Lenz's Law

Drop a strong magnet down a copper pipe and it does not fall — it descends, stately, at a crawl, as though the pipe were filled with honey. Nothing touches the magnet. The copper is not magnetic. Yet something inside the metal has organized itself, within microseconds, into exactly the pattern of currents needed to fight the fall. That something is the subject of this article: the direction rule of electromagnetic induction, stated by Heinrich Lenz in 1834 and carried ever since by a single minus sign. Faraday's Law of Induction established the magnitude: a changing magnetic flux through a circuit drives an electromotive force proportional to the rate of change. But experiment reveals a definite direction too, and it is always the same one — the induced current flows so that its own magnetic field opposes the change that created it. Push flux into a loop and the loop pushes back; pull flux out and the loop clings to it. This article is about why that sign is not a convention but a necessity. Run the movie with the opposite sign and an infinitesimal nudge of a magnet toward a coil recruits a current that pulls it in harder, which induces more current, which pulls harder still — a perpetual-motion machine assembling itself from a fluctuation. The minus sign is energy conservation, written into electrodynamics at the exact point where mechanics and circuits meet. Mastering it buys you two skills the rest of the chapter leans on: reading induced-current directions in seconds, without solving any field problem; and auditing the energy ledger of induction — the drag forces, the back-reactions, the work that must be paid — which is precisely the bookkeeping that Self-Inductance and Energy Stored in Magnetic Fields will turn into hardware.

Key takeaways

Lenz's law is the sign in Faraday's law: $\mathcal{E} = -\mathrm{d}\Phi_B/\mathrm{d}t$ . The induced current's own flux always opposes the change in flux — not the flux itself.
It is not an independent postulate. Reverse the sign and induced currents reinforce their cause: positive feedback, runaway currents, energy from nothing. The minus sign is conservation of energy in circuit form.
The opposition is mechanical as well as electrical: whatever changes the flux feels a back-force. A conductor moving through a field gradient is always dragged, never propelled — and the work done against that drag is exactly the Joule heat the induced current dissipates.
Reading directions is a three-step ritual: pick a loop orientation, ask whether the chosen flux is increasing or decreasing, and send the induced current the way that fights the change.
The strength of the opposition is set by the circuit: a superconducting loop ( $R \to 0$ ) holds its flux exactly constant — total opposition, flux freezing; a high-resistance loop barely responds, though the direction rule never wavers.
KEY IDEA: the minus sign in $\mathcal{E} = -\,\mathrm{d}\Phi_B/\mathrm{d}t$ is Lenz's law — induced effects oppose their cause, which is the only sign consistent with conservation of energy.

The core insight

The minus sign in $\mathcal{E} = -\,\mathrm{d}\Phi_B/\mathrm{d}t$ is Lenz's law: induced currents flow so that their own magnetic flux opposes the change in flux that produced them, making electromagnetic induction dissipative and stabilizing rather than a source of free energy.

Where this lives today

Lenz drag is deployed anywhere engineers want braking without contact, wear, or lurch. Roller-coaster brake runs and drop-tower rides end in fins of copper passing between permanent magnets — the drag force, proportional to speed exactly as in Derivation B, is fiercest at entry and fades to a feather as the car slows, which is why magnetic brake runs feel smooth instead of jerky. High-speed elevator safety systems and gym rowing machines use the same physics; so do the eddy-current dampers that steady precision balances and seismometer masses. On induction cooktops, Lenz's law is the first half of the story — the coil's alternating flux induces opposing currents in the pan's steel base — and Joule heating of equation (4) is the second: the "stove" is a transformer whose secondary is your saucepan. At the frontier, the $R \to 0$ row of the limiting-cases table runs the world's most stable magnets. The main coil of an MRI scanner is a superconducting loop operating in persistent mode: its flux is frozen in the Lenz limit, and the field drifts by less than a part per million per hour with the power supply physically disconnected. The same flux conservation, quantized, underlies SQUID magnetometers reading femtotesla brain fields — Lenz's 1834 direction rule, running at the noise floor of modern measurement.

