Lenz's law — opposing induced current — Physics notes

WHY does Lenz's law exist?

WHAT is the problem it solves? Faraday's law tells us the size of the induced EMF: $\varepsilon = -\frac{d\Phi_B}{dt}$ but the bare magnitude $\left|\dfrac{d\Phi_B}{dt}\right|$ doesn't tell you which way the current goes. Lenz's law is the physical meaning of the minus sign.

WHY must the current oppose the change? Suppose it helped the change instead. Then:

Push a magnet toward a coil → induced current pulls the magnet in faster → it speeds up on its own → current grows → it pulls harder → infinite energy from nothing.

That violates conservation of energy. So the only physically allowed direction is the one that resists the motion / change. You must do work to push the magnet, and that work becomes electrical energy.

HOW to apply it (the 4-step recipe)

Figure — Lenz's law — opposing induced current

Deriving the minus sign from energy

WHY is the sign negative and not positive? Let's build it.

Move a bar magnet (N-pole first) toward a loop at speed $v$ . Flux into the loop grows: $\dfrac{d\Phi_B}{dt}>0$ .

If induced current opposed nothing, no force, free energy — forbidden.
So the loop's face nearest the magnet must become a N-pole to push the magnet back (repel like poles). This requires current in a specific sense.

The mechanical work you do against this repulsion equals the electrical energy dissipated: $W_{\text{you}} = \int F\,dx = \int I^2 R\, dt = \text{heat in resistor}.$ For the bookkeeping to balance ( $P_{\text{mech}} = P_{\text{elec}} > 0$ ), the induced EMF must point so the current opposes the change — hence the minus sign in $\varepsilon = -d\Phi_B/dt$ . The minus sign is energy conservation.

Worked examples

Worked example Example 1 — Magnet approaching a coil

A bar magnet's N-pole moves toward a circular coil. Which way does current flow (viewed from the magnet's side)?

Step 1: Field from N-pole points away from magnet, i.e. into the coil. Why? Field lines leave N.
Step 2: Magnet getting closer → flux into coil increases. Why? $B$ at the coil grows as distance shrinks.
Step 3: Coil must oppose the increase → induced field points out of the coil toward the magnet → coil face is a N-pole (repels). Why? Opposing the increase means cancelling the incoming field.
Step 4: Right-hand rule: field out of coil toward you ⇒ current flows counterclockwise as seen from the magnet. Why? Curl fingers along current, thumb gives field direction.

✔ Result: counterclockwise, and the magnet feels a push-back force.

Worked example Example 3 — Sliding rod (quantitative)

A rod length $L$ slides at speed $v$ on rails in a uniform field $B$ pointing into the page; area enclosed grows.

$\Phi_B = B\,L\,x$ , so $\dfrac{d\Phi_B}{dt}=BL\dfrac{dx}{dt}=BLv$ (increasing into page).
$|\varepsilon| = BLv$ . Why this step? differentiate flux; only $x$ changes.
Flux into page increasing ⇒ induced current makes field out of page inside loop ⇒ current flows counterclockwise.
Current $I=\dfrac{BLv}{R}$ produces force $F=BIL=\dfrac{B^2L^2v}{R}$ opposing the motion. Why? Lenz — the rod must be pushed, and you feel drag.
Power you supply $= Fv = \dfrac{B^2L^2v^2}{R} = I^2R$ ✔ energy balances.

Common mistakes (Steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine you have a sleepy guard dog (the coil). When a magnet sneaks toward it, the dog growls and pushes it away. When the magnet tries to leave, the dog grabs its tail to pull it back. The dog always wants things to stay exactly as they were. That "wanting to stay the same" is Lenz's law — and the dog has to use energy (your pushing) to growl, so you never get free magic energy.

Active-recall flashcards

What does Lenz's law determine that Faraday's magnitude alone does not?

The direction of the induced current (the meaning of the minus sign).

State Lenz's law in one sentence.

Induced current flows so its magnetic field opposes the change in flux that produced it.

Which fundamental principle is Lenz's law equivalent to?

Conservation of energy.

If flux through a loop is decreasing, does the induced field point with or against the external field inside the loop?

With it (same direction), to maintain the flux.

An N-pole approaches a coil; what pole does the near face of the coil become?

A north pole (it repels, opposing the approach).

An N-pole is withdrawn from a coil; near-face pole?

A south pole (it attracts, trying to keep the magnet).

For a rod of length

L

moving at

v

in field

B

, what is

|\varepsilon|

BLv

Why is the force on the sliding rod always opposite to its motion?

Because the induced current must oppose the increasing flux (Lenz), creating a drag

F=B^2L^2v/R

Write Faraday's law with the Lenz sign.

\varepsilon=-\dfrac{d\Phi_B}{dt}

What would happen physically if the induced current aided the change?

Self-amplifying motion → infinite free energy → violates energy conservation.

Connections

Faraday's law of induction — Lenz supplies the sign/direction.
Magnetic flux — the quantity whose change drives induction.
Right-hand rule — converts induced-field direction to current direction.
Motional EMF and sliding rod — quantitative Lenz example.
Eddy currents and magnetic braking — Lenz drag in bulk conductors.
Conservation of energy — the deep reason behind the minus sign.
Self-inductance and back-EMF — Lenz applied to a circuit's own changing current.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Lenz ka law bahut simple idea hai: nature ko change pasand nahi. Jab bhi tum kisi loop ke through magnetic flux ko badalne ki koshish karte ho, loop uska virodh karta hai. Yani induced current hamesha us direction me behta hai jo us change ko oppose kare jisne use paida kiya. Yahi reason hai ki Faraday ke law me $\varepsilon=-\frac{d\Phi}{dt}$ wala minus sign aata hai — wo minus sign actually energy conservation ka signature hai.

Socho: agar magnet ka N-pole coil ke paas aa raha hai, to coil ka saamne wala face bhi N-pole ban jaata hai taaki magnet ko dhakka de (repel kare). Aur agar magnet door ja raha hai, to coil S-pole banke usse pakadne ki koshish karta hai. Matlab loop hamesha chahta hai ki flux jaisa tha waisa hi raha jaaye. Isiliye tumhe magnet ko push/pull karne me mehnat karni padti hai — aur wahi mehnat electrical energy aur heat me convert hoti hai.

Agar induced current help karta change ko, to magnet apne aap tez hota jaata aur free energy mil jaati — jo impossible hai. Isiliye Lenz ka law sirf ek rule nahi, balki conservation of energy ka direct result hai.

Practical tip: pehle decide karo external $\vec B$ kis taraf hai aur flux badh raha hai ya ghat raha hai. Agar badh raha hai → induced field opposite, agar ghat raha hai → induced field same direction (taaki flux bacha rahe). Phir right-hand rule lagao to current ki direction mil jaayegi. Sliding rod example yaad rakho: $|\varepsilon|=BLv$ aur force $F=\frac{B^2L^2v}{R}$ hamesha motion ke against — yahi drag energy balance karta hai.

Lenz's law — opposing induced current

WHY does Lenz's law exist?

HOW to apply it (the 4-step recipe)

Deriving the minus sign from energy

Worked examples

Common mistakes (Steel-manned)

Active-recall flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections