RL Circuits

Open a switch. It is the most innocent act in circuit theory — and if the circuit contains a coil, it can answer with a kilovolt. The fat blue spark that jumps a switch's contacts when you cut power to a motor, a relay, or an old doorbell is not a malfunction; it is an inductor keeping a promise. A capacitor, you now know, refuses to change its voltage in an instant. An inductor makes the mirror-image vow about its current — and unlike the capacitor's gentle stubbornness, the inductor enforces its vow with whatever voltage it takes. This article is the second panel of the transient-circuits diptych. The previous article tamed the RC circuit: a first-order differential equation, an exponential solution, a time constant. Here the same mathematics returns with the roles swapped — the clock is now $\tau = L/R$ , the protected quantity is current, and the resistor's position in the formula flips from numerator to denominator. If you internalized equation (4) of the RC article, you already know every formula here; what is genuinely new is the physics of the swap, the beautiful two-instinct shortcut (an inductor is an open circuit at the instant of switching and a plain wire forever after), and the interrupted-inductor drama — the L di/dt spike that fires your car's spark plugs forty times a second and destroys an unprotected transistor on the first try. By the end you will hold the full duality between the two first-order circuits, a table worth more than a dozen memorized formulas — and the stage will be set for what happens when L and C finally meet in one loop.

Key takeaways

An inductor's current cannot jump: changing it requires $v_L = L\,di/dt$ , and an instantaneous change would demand infinite voltage. Current is to an inductor what voltage is to a capacitor.
Closing a switch on a series EMF–R–L loop: $i(t) = \dfrac{\varepsilon}{R}\left(1 - e^{-t/\tau}\right)$ with time constant $\tau = L/R$ — 63% of final current after one $\tau$ , done by $5\tau$ .
Switching instincts: at $t = 0^+$ an inductor is an open circuit (holds its old current, initially zero); as $t \to \infty$ it is a plain wire (steady current, zero voltage). Exactly opposite to the capacitor.
Interrupting an inductor's current forces $v = L\,di/dt$ as large as the interruption is fast: the switch-break spike — 1 A through 1 H broken in 1 ms is a thousand volts. Ignition coils exploit it; flyback diodes tame it.
Decay through a resistor dissipates $\int i^2R\,dt = \tfrac12 LI_0^2$ exactly — the stored magnetic energy, with $R$ setting the pace, never the total.
Every formula is the RC article's under the swap $v \leftrightarrow i$ , $C \leftrightarrow L$ , $\tau = RC \leftrightarrow \tau = L/R$ : one duality, two circuits.
Key idea: an inductor cannot change its current instantly — so RL circuits relax exponentially on the clock $\tau = L/R$ , and interrupting them abruptly converts stored current into violent voltage.

The core insight

Current through an inductor in a resistive circuit cannot jump; it relaxes exponentially toward its steady value with time constant $\tau = L/R$ , and any attempt to interrupt it abruptly is answered by a voltage spike $L\,di/dt$ as large as the interruption is fast.

Where this lives today

Power electronics is applied RL transients. Every switching converter — the bricks that charge laptops, the regulators on motherboards, the drivetrain inverters in electric cars — works by deliberately energizing an inductor for microseconds and then redirecting its unstoppable current somewhere useful, tens of thousands to millions of times per second; the flyback diode of this article, upgraded to a synchronous transistor, is the central component of the trade. The same $L\,di/dt$ physics polices the unwanted inductances: the few nanohenries of a chip's power-supply pins turn sudden processor current demands into voltage droop ( $v = L\,di/dt$ with $di/dt$ in amps per nanosecond), which is why circuit boards are carpeted with decoupling capacitors standing guard beside every chip. Electric-vehicle contactors and grid DC breakers face the drama at scale — interrupting hundreds of amps of inductive current without sustaining a plasma arc is a genuinely hard, actively researched problem (vacuum interrupters, magnetic arc blowout, semiconductor breakers). And at the bottom of the temperature scale the decay law completes its arc: an MRI magnet is an RL loop with superconducting $R \approx 0$ , $\tau = L/R$ of years — its hundreds of amps circulate unattended until a quench suddenly restores a resistance, at which point $\tfrac12LI^2$ (megajoules) exits as heat on a schedule this article lets you compute.

