= Faraday's law of induction
{c}
{title2=$\nabla\times\mathbf E=-\partial_t\mathbf B$}
{wiki}
= Faraday's law
{c}
{synonym}
= Faraday law
{c}
{synonym}
<Faraday's law of induction> relates a changing <magnetic field> to circulation of the <electric field>. For a fixed <oriented surface> $S$ with boundary $C=\partial S$, choose the positive direction around $C$ by the <right-hand rule>: the fingers follow $C$ while the thumb follows the chosen <unit normal>. The fixed-loop electromagnetic <electromotive force> is
$$
\mathcal E=\oint_C\mathbf E\cdot d\mathbf l
=-\frac{\partial}{\partial t}\int_S\mathbf B\cdot d\mathbf S
=-\frac{d\Phi_B}{dt},
\qquad
\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}.
$$
The final <derivative> of <magnetic flux> is an ordinary time derivative, with the <surface> held fixed. Reversing both the <unit normal> and the boundary direction reverses both signs. The loop may be an imagined path: the field law does not require a wire or an <electric current>. https://www.feynmanlectures.caltech.edu/II_17.html#Ch17-S1[Feynman lectures, II.17.1: electric-field circulation and induction]{ref}.
For a computed example, a fixed single-turn loop has <area> $A=0.020\,\mathrm{m^2}$ and out-of-page <unit normal>. Prescribe an external uniform <magnetic field> in that direction with $B(t)=0.10\,\mathrm T+(0.30\,\mathrm{T/s})t$ for $0\le t\le1.0\,\mathrm s$. Then
$$
\Phi_{\mathrm{ext}}(t)=AB(t)
=0.0020\,\mathrm{Wb}+(0.0060\,\mathrm{Wb/s})t,
\qquad
\mathcal E_{\mathrm{CCW}}=-0.0060\,\mathrm V.
$$
In a quasistatic resistive model with <electrical resistance> $R=2.0\,\Omega$, neglect the loop's <self-inductance> and its own contribution to changing <magnetic flux>. <Ohm's law> gives
$$
I_{\mathrm{CCW}}=\frac{\mathcal E_{\mathrm{CCW}}}{R}=-0.0030\,\mathrm A.
$$
The <magnetic flux> rises from $0.0020\,\mathrm{Wb}$ to $0.0080\,\mathrm{Wb}$ while the induced <electromotive force> stays constant: it measures the rate of change, rather than the amount of <magnetic flux>. The negative sign means clockwise <electric current>; its <magnetic field> through the loop points into the page, as explained by <Lenz's law>. The toy model describes the interval of steady ramping, excluding switching transients at its ends.
\Image[/faraday-changing-flux-and-lenz-current.png]
{title=<Faraday's law of induction> and <Lenz's law> for increasing out-of-page <magnetic flux>}
{description=Counterclockwise is the positive orientation. Dots denote the external out-of-page <magnetic field>, and crosses denote the in-page <magnetic field> produced by the clockwise induced <electric current>. The plots show $\Phi_{\mathrm{ext}}$ increasing from $2$ to $8\,\mathrm{mWb}$, with constant $\mathcal E=-6\,\mathrm{mV}$ and $I=-3\,\mathrm{mA}$ in the resistive approximation.}
For a moving conducting loop $C(t)$, the electromagnetic <electromotive force> instead includes the <Lorentz force> contribution from the local <velocity> $\mathbf v$ of each conductor segment:
$$
\mathcal E(t)=\oint_{C(t)}(\mathbf E+\mathbf v\times\mathbf B)\cdot d\mathbf l
=-\frac{d}{dt}\int_{S(t)}\mathbf B\cdot d\mathbf S.
$$
This form assumes a thin-wire model with a well-defined closed contour and spanning <oriented surface> that move compatibly, a <magnetic field> satisfying <Gauss's law for magnetism>, and conductor speeds small compared with the <speed of light>. Ideal sliding contacts such as those in <motional electromotive force> can be treated piecewise. Arbitrary changes of electrical connections require checking the actual contour and conductor velocities; differentiating the flux through an arbitrary chosen moving path is insufficient. The <velocity> here describes conductor motion, not the charge carriers' drift along the wire. https://www.feynmanlectures.caltech.edu/II_17.html#Ch17-S2[Feynman lectures, II.17.2: limits of the flux rule]{ref}.
A short derivation of the fixed-loop integral statement applies <Stokes theorem> to the local <curl> equation and differentiates under the fixed <surface integral>. For the moving loop, the <magnetic flux transport for a moving circuit> adds the swept-area term $-\oint_C(\mathbf v\times\mathbf B)\cdot d\mathbf l$ to $d\Phi_B/dt$. Combining these two identities gives the moving-loop statement. This derives the integral forms from the local field law; it is not a proof of that empirical law from mathematics alone.
The useful unity is geometric: changing the <magnetic field>, the enclosed <area>, or the orientation can change <magnetic flux> and hence produce an <electromotive force>. A coil's multiple turns are accounted for by <magnetic flux linkage>, rather than by multiplying a single-turn flux without checking which flux each turn receives. https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law[OpenStax University Physics 2, §13.1: flux changes and coils]{ref}.
Faraday's report read on 24 November 1831 describes opposite momentary responses when a neighboring circuit is connected and disconnected, with no continuing response while its steady <electric current> is maintained. These are concrete experimental roots of the changing-flux idea. The report predates the modern <Maxwell equations> notation used above. <Experimental Researches in Electricity, Volume 1 by Michael Faraday>[Faraday, Experimental Researches in Electricity, First Series (1831), §§10–11, reprinted in volume I]{ref}.
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