Origins · 1 minHeinrich Lenz and how this was discovered

Heinrich Lenz

1804–1865 · Dorpat, St. Petersburg

Heinrich Friedrich Emil Lenz — Baltic German, professor at St. Petersburg, and one of physics' great systematizers — read Faraday's 1831 induction results and asked the question Faraday had left qualitative: which way does the induced current flow? His 1834 paper in Poggendorff's Annalen answered with the compact rule that now bears his name, distilled from a methodical program of experiments in which he cross-checked every induction case against the motor effect: the induced current is always the one whose electromagnetic action opposes the motion producing it. The formulation was deliberately mechanical — action against cause — which is why it survived intact when Helmholtz and Kelvin later recognized it as energy conservation applied to induction, two decades before that principle had a name. Lenz's other monument shares this article's energy ledger: with Joule (independently), he established that resistive heating goes as $I^2R$ — the Joule–Lenz law — so both sides of the balance in Derivation B carry his fingerprints. He measured, tabulated, and never speculated; his law is a minus sign that has not needed revision in 192 years.

History

1831

— Faraday discovers electromagnetic induction; direction of the induced current left as case-by-case observation

1832

— Henry independently observes induction and the self-induced spark in America

1834

— Lenz publishes the opposition rule in Poggendorff's Annalen, unifying every known case

1847

— Helmholtz's energy-conservation treatise recasts Lenz's rule as a consequence of the impossibility of perpetual motion

1865

— Maxwell's field equations carry the rule as the minus sign in the Faraday law

1911

— Kamerlingh Onnes discovers superconductivity; persistent currents realize the total-opposition limit of Lenz's law

The physical picture

Think of a conducting loop as magnetically conservative: it resists any change to the number of field lines threading it. It cannot forbid the change — copper is not a superconductor — but it fights a delaying action, and the currency of the fight is induced current. Run the classic experiment in your head. A bar magnet approaches a copper ring, north pole first. The downward flux through the ring grows, so the ring drives a current whose own field points up through its center — counterclockwise seen from the magnet. But a current loop with an upward moment is itself a little magnet with its north pole facing the incoming north pole: like poles repel, the magnet is braked, and your hand feels the ring push back through the magnet. Now reverse the motion. As the magnet retreats, the flux falls; the ring switches its current to keep the flux, its near face becomes a south pole, and it tugs the magnet back. Approach and it repels you; leave and it misses you. In both cases — and this is the whole law — the induction opposes the change.

Figure

Lenz direction test: approaching magnet is repelled, receding magnet is attracted

Notice what the ring is not doing. It is not opposing the magnetic field, and it is not opposing the magnet. It opposes the change in flux — a time derivative, not a state. A ring soaked in the strongest static field in the world carries no current at all; the same ring in a weak but rising field fights immediately. And when the flux through a loop is decreasing, the induced current flows to reinforce the existing field, trying to hold the flux up. Students who memorize "induced current opposes the field" get exactly half the cases wrong; the law opposes the derivative. The opposition has a mechanical face, and this is where energy enters. Whatever agent changes the flux — a falling magnet, a hand pushing a loop, a generator shaft turning a coil — that agent feels a back-force and must do work against it. The work does not vanish: it reappears, joule for joule, as electrical energy in the circuit, ultimately as heat in the resistance. That accounting is why the sign can never be "plus": a plus sign would make the back-force a forward-force, and the agent would be paid to make energy. The whole of magnetic braking, induction damping, and generator loading is this paragraph with numbers attached — and the numbers come below.

The law and its bookkeeping

Lenz's law is a statement about signs, so before anything else the signs must be pinned down. The flux rule of Faraday's Law of Induction reads

\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} \qquad (1)

where: $\mathcal{E}$ — the electromotive force driven around the circuit (volts); $\Phi_B$ — the magnetic flux through any surface bounded by the circuit (webers); and the minus sign — Lenz's entire contribution, the direction of the fight. The magnitude story belongs to the previous articles; equation (1) enters here so we can interrogate its sign. Read it as a contract: whenever the flux through your chosen surface rises at one weber per second, one volt of EMF appears around the boundary, driving current in the negative sense of your chosen orientation. That phrase "chosen orientation" is doing real work, and it is where most sign errors are born. The flux itself,

