Lenz's law — opposing induced current: Question bank

Definition The symbols this page uses — read this first

Before any trap, let's earn every symbol so nothing surprises you.

Magnetic flux $Φ_{B}$ — imagine the magnetic field $B$ as a bundle of arrows piercing a loop. Flux is how many of those arrows thread through the loop's area — a "how much field is caught in the net" number. For a flat loop of area $A$ in a uniform field, $Φ_{B} = B A cos θ$ , where $θ$ is the angle between $B$ and the loop's normal (an arrow sticking straight out of the loop's face). If the field points straight through the loop, $θ = 0$ and $Φ_{B} = B A$ is as big as it gets; if the field skims along the loop's plane, $θ = 9 0^{\circ}$ and $Φ_{B} = 0$ . See Magnetic flux.
Induced EMF $ε$ — the "electrical push" (electromotive force, measured in volts) that a changing flux creates around the loop. It is what drives the induced current. Faraday's law says $ε = - \frac{d Φ _{B}}{d t}$ , i.e. the push equals how fast the flux is changing. See Faraday's law of induction.
Drag force $F$ — the mechanical force the induced current pushes back with when something moves. For a rod of length $L$ moving at speed $v$ in field $B$ across rails of resistance $R$ , this is $F = \frac{B ^{2} L ^{2} v}{R}$ , always pointing against the motion. Because everything else is fixed, $F$ grows in step with $v$ — that is what we mean by $F \propto v$ (" $F$ is proportional to $v$ "). See Motional EMF and sliding rod.
The ==double equals== around a word is just this vault's highlighter — a cue that the word is a key term worth memorising. It has no physics meaning.

The single sentence everything below springs from: the induced current opposes the change in flux, not the field, not the motion directly, and not the current itself. Keep that pinned.

The figure below fixes the convention and the two core scenarios in pictures — glance at it before you start.

Figure — Lenz's law — opposing induced current

True or false — justify

Trap: nearly every one is "almost true." The wrong half is what you must catch.

The induced current always flows opposite to the external field

B

False. It opposes the change in flux. If flux is decreasing, the induced field points the same way as

B

to prop it up.

A steady magnet held motionless inside a coil drives an induced current.

False. No change in flux means

\frac{d Φ _{B}}{d t} = 0

, so the induced EMF

ε = 0

. Lenz opposes change; a constant flux is nothing to oppose.

Lenz's law can be derived from conservation of energy alone.

True. If the current aided the change you'd get self-amplifying motion and free energy; only the opposing direction balances mechanical work against heat.

Doubling the speed of a magnet doubles the magnitude of the induced EMF.

True (roughly).

ε = - \frac{d Φ _{B}}{d t}

scales with the rate of change of flux, and moving twice as fast changes flux about twice as fast.

The drag force from Lenz's law can bring a moving magnet to a dead stop and then push it backward.

False. The drag force

F

\propto v

; as

v \to 0

the force

\to 0

. It bleeds kinetic energy away but never reverses the motion.

If you flip the magnet (S-pole first) approaching a coil, the induced current reverses direction.

True. Flipping the pole reverses

B

through the coil, so the flux change reverses, and the opposing current reverses with it.

The induced current would still flow if the coil were an open (broken) ring.

False. An EMF

ε

is still induced, but with no closed path no current flows — and hence essentially no opposing force. Lenz gives direction, but a current needs a circuit.

A coil moving through a uniform field with all of it always inside the field feels a Lenz drag.

False. If the whole loop stays fully inside a uniform field, the enclosed flux

Φ_{B} = B A

doesn't change, so no induced current and no drag. See Motional EMF and sliding rod.

Spot the error

Each statement contains one broken step. Name it.

"N-pole approaches → coil near-face becomes S-pole to keep flux, so it attracts the magnet."

Error: an approaching magnet increases flux, so the coil must oppose the increase → near-face becomes N and repels. The S-pole/attract case is for a withdrawing magnet.

"Flux into the page is increasing, so the induced field must also point into the page to match it."

Error: to oppose an increase the induced field points out of the page, not with it. "Match" is exactly backwards.

"I found the current direction with the right-hand rule on the wire, then figured out the field."

Error: order reversed. First decide the required field direction inside the loop from "oppose the change," then use the Right-hand rule to read off the current.

"The magnet loses no energy braking in a copper tube — magnetic forces do no work."

Error: the magnetic force on a single charge does no work, but here the source (gravity / your hand) does work against the drag force

F

, and that work becomes heat in the metal. Energy is conserved, not free. See Eddy currents and magnetic braking.

"Faraday's law

ε = - \frac{d Φ _{B}}{d t}

already gives the current direction, so Lenz's law is redundant."

Error: Faraday gives the magnitude of the induced EMF

ε

; the minus sign is meaningless until you fix a sign convention (choice of normal + circulation, as above), and Lenz's law is precisely the rule that interprets that sign physically. See Faraday's law of induction.

"Because the drag force always opposes motion, it must eventually reverse the magnet's velocity."

Error: opposing motion only means decelerating. Since

F \propto v

, the force fades to zero exactly as the speed reaches zero — it can slow but never flip the direction.

Why questions

Answer with the mechanism, not the slogan.

Why does the induced current oppose the change rather than help it?

If it helped, the motion would self-amplify and generate energy from nothing, violating Conservation of energy. Opposition is the only direction that makes you pay for the electrical energy.

Why is there a minus sign in

ε = - \frac{d Φ _{B}}{d t}

Once you fix the normal + circulation convention, the minus is Lenz's law: it forces the current so its own field fights the flux change, so mechanical work in equals heat out. Drop it and energy bookkeeping fails.

Why does a magnet dropped through a copper pipe fall slower than through a plastic one?

The moving magnet changes the flux

Φ_{B}

through each ring of copper, inducing looping eddy currents whose fields oppose the magnet's approach/departure — a velocity-dependent drag force. Plastic can't carry those currents.

Why does the sliding-rod drag force point against the motion regardless of which way the rod moves?

Whatever direction increases the enclosed flux gets opposed, and opposing an area increase always means a force

F

resisting the expansion. Reverse the motion and both the flux trend and the force flip, staying anti-parallel to

v

Why can Lenz's law be described as "the loop wants the flux to stay the same"?

The induced field always points to cancel an increase or replenish a decrease — either way it nudges the total flux

Φ_{B}

back toward its previous value, like a thermostat resisting change.

Why doesn't a faster magnet cause a runaway "infinite braking"?

The opposing force grows with speed (

F \propto v

), but it only dissipates the kinetic energy already present; it can't exceed the energy you supply, so it decelerates smoothly toward

v = 0

where the force also vanishes.

Why does the direction of induced current in a self-inductor oppose its own changing current?

The changing current changes its own flux, and Lenz opposes that change — producing a back-EMF that resists the current's increase or decrease. See Self-inductance and back-EMF.

Edge cases

The scenarios where the naive rule needs care.

What is the induced current at the exact instant the flux is a maximum (turning point of

Φ_{B}

At a maximum

\frac{d Φ _{B}}{d t} = 0

, so

ε = 0

and the current is momentarily zero — even though the flux itself is large. Rate of change, not value, drives induction.

A loop is rotated so its plane stays parallel to

B

(normal perpendicular to

B

). Flux? Induced current?

With the normal

⊥ B

Φ_{B} = B A cos 9 0^{\circ} = 0

at that instant, but as it rotates through that orientation the flux is changing fastest, so the induced current is at a maximum there. See Magnetic flux.

A magnet approaches a loop, passes through the plane, and recedes. What does the current do?

It flows one way on approach (repel), drops to zero as the magnet crosses the plane where flux peaks and

\frac{d Φ _{B}}{d t} = 0

, then reverses on recession (attract). The current changes sign, tracking the sign of the flux rate.

Two identical magnets approach a loop from opposite faces at equal speed. Net induced current?

If both add flux in the same sense through the loop, effects add; if they add flux in opposite senses, the flux changes can cancel to give zero net induced current despite two moving magnets.

A perfectly circular loop expands uniformly in a uniform field perpendicular to it. Direction of induced current?

Growing area means increasing flux

Φ_{B} = B A

, so the induced current flows to make its field oppose

B

inside — the sign depends only on the trend of the flux, not on any wire being "cut."

What happens to the induced current if the resistance

R \to \infty

(open circuit)?

The EMF

ε

is still present (Lenz still assigns a direction) but

I = ε / R \to 0

: no current, no opposing force, no heat. You can move the magnet "for free" because nothing dissipates.

A superconducting loop has resistance

R \to 0

. What does its flux do when you try to change it?

With

R \to 0

any tiny EMF drives a huge induced current, whose field almost perfectly cancels the attempted change — so

Φ_{B}

stays essentially frozen at its initial value. Zero resistance makes the loop the strongest possible flux-defender, the opposite of "flux flows freely."

Connections

Faraday's law of induction — supplies the EMF magnitude; Lenz interprets its sign.
Magnetic flux — the quantity $Φ_{B}$ whose change every question here tracks.
Right-hand rule — converts required field direction into current direction.
Motional EMF and sliding rod — the "loop fully inside field" edge case and the drag force $F$ .
Eddy currents and magnetic braking — the copper-pipe traps.
Conservation of energy — the reason the "opposes" direction is forced.
Self-inductance and back-EMF — Lenz applied to a circuit's own current.