=\mu_0 n^2\ell A$ , Eq. 7.27; energy $\tfrac12 LI^2$ ), Jackson 3rd ed. §5.17 (coax and Neumann energy formula), and Purcell 3rd ed. Ch. 7 (internal inductance $\mu_0/8\pi$ ). A verifier should recompute: (1) the solenoid flux-linkage chain $\Phi=N\Phi_1$ ; (2) the coax energy integral $\int_a^b \mathrm{d}s/s=\ln(b/a)$ ; (3) the internal-inductance integral $\int_0^R s^3\,\mathrm{d}s=R^4/4$ canceling the $1/R^4$ ; and (4) the constant $\mu_0/8\pi\approx5.0\times10^{-8}\,\mathrm{H/m}$ . Vacuum permeability $\mu_0=4\pi\times10^{-7}\,\mathrm{H/m}$ (pre-2019 exact value, now $\approx$ to that precision). Addendum for this edition: the coax is computed here by the flux method (energy method re-scoped by name to Energy Stored in Magnetic Fields, which re-derives it as a cross-check, along with internal inductance); worked numbers recomputed — solenoid ( $n$ = 1000/m, $\ell$ = 20 cm, $r$ = 1.5 cm): $L$ = 0.178 mH, back-EMF at 400 A/s = 71 mV; toroid ( $N$ = 500, $a$ = 2 cm, $b$ = 3 cm, $h$ = 1 cm): $L = \frac{\mu_0 N^2 h}{2\pi}\ln(b/a)$ = 0.203 mH; coax $b/a$ = 7: 0.39 μH/m; kick ( $L$ = 0.50 H, 2.0 A, 1 ms): 1.0 kV, protected decay $\tau$ = 5.0 ms, energy 1.0 J.

Builds on: Faraday's Law of Induction · Lenz's Law · Electromotive Force

Leads to: Mutual Inductance · Energy Stored in Magnetic Fields · The Displacement Current

Is inductance really a property of the circuit alone, if the flux depends on the field filling all space?

Yes — because the field filling all space is itself dictated by the circuit. Biot–Savart makes $\mathbf{B}$ everywhere strictly proportional to $I$ with a purely geometric kernel, so the self-flux is (geometry) × $I$ , and the ratio $L$ carries only the geometry. The proportionality survives any linear magnetic medium (multiply by $\mu/\mu_0$ ) but genuinely fails on iron: there $\mathbf{B}$ is not proportional to $I$ , and "the inductance" of an iron-core coil is a current-dependent effective slope — the honest treatment lives in the magnetism-in-matter chapter (Linear and Nonlinear Magnetic Media, Hysteresis). Air-core formulas like this article's catalog are exact; iron-core numbers on datasheets always carry an operating point.

Why does the inductance of a single loop of thin wire come out infinite in the formulas?

Because a truly filamentary current is a fiction that the self-field punishes. The field near a wire grows as $1/s$ , so the flux integral $\int B\,\mathrm{d}a$ collecting flux right down to a zero-radius filament diverges logarithmically. Real wires have radius $a_w$ , the integration stops at the metal, and single-loop inductances go as $\mu_0 R[\ln(8R/a_w) - 2]$ — finite, but forever remembering the wire's thickness inside a logarithm. This is why the catalog leads with solenoids, toroids, and coaxes: distributed currents and well-defined flux apertures, no divergence, no wire-radius fine print at leading order. Moral: whenever a self-interaction integral diverges, the cure is the finite size of the source — a lesson that returns, with higher stakes, for the electron's self-energy in the radiation chapter.

If the back-EMF opposes the current change, how does the current ever increase at all?

The back-EMF opposes; it does not veto. At every instant the circuit obeys $\mathcal{E}_0 - L\,\mathrm{d}I/\mathrm{d}t - IR = 0$ : the source pushes, the inductor pushes back exactly $L\,\mathrm{d}I/\mathrm{d}t$ , and the current rises at precisely the rate that balances the ledger, $\mathrm{d}I/\mathrm{d}t = (\mathcal{E}_0 - IR)/L$ . Think of pushing a heavy cart: inertia opposes acceleration but the cart still accelerates, at $F/m$ . The opposition is what makes the rise finite and orderly — and the work done pushing against it is not lost; it is banked, joule for joule, as the $\tfrac12 LI^2$ that Energy Stored in Magnetic Fields locates in the field.

Why does breaking an inductive circuit make a spark, but connecting it doesn't?

Asymmetry of what is being demanded. At switch-on, the circuit is asked to raise current from zero — the inductor answers with a back-EMF capped by the supply voltage, and the current simply ramps at $\mathcal{E}_0/L$ ; nothing exceeds the supply anywhere. At switch-off, the circuit is asked to destroy current in the contact-opening time — but current through an inductor cannot jump, so the coil generates whatever EMF is needed to keep it flowing, and "whatever is needed" across a widening air gap is kilovolts within microseconds: the air ionizes and the spark is the current continuing. Henry saw it in 1832; every unprotected relay driver rediscovers it. The flyback diode removes the asymmetry by giving switch-off a lawful current path.

Do straight wires and PCB traces have inductance, or only coils?

Everything that carries current has inductance — a coil merely concentrates it. A straight wire runs to roughly a microhenry per metre (with the usual logarithm of its length-to-radius ratio); a PCB trace above a ground plane, nanohenries per centimetre. At DC and audio frequencies these are negligible; at modern switching speeds they run the show: 10 nH fed with 1 A per 10 ns generates a full volt of $L\,\mathrm{d}I/\mathrm{d}t$ — the "ground bounce" and supply-rail ringing of high-speed electronics. Whole subfields (power-distribution-network design, decoupling-capacitor placement) exist to manage the inductance of wiring nobody meant to be an inductor. The rule of thumb every layout engineer carries: at nanosecond edges, every conductor is a coil.

What exactly is the henry, dimensionally, and why is it such a huge unit in practice?

One henry is one weber per ampere, equivalently one volt-second per ampere: a 1 H inductor develops 1 V across it when its current changes at 1 A/s. It is "huge" for air-core coils because $\mu_0$ is small: the catalog's bench solenoid manages 0.18 mH with 200 turns, and getting to a full henry in air takes serious wire. Iron multiplies flux thousands-fold, which is why relay coils and chokes reach henries at pocket size — at the price of the nonlinearity fine print. Typical working scales: nanohenries (traces, interconnects), microhenries (RF and power-supply inductors), millihenries (audio chokes, air solenoids), henries (iron-core relays, transformers' magnetizing inductance, MRI magnets).

Is there a deeper reason inductance behaves like mass, or is the analogy a coincidence?

Deeper reason. In the Lagrangian formulation of circuits, charge $q$ plays the coordinate, current $\dot q$ the velocity — and the magnetic energy $\tfrac12 L\dot q^2$ sits in the Lagrangian precisely where kinetic energy $\tfrac12 m\dot x^2$ does. Inductance is the mass of the charge coordinate, exactly, and everything follows: back-EMF as inertial reaction, $L/R$ as the viscous relaxation time (matching the mechanics chapter's $m/b$ ), and LC oscillation as the mass-on-spring with $1/C$ the stiffness. The correspondence is load-bearing in modern physics: superconducting-circuit quantization proceeds by writing exactly this Lagrangian and promoting $q$ and $L\dot q$ to operators — the "electrical inertia" metaphor, taken literally, is how qubits are engineered.