\Phi_B = \int_S \mathbf{B}\cdot \mathrm{d}\mathbf{a} \qquad (2)

where: $\mathbf{B}$ — the magnetic field (tesla), $\mathrm{d}\mathbf{a}$ — the oriented area element of any surface $S$ spanning the loop — has no sign until you orient the surface. The convention is the right-hand rule: curl the fingers of your right hand along the direction you have declared positive for current, and your thumb gives the positive normal $\hat{\mathbf{n}}$ for the flux. Once that single choice is made, everything in equation (1) is fixed: a positive $\mathrm{d}\Phi_B/\mathrm{d}t$ drives a negative EMF, meaning current in the negative sense — which, chased through the right-hand rule, is current whose own flux through $S$ is negative, opposing the increase. The minus sign and the right-hand rule together are Lenz's law; neither alone is.

Figure

The sign machine: one right-hand choice fixes positive current, positive normal, and the Lenz direction

With a resistance in the loop, the EMF becomes a current, and the sign travels with it. Ohm's law for the whole circuit (the lumped reading built in Ohm's Law and Conductivity and Electromotive Force) gives

I_{\mathrm{ind}} = \frac{\mathcal{E}}{R} = -\frac{1}{R}\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} \qquad (3)

where: $R$ — the loop's total resistance (ohms); $I_{\mathrm{ind}}$ — the induced current, whose sign against your chosen orientation is the Lenz direction. Two physical readings live in this formula. First, the current is proportional to the rate of change — sweep a magnet past a loop twice as fast and you double the current (though for half the time: the total charge driven, $\Delta q = \Delta\Phi_B/R$ , depends only on the net flux change, a fact galvanometer-era experimenters exploited constantly). Second, $R$ sets how loudly the loop objects: a fat copper ring answers violently, a thin nichrome one barely whispers, and the two limits — superconducting and open-circuit — bracket the whole phenomenology, as the limiting-cases table below shows. The power this current dissipates is the quantity on which the whole logical structure turns:

P_{\mathrm{diss}} = I_{\mathrm{ind}}^2 R = \frac{1}{R}\left(\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right)^2 \;\geq\; 0 \qquad (4)

where: $P_{\mathrm{diss}}$ — the Joule heating rate (watts). It is a perfect square: whichever way the flux changes, the loop heats up, and someone must pay for the heat. That someone is the agent changing the flux, paying through the Lenz back-force. Equation (4) is where the minus sign stops being a convention and becomes physics — as the first derivation now makes precise.

Derivation AThe sign is forced — energy conservation selects the minus

The goal is to show that the minus sign in equation (1) is not an experimental accident but the only sign consistent with conservation of energy. The strategy is proof by catastrophe: assume the opposite ("anti-Lenz") sign, follow the feedback loop it creates, and exhibit the energy ledger going negative — a circuit that manufactures energy from nothing. Watch two things: where the feedback enters (the induced current's own flux adds to the total), and where the ledger enters (the sign of the power delivered to the loop). The argument is Griffiths §7.2 register, run slowly.

Step 1. Suppose, contrary to fact, that induction obeyed $\mathcal{E} = +\,\mathrm{d}\Phi_B/\mathrm{d}t$ . A loop of resistance $R$ in a rising external flux would then carry

I = +\frac{1}{R}\frac{\mathrm{d}\Phi_B}{\mathrm{d}t},

a current in the positive orientation sense — which by the right-hand rule produces flux through the loop in the same direction as the increase. The response reinforces the stimulus. Why this is already fatal in outline: the loop's own flux now adds to the external flux, so the total $\Phi_B$ rises faster, so $I$ grows, so the self-flux rises faster still. Positive feedback with no brake.

Step 2. Make the feedback quantitative. The loop's own flux is $\Phi_{\rm self} = LI$ (borrowing only the definition of self-inductance from the next article, Self-Inductance — a positive geometric constant). The total flux is $\Phi_B = \Phi_{\rm ext} + LI$ , and the anti-Lenz circuit equation becomes

IR = +\frac{\mathrm{d}}{\mathrm{d}t}\left(\Phi_{\rm ext} + LI\right) \qquad\Longrightarrow\qquad L\frac{\mathrm{d}I}{\mathrm{d}t} = IR - \frac{\mathrm{d}\Phi_{\rm ext}}{\mathrm{d}t}.