The physical picture

The capacitor of the last article was a tyre being pumped full; the inductor is a flywheel. Set a heavy wheel spinning and it neither starts nor stops willingly: torque applied gradually spins it up along a smooth curve, and grabbing it to an instant halt is simply not available — something will break first. The Self-Inductance article of the Maxwell chapter built the reason: a coil's current threads flux through its own windings, and any change in that current induces a back-EMF $L\,di/dt$ that opposes the change (Faraday's law wearing Lenz's minus sign). Current in an inductor carries inertia, with $L$ playing the role of mass — the same analogy, made exact, that maps this whole circuit onto the mechanics track's damped systems, as the next article will exploit in full.

Figure

The RL rise: current climbs like a flywheel spinning up, while the inductor's opposing voltage dies away — mirror twins of the RC curves

The two switching instincts, stated once and used forever, are the capacitor's instincts with the labels swapped. At the instant of switching, an inductor is an open circuit — more precisely, a current source frozen at whatever current it carried the moment before. Starting from rest that current is zero, so at $t=0^+$ no current flows anywhere the inductor sits in series, however loudly the battery insists. After a long time, an ideal inductor is a plain wire: in steady state $di/dt = 0$ , so $v_L = 0$ , and the coil contributes nothing but its (idealized away) winding resistance. Wire at the end, open at the start — precisely where the capacitor is open at the end and a wire at the start. Every $t=0^+$ and $t\to\infty$ analysis in this chapter reduces to bookkeeping with these four instincts. One more picture before the mathematics: where the danger lives. A capacitor left alone holds its energy quietly as separated charge. An inductor's energy $\tfrac12 Li^2$ exists only while current flows — it is energy of motion, like the flywheel's. You can walk away from a charged capacitor; you cannot walk away from an energized inductor without providing its current a path. That asymmetry is the entire story of the switch-break spike.

The rise, the decay, and the clock

Close the switch on the series loop — EMF $\varepsilon$ , resistor $R$ , ideal inductor $L$ . Kirchhoff's loop rule (this chapter's rules article), with the inductor's terminal relation from the Self-Inductance article, reads

\varepsilon - iR - L\frac{di}{dt} = 0 \qquad (1)

where: $L\,di/dt$ — the back-EMF, entering with the sign of opposition (Lenz), so it subtracts while the current grows. Structurally this is the RC loop equation with $q/C$ replaced by the term that resists change of current rather than accumulated charge. The natural clock of the equation is the only time you can build from $L$ and $R$ :

\tau = \frac{L}{R} \qquad (2)

where: henries over ohms are seconds ( $\text{H}/\Omega = \tfrac{\text{V}\cdot\text{s}}{\text{A}}\cdot\tfrac{\text{A}}{\text{V}} = \text{s}$ ). Note the flip: resistance now sits in the denominator. A larger $R$ makes an RL circuit faster — it brings the final current $\varepsilon/R$ down toward what the inductor already carries — while it made the RC circuit slower. This inversion is the most commonly fumbled fact in the pair, and the duality table below is the cure.

Derivation ACurrent rise in the RL loop

We want $i(t)$ from equation (1) with $i(0) = 0$ , and we want it by the same separation-of-variables route as the RC charging derivation — partly for the answer, mostly to expose that it is the same equation, so that the solution transfers rather than multiplies. Watch the shortfall structure reappear: the rate of change of current is proportional to how far the current still is from its destination. The one physical novelty to watch: what the inductor's voltage does at $t = 0^+$ , which is the seed of the interrupted-inductor drama.

Step 1. Rearrange (1) around the final current $I_f = \varepsilon/R$ (the plain-wire steady state):

\frac{di}{dt} = \frac{R}{L}\left(I_f - i\right) = \frac{I_f - i}{\tau}

Why is $I_f = \varepsilon/R$ ? In steady state $di/dt = 0$ , the back-EMF dies, and the loop is just battery and resistor — the inductor has become a wire. The equation says: rate of approach proportional to remaining distance, the self-throttling signature.

