Lenz's law and conservation of energy

Push a bar magnet toward a coil and something strange resists you. Not friction, not gravity — a force that seems to know, the instant you start pushing, exactly which way to push back. Pull the same magnet away and the resistance flips, now trying to drag it back in. The coil behaves like a silent guardian that always leans against whatever change you attempt, never helping it along. That guardian has a name: Lenz's law, and it is really conservation of energy wearing an electromagnetic disguise. Faraday's law already tells us an EMF appears whenever flux through a loop changes, $\varepsilon = -N\,d\phi_B/dt$ . But a formula with a plus sign instead of that minus would still balance dimensionally — so why must nature choose the minus? Lenz's law answers by fixing the direction: the induced current always flows so as to oppose the very change that created it. Get this backwards and you get something absurd — a magnet that accelerates itself while simultaneously heating a resistor, manufacturing both kinetic energy and thermal energy from nothing. The universe forbids that, and the induced current's opposition is precisely how it enforces the ban. This article builds the idea from the ground up: what "opposing the change" concretely means for an approaching or receding magnet, why the opposition is not optional but mathematically forced by the negative sign already sitting inside Faraday's law, and why every joule of electrical energy delivered by an induced current can be traced to an equal joule of mechanical work done against that opposition. We keep the two worked scenarios from the original notes — the approaching magnet and the growing primary-coil current — and extend them with sharper energy bookkeeping, a labelled figure, two fresh numeric-style examples, and the traps that quietly cost marks on this exact topic every board season.

Key takeaways

Lenz's law fixes the direction of induced current: it always opposes the change in flux that produces it, never assists it.
It is the physical reading of the minus sign already present in Faraday's law, $\varepsilon = -N\,d\phi_B/dt$ — not a separate, independent rule bolted on afterward.
Approaching magnet, repelling coil: an approaching north pole makes the coil's near face behave like a north pole (repulsion); a receding north pole makes the near face a south pole (attraction).
The polarity fixes the current's sense, and the right-hand rule then reads off clockwise or anticlockwise from a stated viewing side.
Energy conservation is the enforcer: if induced current instead helped the change, kinetic energy and $I^2R$ heat would both appear with no energy input — forbidden by the first law of thermodynamics.
External work in equals electrical energy out: the work done pushing against the opposing force is exactly what the circuit later delivers as electrical energy, including whatever is dissipated as heat.
Key idea: the induced current in a circuit always flows in the direction that opposes the change of flux producing it, because any "helping" current would create energy from nothing, so the work done against this opposition is precisely what conservation of energy demands appear as electrical energy and heat.

The core insight

Lenz's law says the induced EMF always opposes the flux change that creates it — the minus sign in Faraday's law made physical — and this opposition is required, not optional, because it is exactly what conservation of energy demands: mechanical work in equals electrical energy plus heat out.

Where it fits in the syllabus

Lenz's law sits directly inside the CBSE Class 12 Unit VI chapter "Electromagnetic Induction," listed immediately after Faraday's laws in the prescribed scope, alongside the minus sign in $\varepsilon = -N\,d\phi_B/dt$ . The board expects a clean statement of the law, its application to find current direction in magnet-and-coil or two-coil situations, and a qualitative but precise argument connecting it to energy conservation — no calculus beyond what Faraday's law already used, and no new formula. This chapter's next topics, self- and mutual inductance, both lean on this same opposing-current idea, so getting the direction argument solid here pays off immediately afterward.

The physical picture

Flux through a loop is the "amount" of magnetic field threading it, and any change in that amount — whether from a moving magnet, a growing current in a nearby coil, or a shrinking loop area — is what triggers an induced EMF.

\phi_B = \vec B \cdot \vec A, \qquad \varepsilon = -N\,\frac{d\phi_B}{dt}. \qquad (1)

where: $\phi_B$ is the magnetic flux through one turn (weber, Wb), $\vec B$ is the magnetic field (tesla, T), $\vec A$ is the loop's area vector (square metres, m², direction along the outward normal), $N$ is the number of turns, $\varepsilon$ is the induced EMF (volt, V), and $t$ is time (second, s). The physical reading of the minus sign is simply Lenz's law in symbols: the induced EMF's sense is such that any current it drives creates a flux that fights the change $d\phi_B/dt$ , not one that adds to it. For a magnet-and-coil pair, the rule collapses to a face-polarity trick. If the north pole of a bar magnet approaches a coil, the flux entering the coil through the near face is increasing, so the near face must itself turn into a north pole — repelling the intruder and thereby opposing the increase. If the same pole is withdrawn, the flux is decreasing, so the near face becomes a south pole, attracting the magnet back and opposing the decrease. Once the required face polarity is known, the right-hand curl rule (fingers curl in the current's sense, thumb points along the pole that face becomes) gives the actual clockwise or anticlockwise direction, viewed from whichever side is stated.