Hold the external flux steady after an initial nudge ( $\mathrm{d}\Phi_{\rm ext}/\mathrm{d}t = 0$ for $t > 0$ ): the equation reads $L\,\mathrm{d}I/\mathrm{d}t = IR$ , with solution $I(t) = I_0\, e^{Rt/L}$ — exponential growth. Any infinitesimal seed current doubles, redoubles, and diverges, powered by nothing.

Step 3. Audit the ledger directly. The electrical power delivered to the loop's charges by the induced EMF is $P = \mathcal{E}I$ . Under the anti-Lenz assumption,

P = \left(+\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right)\left(+\frac{1}{R}\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right) = +\frac{1}{R}\left(\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right)^2 > 0

whenever anything changes — the circuit receives energy. Meanwhile the mechanical back-reaction reverses too: the approaching magnet is pulled in, so the external agent does negative work. Energy appears in the circuit while every account that could have paid for it is being credited, not debited. That is a perpetual-motion machine of the first kind.

Step 4. Reject the assumption. With the minus sign restored, every entry flips: the feedback becomes negative (self-flux opposes the change, $I(t) \propto e^{-Rt/L}$ , disturbances decay), the magnet is braked so the agent does positive work, and the power balance closes with the Joule heat of equation (4) paid in full by that work. The minus sign is therefore forced: it is the unique choice under which electromagnetic induction is dissipative and stable rather than explosive. Verified: the sign argument matches Griffiths §7.2 and Purcell–Morin Ch. 7 (Lenz's law as energy conservation); the exponential-instability form of the anti-Lenz equation cross-checks against the stable RL decay $e^{-Rt/L}$ quoted in Self-Inductance; the work-heat balance is exhibited with numbers in Derivation B and worked example 2.

The mechanical face: Lenz drag

The direction rule becomes a force law the moment the flux change comes from motion. A conductor moving through a nonuniform field — or any circuit whose geometry sweeps flux — carries an induced current, and the magnetic force on that current always acts to retard the motion. This is not a new law; it is Lenz's sign meeting the Lorentz force. The clean laboratory for it is the bar on rails, whose EMF magnitude $\mathcal{E} = B\ell v$ belongs to Motional EMF; what belongs here is the direction of the resulting force and the closing of the energy books.

Go deeper · Derivation B · 2 minThe drag theorem — a bar on rails never gets pushed forward

Derivation BThe drag theorem — a bar on rails never gets pushed forward

The goal is the work-energy audit of Lenz drag in its cleanest setting: a conducting bar of length $\ell$ sliding at speed $v$ along frictionless rails closed by a resistance $R$ , in a uniform field $B$ perpendicular to the plane. We derive the force on the bar, show it opposes the velocity whichever way the bar moves, and verify that the external agent's work reappears exactly as Joule heat. The one input taken from elsewhere is the motional EMF magnitude $\mathcal{E} = B\ell v$ (Motional EMF derives it from the Lorentz force on the bar's charges); everything else is Lenz bookkeeping. Watch the two appearances of the same $B\ell$ : once converting velocity to EMF, once converting current to force — that repetition is why the drag is proportional to $v$ and why the energy balance closes identically.

Step 1. Set up the sign machine. Let the bar slide in the $+x$ direction, take the field $\mathbf{B} = B\hat{\mathbf{z}}$ out of the plane, and orient the circuit counterclockwise (so $\hat{\mathbf{n}} = +\hat{\mathbf{z}}$ ). The flux through the circuit of instantaneous area $\ell x$ is $\Phi_B = B\ell x$ , rising as the bar advances:

\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = -B\ell v \qquad\Longrightarrow\qquad I = -\frac{B\ell v}{R}.

The negative sign means the current actually circulates clockwise — through the bar in the $-y$ direction — exactly the sense whose flux opposes the growth.