Step 2. Separate and integrate from $(0, 0)$ to $(t, i)$ :

\int_0^i \frac{di'}{I_f - i'} = \int_0^t \frac{dt'}{\tau} \quad\Longrightarrow\quad -\ln\frac{I_f - i}{I_f} = \frac{t}{\tau}

Step 3. Exponentiate and solve:

i(t) = \frac{\varepsilon}{R}\left(1 - e^{-t/\tau}\right), \qquad \tau = \frac{L}{R}

The 63–86–95–99.3% ladder at $\tau, 2\tau, 3\tau, 5\tau$ carries over verbatim from the RC article — it is a property of the exponential, not of the components.

Step 4. The inductor's voltage, by differentiating: $v_L = L\,di/dt = \varepsilon\,e^{-t/\tau}$ . At $t = 0^+$ the entire battery voltage sits across the coil — the current is zero, so the resistor drops nothing, and the inductor momentarily behaves as the open circuit of our instinct. The current may not jump, but the inductor's voltage jumped from 0 to $\varepsilon$ the moment the switch closed. Why do voltage and current trade the right to jump? Each storage element protects the quantity its energy depends on: $\tfrac12Cv^2$ protects $v$ ; $\tfrac12Li^2$ protects $i$ . The other variable is unprotected and moves as abruptly as the topology demands. This single sentence generates all four switching instincts. Verified: substitution back into (1) is an identity; at $t=\tau$ , $i/I_f = 1 - e^{-1} = 0.632$ ; numerical integration of $di/dt = (I_f - i)/\tau$ (relay coil: $L = 0.5$ H, $R = 100\ \Omega$ , $\tau = 5.0$ ms, $\varepsilon = 12$ V) reproduces the closed form to five figures, with $i$ reaching 95% of 120 mA at $t = 15$ ms $= 3\tau$ .

The decay problem is the mirror: a coil carrying $I_0$ is switched away from its source and left to spend its current through a resistor. The loop rule gives $L\,di/dt + iR = 0$ , hence $i(t) = I_0\,e^{-t/\tau}$ — and the general first-order template of the RC article,

x(t) = x_\infty + \left(x_0 - x_\infty\right)e^{-t/\tau} \qquad (3)

where: $x_0$ fixed by "inductor current cannot jump," $x_\infty$ by "inductor = wire," $\tau = L/R_{\text{Th}}$ with $R_{\text{Th}}$ the Thévenin resistance the inductor sees (the Series and Parallel Networks collapse, exactly as for the capacitor) — covers every one-inductor circuit ever set on an exam. Same template, swapped instincts, inverted clock.

The interrupted inductor: sparks, coils, and the diode that saves the transistor

Now the drama. Decay through a resistor is gentle. But what if you simply open the switch — removing every path? The current must go from $I_0$ to zero in the switch's opening time $\Delta t$ , and the inductor answers with

v_L = L\frac{di}{dt} \approx L\frac{I_0}{\Delta t} \qquad (4)

where: $\Delta t$ — however fast the contacts part. The inductor does not negotiate: one ampere through one henry interrupted in one millisecond is a thousand volts, and a faster break is proportionally worse. Physically, the collapsing flux must induce this EMF (Faraday's law is not optional), and the voltage rises until it finds a path — usually by ionizing the air gap at the parting contacts. That blue flash is the inductor completing its own circuit through a plasma of its own making.

Go deeper · Derivation B · 2 minThe switch-break spike and the energy audit of decay

Derivation BThe switch-break spike and the energy audit of decay

We want two quantitative handles on the interruption drama: the spike voltage for a real device (a car's ignition coil, where the spike is the product), and the proof that a resistive decay path dissipates exactly the stored magnetic energy $\tfrac12LI_0^2$ — the RL twin of the RC article's half-theorem, with one honest difference worth exposing. The strategy: estimate the spike from (4) with real numbers, then integrate Joule heating over the decay exponential. Watch $R$ cancel from the energy total exactly as it did for the capacitor.

Step 1. Ignition coil, primary side: $L \approx 8$ mH carrying $I_0 = 4$ A, interrupted in $\Delta t \approx 0.1$ ms by the switching transistor (in Kettering's day, the mechanical "points"). Equation (4):

v_{\text{primary}} \approx \frac{(8\times10^{-3})(4)}{10^{-4}} = 320\ \text{V}

Step 2. The coil is also a transformer (the Mutual Inductance article's physics; the Transformers article completes it) with a ~100:1 secondary winding, multiplying the spike to $\sim 32$ kV — comfortably above the ~10–20 kV needed to ionize a millimetre spark-plug gap under compression. Your engine fires on equation (4), thousands of times a minute, for years. Why doesn't the primary circuit's battery voltage (12 V) matter? The spike comes from the collapse rate, not the source: during interruption the inductor is the only EMF that counts, and it manufactures whatever voltage the demanded $di/dt$ requires. The battery merely set up $I_0$ beforehand.

Step 3. Energy audit for decay through a resistor: with $i = I_0e^{-t/\tau}$ ,

E_{\text{heat}} = \int_0^\infty i^2R\,dt = I_0^2R\cdot\frac{\tau}{2} = I_0^2R\cdot\frac{L}{2R} = \tfrac12 L I_0^2

— the full magnetic stockpile, independent of $R$ . (For the field-side derivation of $\tfrac12LI^2$ itself, this chapter defers by name to Energy Stored in Magnetic Fields, which built it from the field's energy density; here we have just watched the same joules exit through the resistor, a satisfying closed loop between the field picture and the circuit picture.)

Step 4. One honest asymmetry with the RC case: charging a capacitor wasted an unavoidable extra half in the resistor. Energizing an inductor from a battery has no such theorem — in the idealized $R\to0$ limit the inductor can be spun up losslessly (which is exactly why switching power supplies energize inductors, not resistively-charged capacitors, to move energy efficiently). With a real series $R$ , heat flows for as long as the current does, growing without bound in steady state; the "half" was a capacitor-specific ledger, not a universal toll. Verified: spike arithmetic $8\,\text{mH}\times4\,\text{A}/0.1\,\text{ms} = 320$ V and $\times100 \Rightarrow 32$ kV; decay integral $\int_0^\infty e^{-2t/\tau}dt = \tau/2$ gives $\tfrac12LI_0^2 = 0.064$ J for the coil's 8 mH at 4 A; numerical check of the energy integral for $R$ spanning $1\ \Omega$ – $1\ \text{k}\Omega$ returns identical totals to machine precision.

Engineering tames the spike with a flyback diode (also freewheeling diode): a diode placed across the coil, oriented to be dead while the source drives current, but forward-biased the instant the inductor's voltage reverses at switch-off. The collapsing current then freewheels harmlessly around the coil–diode loop, decaying with $\tau = L/R_{\text{coil}}$ while the voltage across the switch never exceeds the supply plus one diode drop. Every relay, motor driver, and solenoid valve on a circuit board carries one; forgetting it is a rite of passage that costs one transistor. (The tradeoff is honest: clamping the voltage means the relay releases more slowly — the energy must still drain, and a gentler voltage takes longer. Fast-release designs clamp at a higher voltage with a Zener instead. Pace versus peak, never neither.)

Figure

Opening the switch: without a diode the coil manufactures kilovolts across the gap; with a flyback diode the current freewheels and the switch sees one diode drop

The duality, laid on the table

Every result of this article and the last is one theorem viewed twice. The dictionary:

	RC circuit	RL circuit
protected quantity (cannot jump)	capacitor voltage $v_C$	inductor current $i_L$
free-to-jump quantity	current	voltage
stored energy	$\tfrac12Cv^2$ (charge at rest)	$\tfrac12Li^2$ (charge in motion)
time constant	$\tau = R_{\text{Th}}C$	$\tau = L/R_{\text{Th}}$
bigger $R$ means	slower	faster
at $t=0^+$ element acts as	wire (if uncharged)	open circuit (if unenergized)
as $t\to\infty$ element acts as	open circuit	wire
danger scenario	short a charged C: current surge	open an energized L: voltage spike
universal solution	$x_\infty + (x_0 - x_\infty)e^{-t/\tau}$	the very same equation (3)

Read the last row twice: there is one first-order transient theory, not two. The deep reason — that $(q, v, C)$ and $(\Phi_B/L\cdot L = \text{flux}, i, L)$ play formally identical roles under the exchange of voltage and current — will graduate into the full mechanical analogy in the next article, where $L$ becomes mass, $1/C$ becomes stiffness, and the two protected quantities become position and momentum.

Limiting cases

case	result	meaning
$t \ll \tau$	$i \approx (\varepsilon/L)\,t$ : linear ramp	back-EMF ≈ ε; the inductor alone sets the initial slope
$t = \tau$	63.2% of $\varepsilon/R$ ; $v_L$ down to 36.8%	the universal landmark, RL edition
$t \gtrsim 5\tau$	steady: $i = \varepsilon/R$ , $v_L = 0$	inductor = wire
$R \to \infty$ (opened switch)	$v_L = L\,I_0/\Delta t \to$ spike	the interruption drama, eq. (4)
$R \to 0$ decay loop	$\tau \to \infty$ : current persists	superconducting loops carry current for years
$L \to 0$	$\tau \to 0$ : plain resistor circuit	no inertia, no story
network, one L	$\tau = L/R_{\text{Th}}$ , eq. (3)	Thévenin collapse, exactly as for C

Worked examples

1 · Relay coil: on, and safely off (confidence builder)

A relay coil has $L = 0.5$ H and winding resistance $R = 100\ \Omega$ , driven by 12 V. (a) Find the time constant, final current, and time to reach the 80 mA pull-in threshold. (b) A flyback diode (drop 0.7 V) is fitted; estimate the release decay clock.

Step 1. Clock and destination: $\tau = L/R = 0.5/100 = 5.0$ ms; $I_f = 12/100 = 120$ mA.

Step 2. Pull-in time: solve $120\,(1 - e^{-t/\tau}) = 80$ mA, i.e. $e^{-t/\tau} = 1/3$ : $t = \tau\ln 3 = 5.0 \times 1.099 = 5.5$ ms. The relay clicks about one time constant after the drive.

Step 3. Switch-off with the diode: the current freewheels through the coil's own $100\ \Omega$ , so the decay clock is again $\tau = 5.0$ ms — the armature releases within ~2–3 time constants (10–15 ms), and the driving transistor sees only $12 + 0.7 = 12.7$ V.

Step 4. Without the diode: parting contacts in $\sim 10\ \mu$ s would demand $v \approx L I_0/\Delta t = (0.5)(0.12)/(10^{-5}) = 6$ kV — the arithmetic of a destroyed transistor.

Answer: $\tau = 5.0$ ms, $I_f = 120$ mA, pull-in at 5.5 ms; with the diode the release rides the same 5 ms clock at 12.7 V, instead of a 6 kV spike.

2 · Two resistors, one inductor: different clocks for rise and decay (exam level)

A 12 V battery with $R_1 = 4\ \Omega$ in series feeds node A through a switch. From node A, an ideal inductor $L = 6$ mH goes to ground, and $R_2 = 12\ \Omega$ also connects A to ground (in parallel with the inductor). The switch closes at $t = 0$ and, after steady state, opens again. Find the rise clock, the steady currents, the decay clock, and the voltage across $R_2$ just after opening.

Step 1. Rise clock: the inductor sees the Thévenin resistance of everything else — battery quenched, $R_1$ and $R_2$ in parallel from node A: $R_{\text{Th}} = (4)(12)/16 = 3\ \Omega$ , so $\tau_{\text{rise}} = L/R_{\text{Th}} = 6\times10^{-3}/3 = 2.0$ ms.

Step 2. Steady state: the inductor is a wire, clamping node A to ground. All current flows through $R_1$ into it: $I_L(\infty) = 12/4 = 3.0$ A, while $R_2$ (zero volts across it) carries nothing. Assemble by (3): $i_L(t) = 3.0\,(1 - e^{-t/2.0\,\text{ms}})$ A.

Step 3. Switch opens: $R_1$ is disconnected; the inductor's 3.0 A cannot jump and now must flow through $R_2$ alone. New clock: $\tau_{\text{decay}} = L/R_2 = 6\times10^{-3}/12 = 0.5$ ms — four times faster than the rise, because the freewheel path is more resistive than the Thévenin drive was.

Step 4. Voltage across $R_2$ at $t = 0^+$ of the decay: $v = I_L R_2 = (3.0)(12) = 36$ V — three times the battery that built the current, with the node's polarity flipped (the inductor now pushes current up through $R_2$ ). A "12-volt circuit" just presented 36 V to anything attached at node A; equation (4) in miniature, tamed to a finite value by the resistor path.

Answer: $\tau_{\text{rise}} = 2.0$ ms to $I_L = 3.0$ A; $\tau_{\text{decay}} = 0.5$ ms; at switch-off, $R_2$ momentarily carries 36 V. Rise and decay run on different clocks because the inductor sees different resistances.

Origins · 1 minJoseph Henry and how this was discovered

Joseph Henry

1797–1878 · Albany, Princeton, the Smithsonian

The unit of inductance honours the Albany schoolteacher who — as the Self-Inductance article recounts — caught the self-induction spark in 1832. This article meets Henry in his other role: the great engineer of the inductive transient. To ring a bell through a mile of wire he invented what was in effect the electromagnetic relay (1835), the device whose coil-plus-switch physics is exactly this article's worked example — and whose cascading action (a weak current switching a stronger one) made long-distance telegraphy possible; Morse leaned directly on Henry's counsel and never adequately credited him. Henry measured the "extra current" at break by the ferocity of the spark, understood that winding geometry set its strength, and built graded coils for maximal effect — engineering the L di/dt spike a century before anyone wrote a flyback-diode datasheet. As founding Secretary of the Smithsonian he became the young republic's chief scientific statesman; international honour arrived in 1893, fifteen years after his death, when the Chicago Electrical Congress fixed "the henry" as the world's unit of inductance — so that every $\tau = L/R$ in this article silently carries his name in its numerator.

History

1832

— Henry publishes self-induction, discovered by the spark at a breaking circuit — the first recorded RL transient

1835

— Henry's relay: a coil's transient throws a switch, and signals leapfrog across continents

1855

— Thomson's telegraph analysis treats distributed L, R, C together — transients become quantitative engineering

1886

— Heaviside's operational calculus turns switching problems into algebra; the τ = L/R idiom enters practice

1910

— Kettering's ignition system makes the interrupted-inductor spike a mass-produced product line

1962

— Josephson analyses superconducting loops: L/R with R = 0, currents that persist indefinitely

Going further

References. Halliday, Resnick & Walker, Fundamentals of Physics, ch. 30 (RL circuits) · Purcell & Morin, Electricity and Magnetism, 3rd ed., §7.9–7.10 · Horowitz & Hill, The Art of Electronics, 3rd ed., ch. 1 (inductive kick and clamp diodes — the practitioner's liturgy) · Griffiths, Introduction to Electrodynamics, 4th ed., §7.2.3–7.2.4 for the field-side energy story.

Provenance. Authored fresh for the SuperSTEM E&M track's circuits chapter (recon-§d gap heal); sources: v3 circuit pilots (rewritten to UG register) + legacy wiki circuit articles (narrative) — no AllQFT base exists for this topic. A verifier should recompute: the rise solution and its $t=0^+$ voltage $v_L = \varepsilon$ ; the spike arithmetic 8 mH × 4 A / 0.1 ms = 320 V (×100 → 32 kV); the decay integral $\int i^2R\,dt = \tfrac12LI_0^2$ independent of $R$ ; example 1's $\tau = 5.0$ ms, $t_{\text{pull}} = \tau\ln3 = 5.5$ ms, 6 kV unprotected estimate; example 2's $\tau_{\text{rise}} = 2.0$ ms ( $R_{\text{Th}} = 3\ \Omega$ ), $I_L = 3.0$ A, $\tau_{\text{decay}} = 0.5$ ms, 36 V across $R_2$ .

