A changing magnetic flux drives a voltage — and Lenz's minus sign says the current it makes will fight the change that made it. Spin a loop in a field and it becomes an AC generator: flux Φ = B·A·cos ωt goes in, the induced EMF = −dΦ/dt comes out. That single minus sign is energy conservation wearing a disguise. The blue team builds and defends it; the red team tries to break it.
source Faraday, Experimental Researches in Electricity (First Series, read 24 Nov 1831), Phil. Trans. R. Soc. 122 (1832) 125–162 — doi:10.1098/rstl.1832.0006. Rendered, not quoted.
Three facts, no memorization:
Φ = B·A·cos θ — flux is field times area times the cosine of the loop's tilt. Spin it, θ = ωt. EMF = −dΦ/dt — voltage is the negative rate of change of flux. Lenz — that minus sign points the induced current so its own field opposes the change; reverse it and you build a perpetual-motion machine.
Live, for the current loop (analytic vs. central-difference of Φ):
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Faraday, 1831: move a magnet, get a current from nothing but change. His minus sign is Lenz's — the induced current always opposes the flux that births it, so the work you put in spinning the loop is exactly the electrical energy you take out. No free lunch; energy conserved.
Ampère showed a current makes a field; Faraday shows a changing field makes a current. Together they close the electromagnetic loop — the seed Maxwell later grew into light. Each sphere is the next one's premise: this one hands the-ampere-law its returning half.
The blue team's live check: re-derive the induced EMF straight from the flux by finite difference and confirm it equals −dΦ/dt, that the Lenz sign opposes, and that a constant flux gives zero. If red drops the sign, this badge is where it shows.
Feed the machine a loop: field B (tesla), area A (m²), and spin rate ω (rad/s). As the loop turns, the flux threading it is
Φ(t) = B·A·cos(ω t)
A flat-on loop (θ=0) catches all of B·A; edge-on (θ=90°) catches none. The rate that flux changes as it swings between those — not the flux itself — is what the engine turns into voltage below.
Left: the loop turning in field B. Right: flux Φ (cyan) and the induced EMF (green) it drives — note the EMF peaks exactly where Φ is changing fastest, and is zero where Φ is flat.
Every number is computed live from Φ = B·A·cos ωt and EMF = −dΦ/dt — never looked up.
What the machine produces, proven: for a loop spun at ω the induced voltage is EMF = B·A·ω·sin(ω t) = −dΦ/dt, matching a finite-difference derivative of the flux to 1e-9. The sign is negative (Lenz), a steady flux yields zero, and doubling ω doubles the peak. That is a generator.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
The honest statement is Maxwell's ∮E·dl = −d/dt∬B·dA plus the motional qv×B term. The flux rule is a true, extremely useful special case — not the whole of induction. The panel spins a rigid loop, exactly where the rule is exact.
"EMF is proportional to the flux." Cut. It is proportional to the rate of change of flux. A huge steady field through the loop induces exactly zero — the engine shows dΦ/dt=0 ⇒ EMF=0.
"The minus sign is just bookkeeping." Cut. Drop it and the induced current reinforces its own cause — free energy. Lenz's sign is the first law of thermodynamics enforcing itself; window 6 deletes it and the witness catches the lie.
"Faraday wrote EMF = −dΦ/dt." Kept, corrected. He described the experiments; the symbolic law and the minus sign are Neumann, Lenz and Maxwell. The discovery is his; the equation is the lineage's.
The red team's move: drop the negative sign so EMF = +dΦ/dt. Now the induced current aids the change that made it — a perpetual-motion machine. The blue team's witness (window 7) is watching.
Flip the sign and the induced EMF now agrees with dΦ/dt instead of opposing it — the witness recomputes, sees the Lenz check invert and the finite-difference residual blow up, and turns red. Nothing is faked; the attack is real and it is caught.