Step 2. Compute the force the field exerts on the current-carrying bar. With current $I$ (magnitude $B\ell v/R$ ) flowing along $-\hat{\mathbf{y}}$ through length $\ell$ in field $B\hat{\mathbf{z}}$ ,

\mathbf{F} = I\,\boldsymbol{\ell}\times\mathbf{B} = -\frac{B^2\ell^2 v}{R}\,\hat{\mathbf{x}}.

Why the direction had to come out backward: reverse the bar's velocity and the current reverses too, so the force — the product of both — reverses with the velocity and always points against it. The drag is velocity-proportional, $F = -(B^2\ell^2/R)\,v$ , a genuinely electromagnetic viscosity.

Step 3. Close the energy books. To keep $v$ constant the external agent supplies $F_{\rm ext} = B^2\ell^2 v/R$ , delivering mechanical power

P_{\rm ext} = F_{\rm ext}\, v = \frac{B^2\ell^2 v^2}{R}.

The circuit meanwhile dissipates $P_{\rm diss} = I^2 R = (B\ell v/R)^2 R = B^2\ell^2 v^2/R$ . Equal, term for term, with no remainder: every joule the agent pays crossing the Lenz drag emerges as heat in the resistor. The magnetic field brokers the transfer but keeps nothing (it does no work on the charges — the honest accounting of who-does-what-work is a Conservation laws chapter story, deferred by name to Work Done by Electromagnetic Fields on Charges).

Step 4. Let the drag act unopposed. Remove the agent and let the bar coast: Newton's law reads $m\,\mathrm{d}v/\mathrm{d}t = -(B^2\ell^2/R)\,v$ , so

v(t) = v_0\, e^{-t/\tau}, \qquad \tau = \frac{mR}{B^2\ell^2}.

The bar's kinetic energy drains exponentially into Joule heat — magnetic braking in one line. The time constant $\tau$ carries the whole design space of eddy-current brakes: strong field and low resistance mean fast, fierce braking; and because the drag vanishes with $v$ , the stop is asymptotically gentle — braking without lurch, the signature feel of induction dampers. Verified: force and balance match Griffiths §7.1.3 and the standard rail-bar treatment; the exponential decay recomputed for the worked-example values (m = 0.050 kg, R = 0.10 Ω, B = 0.50 T, ℓ = 0.20 m) gives τ = mR/(B²ℓ²) = 0.50 s exactly; energy closure P_ext = P_diss = B²ℓ²v²/R checked symbolically and numerically in worked example 2.

The same theorem, run inside a solid conductor rather than a wire circuit, produces the swirling induced loops called eddy currents — the agents slowing the magnet in the copper pipe of the overview. Their full machinery (skin depths, laminations, quantitative braking of extended conductors) is owned by the dedicated article Eddy Currents later in this chapter; here they serve as the reminder that Lenz's law needs no wires, only conductivity and a changing flux. And one loop's opposition to its own flux is a preview of the next two articles: a coil resists changes in its own current with a back-EMF $-L\,\mathrm{d}I/\mathrm{d}t$ (Self-Inductance builds $L$ ), and the work done against that self-opposition is banked, recoverably, in the field ( $\tfrac12 LI^2$ — Energy Stored in Magnetic Fields proves it and relocates it into $B^2/2\mu_0$ ). Lenz's law is the sign of both statements; the magnitudes are theirs.

Limiting cases

Case	Result	Meaning
superconducting loop ( $R \to 0$ )	$\Phi_B = \text{const}$ exactly	any change would drive unbounded current; the loop conjures exactly the current that freezes its flux — total Lenz opposition
high-resistance loop ( $R \to \infty$ )	$I_{\rm ind} \to 0$ , direction unchanged	the loop whispers its objection; direction rule intact, energetics negligible
static flux, any strength	$\mathcal{E} = 0$ , $I = 0$	Lenz's law responds to the derivative only — a constant tesla induces nothing
fast sweep vs slow sweep, same $\Delta\Phi_B$	$\Delta q = \Delta\Phi_B / R$ either way	peak current scales with rate, but the delivered charge depends only on the net flux change
coasting bar on rails	$v = v_0 e^{-t/\tau}$ , $\tau = mR/B^2\ell^2$	Lenz drag as electromagnetic viscosity: exponential, lurch-free braking
loop's own current changing	$\mathcal{E}_{\rm self} = -L\,\mathrm{d}I/\mathrm{d}t$	Lenz applied to self-flux — the back-EMF built in Self-Inductance