Builds on: Self-Inductance · RC Circuits · Series and Parallel Networks (Thévenin) · Faraday's Law of Induction Leads to: LC and RLC Oscillations (L meets C: the current inertia meets the voltage spring)

Why exactly can't an inductor's current change instantly?

Because current through an inductor stores energy $\tfrac12Li^2$ in its magnetic field, and changing it in zero time would require infinite power — visible in the terminal law $v = L\,di/dt$ , where an instantaneous step in $i$ demands infinite voltage. Real circuits enforce this smoothly: any attempt at an abrupt change generates exactly the EMF needed to oppose it (Lenz's law), so the current bends rather than breaks. The practical statement: across any switching event, write $i_L(0^+) = i_L(0^-)$ and let every other variable jump as needed.

If the inductor acts like a wire in steady state, why put it in the circuit at all?

Because circuits are rarely interested only in steady state. The inductor's entire value is its behaviour during change: it smooths current ripple in power supplies (opposing the fast switching), blocks high-frequency noise while passing DC (a choke), stores energy between switching cycles in converters, and rings against capacitors to make oscillators and filters. "Wire at DC, stubborn at speed" is not a bug — it is a frequency-dependent personality, and the AC Circuits and Phasors article will price it precisely as the reactance $X_L = \omega L$ .

Where does the spike's energy come from — the battery is only 12 V?

From the inductor's own stockpile $\tfrac12LI_0^2$ , banked during the on-time. The battery's role ended when it established $I_0$ ; at interruption the collapsing magnetic field is the sole source, and Faraday's law sets the voltage at whatever value moves that energy somewhere on the timescale of the interruption. Fast interruption, huge power, huge voltage; slow or diode-clamped interruption, modest voltage over a longer drain. The energy is fixed — you are only choosing the exchange rate between volts and seconds.

Why do the rise and decay of the same inductor run on different time constants?

Because τ = L/R uses the resistance the inductor actually sees in each configuration, and switching changes the topology. During the rise the source network (Thévenin-collapsed) drives the coil; during freewheel decay the coil sees only its clamp path. Example 2's factor of four is typical, and it is a design lever: a relay that must release quickly gets a high-resistance (or Zener) clamp for a short decay τ, at the cost of a bigger voltage excursion. Same L, different R, different clock — the formula was never "the inductor's time constant," but the circuit's.

Is the inductor-as-mass analogy exact or just a mnemonic?

Formally exact for circuit dynamics. The loop equation L(di/dt) + Ri = ε maps term-by-term onto m(dv/dt) + bv = F: inductance is inertial mass, resistance is viscous drag, EMF is applied force, current is velocity. Stored energy ½Li² is kinetic energy; the switch-break spike is the crash force of stopping a mass abruptly; the flyback diode is a crumple zone. The next article completes the dictionary — adding a capacitor supplies the spring (1/C as stiffness), and the mechanics track's damped-oscillator theory transfers wholesale. The analogy earns "exact" because both systems are the same differential equation, not merely similar stories.

What happens if I open the switch on an inductor with genuinely nothing across it — no diode, no resistor, vacuum contacts?

The voltage rises until something conducts, and something always does. Air breaks down near 3 kV/mm, so an arc forms at the contacts and the current completes its decay through plasma; in vacuum interrupters the metal vapour from the contacts themselves sustains a brief arc. If the insulation everywhere held perfectly (a physical idealization), the tiny stray capacitance between the coil's own windings would take the current, forming a parasitic LC oscillator ringing at high frequency and radiating — which is what you actually see on an oscilloscope: a damped ring, not an infinite spike. Equation (4) is an estimate whose Δt is set by whichever escape route wins.

My relay clicks on quickly but releases sluggishly since I added the protection diode — did I wire it wrong?

No — that is the honest price of the clamp. With the diode, the coil's current decays through the winding resistance at one diode drop of back-pressure, so τ = L/R is the same as the rise clock, and the armature holds until the current sags below the drop-out threshold. Without the diode the coil voltage would spike high and dump its energy in microseconds (fast release, endangered transistor). Middle grounds exist: a resistor or Zener in series with the diode raises the clamp voltage, trading a controlled amount of spike for a faster release — pace versus peak, chosen by component values rather than luck.