Figure

A bar magnet with north pole approaching a coil from the left, the coil face nearest the magnet labelled as an induced north pole repelling the magnet, and the induced current direction shown as an anticlockwise arrow when viewed from the magnet side

Derivation AWhy the opposing direction of Lenz's law is forced by conservation of energy

The goal here is not to derive a new formula — CBSE explicitly wants a qualitative argument — but to show, step by step and without hand-waving, that the opposing sense of the induced current is not a convenient convention but a strict logical consequence of the first law of thermodynamics applied to an electromechanical system with a moving magnet and a resistive coil. We do this by supposing the opposite direction and tracking energy through one push-in cycle.

Step 1. Set up the energy accounts. Let the person pushing a magnet toward a coil do external work $W$ over some short interval. The coil has resistance $R$ and the induced current dissipates heat $Q = I^2R\,\Delta t$ in that interval. The magnet also has some kinetic energy $K$ if it changes speed. Conservation of energy for the whole system (person plus magnet plus coil) requires $W = \Delta K + Q$ — external work supplied must equal whatever kinetic energy is added to the magnet plus whatever heat is generated, with nothing left over and nothing missing.

Step 2. Suppose, for contradiction, that the induced current instead flowed in the "helping" direction — the coil's near face becomes a south pole as the north pole approaches, attracting rather than repelling it. An attractive force acting in the direction of the magnet's motion does positive work on the magnet, so the magnet would speed up on its own, gaining kinetic energy $\Delta K > 0$ , even if the person supplies zero external work, $W = 0$ .

Step 3. But the same current, being nonzero, still dissipates heat $Q = I^2R\,\Delta t > 0$ in the coil's resistance regardless of its direction — resistive heating depends on $I^2$ , not on the sign of $I$ . So the energy budget from Step 1 would demand $0 = \Delta K + Q$ with both terms on the right strictly positive, which is impossible. The "helping" direction manufactures both kinetic energy and heat from an external work input of zero — a violation of energy conservation, a genuine free-energy machine.

Step 4. Therefore the induced current must flow in the opposing direction: the coil repels the approaching magnet. Now the force on the magnet acts against its motion, so the person must supply positive external work $W>0$ to keep the magnet moving in. That work splits exactly as $W = \Delta K + Q$ : part goes into any kinetic energy retained by the magnet, and part becomes the $I^2R$ heat in the coil, plus (in a general circuit) whatever useful electrical energy is delivered elsewhere. Every joule out is accounted for by a joule of work in — Lenz's law is simply the direction that keeps the books balanced. Read the argument again for the receding-magnet case: pulling a magnet away also requires the person to do positive work against the coil's attraction, and that same reasoning applies symmetrically, confirming Lenz's law is direction-independent in its energy logic.

Worked examples

Board level - approaching magnet, direction and energy check

Question: A magnet's north pole is moved steadily toward a coil, viewed from the magnet's side. Use Lenz's law to find the direction of the induced current, and justify with an energy argument why the coil must repel the approaching magnet.

Step 1. As the north pole approaches, the flux entering the coil from the magnet's side is increasing with time.

Step 2. By Lenz's law the induced current must oppose this increase, so the coil's near face must become a north pole, repelling the incoming magnet.

Step 3. By the right-hand curl rule, for the near face (as viewed from the magnet's side) to behave as a north pole, the current must circulate anticlockwise when viewed from that same side.

Step 4. Energy check: if the coil instead attracted the magnet, the magnet would accelerate on its own (gaining kinetic energy) while the coil simultaneously dissipates $I^2R$ heat, with no external work supplied — energy created from nothing, which is impossible. Since the coil actually repels the magnet, the person pushing it must do positive external work, and this work supplies both the magnet's retained kinetic energy and the heat dissipated in the coil.

Answer: The induced current flows anticlockwise as seen from the magnet's side, producing a repulsive force on the approaching magnet, exactly as conservation of energy requires.

Competitive level - two-coil mutual growth and the primary's delayed rise

Question: A primary coil is connected to a battery through a switch; a secondary coil sits nearby, connected only to a galvanometer. When the switch is closed, the primary current rises from $0$ to a steady value $I_0$ over a short interval $\Delta t$ . Explain, using Lenz's law, why the primary current cannot jump instantaneously to $I_0$ , and what this means for the energy the battery must supply during the rise.

Step 1. Closing the switch makes the primary's current — and hence its own flux, and the flux it sends through the secondary — start rising from zero.

Step 2. The changing flux linking the secondary coil induces a current in it. By Lenz's law, this secondary current creates a magnetic field that opposes the increasing flux from the primary — it tries to keep the total linked flux from rising as fast as the primary current alone would produce.