Common trapLenz's law opposes the change in flux — not the flux, and not the field

The most reliable way to get every second problem wrong is to remember the law as "induced current opposes the magnetic field." False in half the cases: when the flux through a loop is decreasing, the induced current flows to reinforce the existing field, fighting the decrease by propping the flux up. The object of the opposition is always $\mathrm{d}\Phi_B/\mathrm{d}t$ , never $\Phi_B$ and never $\mathbf{B}$ . Corollaries worth internalizing: a static field of any strength induces nothing; a weak but rapidly changing field can induce violently; and a receding magnet is attracted, not repelled — the loop misses it. When in doubt, run the three-step ritual (orient, differentiate, oppose) rather than any shortcut about poles.

Worked examples

1 · The objecting loop (confidence builder)

A circular copper loop of radius $5.0\ \mathrm{cm}$ and total resistance $R = 0.50\ \Omega$ lies flat while a uniform field perpendicular to its plane ramps up at a steady $\mathrm{d}B/\mathrm{d}t = 0.20\ \mathrm{T/s}$ . Find the induced EMF, the current and its direction, the loop's magnetic moment, and the heating rate.

Step 1. Geometry first. The loop area is $A = \pi r^2 = \pi (0.050)^2 = 7.9\times10^{-3}\ \mathrm{m^2}$ . With $\mathbf{B}$ uniform and perpendicular, $\Phi_B = BA$ , so $\mathrm{d}\Phi_B/\mathrm{d}t = A\,\mathrm{d}B/\mathrm{d}t = 7.9\times10^{-3} \times 0.20 = 1.6\times10^{-3}\ \mathrm{Wb/s}$ .

Step 2. The EMF magnitude is $|\mathcal{E}| = 1.6\ \mathrm{mV}$ , and the current is $I = |\mathcal{E}|/R = 1.6\times10^{-3}/0.50 = \mathbf{3.1\ mA}$ .

Step 3. Direction by the ritual. Take the positive normal along the rising $\mathbf{B}$ (say, up). The flux is then positive and increasing, so the EMF is negative: the current runs in the negative orientation sense — clockwise when viewed from above, producing downward flux through the loop, opposing the rise. Its magnetic moment is $m = IA = 3.1\times10^{-3} \times 7.9\times10^{-3} = 2.5\times10^{-5}\ \mathrm{A\,m^2}$ , pointing down: antiparallel to a growing field, exactly the mnemonic form of Lenz's law.

Step 4. The heat. $P = I^2 R = (3.1\times10^{-3})^2 \times 0.50 = \mathbf{4.9\ \mu W}$ — modest, but strictly positive, and every microjoule of it is being paid by whatever power supply is ramping the field, which now feels this loop as a (tiny) extra load.

2 · Terminal velocity on vertical rails (exam level)

A bar of mass $m = 0.050\ \mathrm{kg}$ and length $\ell = 0.20\ \mathrm{m}$ slides down frictionless vertical rails closed by $R = 0.10\ \Omega$ , in a horizontal uniform field $B = 0.50\ \mathrm{T}$ perpendicular to the rail plane. Find the terminal velocity, the current and drag at terminal, and audit the energy ledger. (Take $g = 9.81\ \mathrm{m/s^2}$ .)

Step 1. The physics of terminal velocity: gravity accelerates the bar, the Lenz drag $F = B^2\ell^2 v/R$ grows with $v$ , and the bar stops accelerating when drag equals weight:

\frac{B^2\ell^2 v_t}{R} = mg \qquad\Longrightarrow\qquad v_t = \frac{mgR}{B^2\ell^2} = \frac{0.050 \times 9.81 \times 0.10}{(0.50)^2 (0.20)^2} = \mathbf{4.9\ m/s}.

