= Lenz's law
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<Lenz's law> expresses the minus sign in <Faraday's law of induction>: the induced <electromotive force> drives a passive resistive loop's <electric current> in the direction whose own <magnetic field> opposes the imposed change in <magnetic flux>. \b[It opposes the change of flux, not invariably the external magnetic field.] Choose an <oriented surface> and its positive boundary direction first, find the sign of $d\Phi_{\mathrm{ext}}/dt$, and then use the <right-hand rule> to translate the signed <electric current> into a physical direction. https://openstax.org/books/university-physics-volume-2/pages/13-2-lenzs-law[OpenStax University Physics 2, §13.2: current direction and changing flux]{ref}.
In the fixed-loop example in <Faraday's law of induction>, out-of-page <magnetic flux> increases at $0.0060\,\mathrm{Wb/s}$. With counterclockwise positive, $I=-0.0030\,\mathrm A$ is clockwise, producing an in-page <magnetic field> through the loop. Reverse the ramp to $B(t)=0.40\,\mathrm T-(0.30\,\mathrm{T/s})t$ on the same one-second interval: now $\mathcal E=+0.0060\,\mathrm V$ and $I=+0.0030\,\mathrm A$. The counterclockwise <electric current> produces an out-of-page <magnetic field>, supporting the decreasing external <magnetic flux>. A steady out-of-page <magnetic field> with fixed loop geometry gives no induced <electromotive force> in either direction.
For the increasing-area example in <motional electromotive force>, clockwise <electric current> runs down the right-hand rod. Its <magnetic force on a current-carrying wire> points left, opposing the imposed motion to the right. Writing $I_{\mathrm{mag}}=B\ell v/R$ for the current magnitude, the <power> required to maintain constant <speed> is
$$
P_{\mathrm{drive}}=(I_{\mathrm{mag}}\ell B)v
=\frac{(B\ell v)^2}{R}=I_{\mathrm{mag}}^2R=P_{\mathrm{Joule}}.
$$
Thus the sign agrees with <conservation of energy>: the external drive supplies the <Joule heating>. The <magnetic forces do no work> on individual charges; mechanical <work> by the drive supplies the electrical <energy>. This also explains the usefulness of <electromagnetic braking of a sliding rod> and the <terminal speed of a falling conducting loop>. https://openstax.org/books/university-physics-volume-2/pages/13-3-motional-emf[OpenStax University Physics 2, §13.3: mechanical power and resistive heating]{ref}.
The sign of induced <electromotive force> alone does not determine an arbitrary circuit's instantaneous <electric current>. Retaining constant <self-inductance> $L$ gives $L\,dI/dt+RI=-d\Phi_{\mathrm{ext}}/dt$, so a pre-existing <electric current> can persist while the induced drive opposes its change. The distinction matters in the <lossless limit of magnetic loop braking>. The resistive toy examples use $L=0$ as an approximation, not a general consequence of <Lenz's law>.
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