Step 3. This opposition is mutual: the changing flux also links back into the primary coil itself (the self-induction effect that appears formally as $\varepsilon = -L\,dI/dt$ later in this chapter), so the primary circuit resists its own current's rise too. The net effect is that the primary current climbs smoothly toward $I_0$ over the finite interval $\Delta t$ rather than jumping to $I_0$ the instant the switch closes.

Step 4. Energy accounting during the rise: the battery must supply not only the eventual steady-state $I_0^2R$ heating but additional work during $\Delta t$ to push current against this self-opposing back-EMF — energy that ends up stored in the magnetic field of the coils and dissipated as heat, never appearing for free.

Answer: The primary current rises smoothly to $I_0$ over the finite time $\Delta t$ rather than instantaneously, because Lenz's law guarantees an opposing back-EMF during any flux change; the battery supplies extra work during this rise precisely to cover the stored-field and dissipated energy that opposition demands. Timing note: In a typical board or competitive numerical, this qualitative reasoning is often paired with a quantitative self-inductance calculation from the next section of the chapter (mutual/self-inductance) — recognise here that Lenz's law is what guarantees the delay exists, while $L$ and $dI/dt$ are what fix how long it actually takes; do not try to compute $\Delta t$ from Lenz's law alone, as it supplies direction and existence, not a numeric rate.

Common traps

Board answer scripts lose marks on this topic in very repeatable ways. First, students often jump to naming clockwise/anticlockwise without first pinning down whether the flux is increasing or decreasing — get that sign wrong and every subsequent direction is flipped. Second, a common reversal is applying Lenz's law backwards, assuming the induced effect assists rather than opposes the change, especially under time pressure. Third, many treat Lenz's law and Faraday's law as two separate rules to memorise rather than recognising that Lenz's law is literally the meaning of the minus sign already inside $\varepsilon = -N\,d\phi_B/dt$ — this shows up as inconsistent signs across a multi-part answer. Fourth, the energy argument is frequently stated as a slogan ("it opposes because of energy conservation") without the actual logical chain — examiners specifically reward showing that a helping current would create both kinetic energy and heat from zero external work, which is the actual contradiction, not just an assertion.

You should be able to…

State Lenz's law precisely and identify it as the physical meaning of the negative sign in Faraday's law.
Determine whether flux through a coil is increasing or decreasing in a given magnet-and-coil or two-coil scenario before assigning a current direction.
Use the induced face-polarity rule plus the right-hand curl rule to find clockwise/anticlockwise current direction, viewed from a stated side.
Construct the specific energy-conservation contradiction that rules out a "helping" induced current (simultaneous kinetic energy and heat with zero external work).
Explain qualitatively why a changing current in one coil delays the rise of current in a nearby coupled coil, and why the battery must supply extra work during that rise.

Ask the tutor: want to practise finding induced-current directions for a few different magnet, coil, and two-coil configurations, or walk through the energy-balance argument once more with numbers plugged in?

Why does the induced current oppose the change instead of just being random in direction?

Because "random" is not a physically allowed option here — the direction is fixed by conservation of energy, not by chance. If the current opposed the change, external work must be done to sustain the change, and that work is exactly what appears as electrical and thermal energy in the circuit. If it instead assisted the change, energy would appear from nowhere, which nature does not allow. So there are really only two directions physically possible for the induced field to point, and energy conservation eliminates one of them completely, leaving Lenz's law as the only consistent choice.

Is Lenz's law a separate law from Faraday's law, or the same thing?

They are two faces of one statement. Faraday's law, $\varepsilon = -N\,d\phi_B/dt$ , tells you the magnitude of the induced EMF from how fast flux changes, and the minus sign inside it already encodes the direction. Lenz's law is simply the plain-language explanation of what that minus sign means physically: the induced effect opposes its cause. You will sometimes see Faraday's law written without the sign, with Lenz's law quoted separately to fix direction — that is a teaching convenience, not evidence they are independent.

Does Lenz's law only apply to magnets moving near coils?

No — it applies to any situation where flux through a circuit changes, for whatever reason: a magnet moving, a nearby coil's current changing (as in the two-coil example), a loop's area shrinking, or a loop rotating in a fixed field. In every case, the induced current's direction is whatever direction makes its own magnetic field fight the change in flux. The magnet-and-coil picture is just the easiest case to visualise the face-polarity trick in, but the underlying rule from energy conservation is completely general.

In the two-coil example, does the secondary coil actually slow down the primary circuit's battery?

Not exactly "slow down" the battery, but it does mean the battery must do more work during the brief rise than it would in an idealised, flux-free circuit. The back-EMF induced (both from the primary's own changing flux and any coupling to the secondary) opposes the growth of the primary current, so the battery pushes against that opposition while the current climbs to its steady value. Once the current is steady, the flux stops changing, the opposing back-EMF disappears, and the circuit behaves like an ordinary resistive circuit again.