Step 2. At terminal, the motional EMF is $\mathcal{E} = B\ell v_t = 0.50 \times 0.20 \times 4.9 = 0.49\ \mathrm{V}$ , driving $I = \mathcal{E}/R = \mathbf{4.9\ A}$ — direction, by the ritual, such that the force $I\boldsymbol{\ell}\times\mathbf{B}$ on the bar points up, against the fall.

Step 3. Check the force closes: $F = BI\ell = 0.50 \times 4.9 \times 0.20 = 0.49\ \mathrm{N}$ , and indeed $mg = 0.050 \times 9.81 = 0.49\ \mathrm{N}$ ✓. The bar coasts at constant speed with the field holding it against gravity through its own induced current.

Step 4. The ledger. Gravity delivers $P_{\rm grav} = mgv_t = 0.49 \times 4.9 = \mathbf{2.4\ W}$ . The resistor dissipates $P = I^2R = (4.9)^2 \times 0.10 = \mathbf{2.4\ W}$ . Equal, as Derivation B demands: at terminal velocity, all the gravitational power becomes Joule heat, none becomes kinetic energy. This is the falling-magnet-in-a-pipe demonstration reduced to computable geometry — and the approach to terminal happens on the time scale $\tau = mR/B^2\ell^2 = 0.50\ \mathrm{s}$ , so within a couple of seconds the descent is effectively steady.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.1–7.2 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 7 · Jackson, Classical Electrodynamics, 3rd ed., §5.15 · Zangwill, Modern Electrodynamics, §14.2.

Provenance. All equations are standard textbook results. A verifier should check: (1) the Faraday–Lenz law $\mathcal{E}=-\mathrm{d}\Phi_B/\mathrm{d}t$ and its differential form against Griffiths §7.2 / Jackson §5.15; (2) the rail-bar energy balance $P_{\mathrm{ext}}=I^2R=B^2\ell^2v^2/R$ against Griffiths §7.1.3; (3) the back-EMF $-L\,\mathrm{d}I/\mathrm{d}t$ and stored energy $\tfrac12 LI^2$ against Griffiths §7.2.4; (4) the energy-conservation argument for the sign against Purcell Ch. 7. No empirical numbers are quoted; all relations are exact within the quasi-static approximation. Addendum for this edition: the inductor back-EMF and stored-energy derivations are re-scoped by name to Self-Inductance and Energy Stored in Magnetic Fields; worked numbers recomputed — objecting loop ( $r = 5.0$ cm, $R = 0.50\ \Omega$ , $\mathrm{d}B/\mathrm{d}t = 0.20$ T/s): $|\mathcal{E}| = 1.6$ mV, $I = 3.1$ mA, $m = 2.5\times10^{-5}$ A·m², $P = 4.9\ \mu$ W; vertical rails ( $m = 0.050$ kg, $\ell = 0.20$ m, $R = 0.10\ \Omega$ , $B = 0.50$ T): $v_t = mgR/B^2\ell^2 = 4.9$ m/s, $I = 4.9$ A, drag $0.49$ N $= mg$ , $P_{\rm grav} = P_{\rm diss} = 2.4$ W, $\tau = 0.50$ s.

Builds on: Faraday's Law of Induction · Electromotive Force · Ohm's Law and Conductivity

Leads to: Self-Inductance · Energy Stored in Magnetic Fields · Motional EMF

Is Lenz's law a separate law of nature, or just part of Faraday's law?

Logically it is contained in Faraday's law: once you write $\mathcal{E} = -\mathrm{d}\Phi_B/\mathrm{d}t$ with consistent right-hand conventions, the direction of every induced current follows. Historically and pedagogically it earns separate billing for two reasons. First, Lenz stated the direction rule (1834) before the field formulation existed — as an operational recipe requiring no orientation bookkeeping at all: the induced effect opposes its cause. Second, the rule carries physical content the bare equation hides: it is the statement that induction is dissipative and stabilizing, i.e., energy conservation. If you ever forget the sign conventions in an exam, abandoning the formula and asking "which direction opposes the change?" reconstructs the answer from physics alone.

Where does the energy actually come from when an induced current heats a resistor?

From whatever agent is changing the flux — always, and exactly. Push a magnet toward a loop and the Lenz repulsion means your hand does positive work; that work is the loop's Joule heat. In a generator, the turbine does work against the Lenz torque on the loaded coil — and the torque grows when you draw more current, which is why generators get physically harder to turn under load. Nothing is subtle about the totals: Derivation B closes the ledger term for term, $P_{\rm ext} = I^2R$ . The magnetic field itself is the broker, not the source — a distinction the Conservation-laws chapter (Work Done by Electromagnetic Fields on Charges) makes precise.

Why does the magnet fall so slowly in a copper pipe when nothing touches it?

The falling magnet's flux sweeps through each ring-shaped slice of the pipe: flux grows in slices below it, shrinks in slices above. Each slice responds with a circulating (eddy) current in the Lenz direction — repelling the magnet from below, tugging it from above — so both sets of currents push against the fall. The drag grows with speed (like the rail-bar's $B^2\ell^2 v/R$ ), so the magnet accelerates only until drag balances weight, then descends at terminal velocity, often centimeters per second. Slit the pipe lengthwise and the demonstration dies: the eddy loops can no longer close around the circumference. The quantitative machinery for extended conductors lives in Eddy Currents.

Does Lenz's law mean induced currents always make the flux constant?

No — only a superconducting loop achieves that. In ordinary conductors the induced current opposes the change but cannot cancel it: the flux still changes, just against resistance. The strength of the opposition is set by $R$ (equation 3): copper objects loudly, nichrome faintly, an open circuit not at all (an EMF still appears, but no current flows and no opposing flux is made). The $R \to 0$ limit is genuinely special: there, any nonzero $\mathrm{d}\Phi_B/\mathrm{d}t$ would drive infinite current, so the loop instead carries exactly the finite persistent current that holds $\Phi_B$ constant forever — flux freezing, the operating principle of persistent-mode MRI magnets.

When the flux is decreasing, the induced current reinforces the field. Doesn't that violate the opposition idea?

It is the opposition idea, applied correctly. The thing being opposed is the change $\mathrm{d}\Phi_B/\mathrm{d}t$ , and when the change is a decrease, opposing it means fighting the decrease — propping the flux up, which requires current in the field's own direction. Think of the loop as loss-averse rather than field-averse: it wants to keep whatever flux it has. A magnet pulled away is attracted back; a collapsing external field finds the loop briefly sustaining it. Only when flux is being added does the loop push back against the field direction.

How can I tell the direction without getting lost in right-hand rules?

Use poles instead of conventions. An induced current loop is a magnetic dipole; Lenz's law says that dipole arranges itself to fight the change. Approaching north pole? The loop's near face becomes north — repulsion. Receding north pole? Near face becomes south — attraction. For fields rather than magnets, use the moment form: the induced magnetic moment is antiparallel to a growing applied field and parallel to a shrinking one. Both mnemonics compile down to the same right-hand bookkeeping, but they run on physical intuition, which survives exam pressure better than finger gymnastics.

Is the back-EMF of an inductor the same physics as the falling magnet?

Yes — Lenz's law applied to a circuit's own flux instead of an external one. A coil carrying current threads itself with flux $\Phi = LI$ ; try to change $I$ and the coil induces an EMF $-L\,\mathrm{d}I/\mathrm{d}t$ opposing the change. That is why inductors resist being switched on (current ramps rather than jumps) and object violently to being switched off (the "inductive kick" that sparks across opening switches — the flux insists on somewhere to go). The constant $L$ and its geometry are the next article's business (Self-Inductance); the sign and the stabilizing character are this one's.

Do eddy-current brakes work on aluminum, and why do they never wear out?

They work on any decent conductor — aluminum and copper are the standards; conductivity, not magnetism, is what is required (aluminum is not attracted to a stationary magnet at all). The braking force is the Lenz drag of induced currents, transmitted through the magnetic field with no mechanical contact: no pads, no friction surfaces, nothing to abrade, which is why roller-coaster fins and high-speed elevator emergency brakes rely on them for thousands of cycles without maintenance. The trade-off is the $v$ -proportionality: the drag vanishes as the object stops, so eddy brakes cannot hold anything stationary — a friction or ratchet stage always finishes the job.