Faraday's law (electromagnetism)

In electromagnetism Faraday's law of magnetic induction states that a change in magnetic flux generates an electromotive force. The law is named after the English scientist Michael Faraday.

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Magnetic flux. The flux through surface S₁ (yellow) is equal to the flux through surface S₂ (gray). Both surfaces have the contour C (red) as boundary.

The magnetic flux Φ through a surface S is defined as the surface integral

$\Phi \equiv \iint _{S}\mathbf {B} \cdot d\mathbf {S} ,$

where dS is a vector normal to the infinitesimal surface element dS and dS is of length dS. The dot stands for the inner product between the magnetic induction B and dS. In vacuum the magnetic induction B is proportional to the magnetic field H. (In SI units: B = μ₀ H with μ₀ the magnetic constant of the vacuum; in Gaussian units: B = H.)

The flux through two surfaces that together form a closed surface is equal because of Gauss' law. Indeed, in the figure on the right the surfaces S₁ and S₂, which have the boundary C in common, form together a closed surface. Hence Gauss' law states that

$0=-\iint _{S_{1}}\mathbf {B} \cdot d\mathbf {S} +\iint _{S_{2}}\mathbf {B} \cdot d\mathbf {S} ,$

where the minus sign of the first term is due to the fact that the flux is into the volume enveloped by the two surfaces. It follows that

$\Phi =\iint _{S_{1}}\mathbf {B} \cdot d\mathbf {S} =\iint _{S_{2}}\mathbf {B} \cdot d\mathbf {S} ,$

and that the magnetic flux Φ can be computed with respect to any surface that has C as boundary.

The electromotive force (EMF) is defined as

${\mathcal {E}}\equiv \oint _{C}\mathbf {E} \cdot d\mathbf {l} ,$

where the electric field E is integrated around the closed path C.

Faraday's law of magnetic induction relates the EMF $\scriptstyle {\mathcal {E}}$ to the time derivative of the magnetic flux, it reads:

${\mathcal {E}}=-k{\frac {d\Phi }{dt}}\quad {\textrm {with}}\quad k={\begin{cases}1\quad &{\hbox{SI units}}\\1/c&{\hbox{Gaussian units}},\end{cases}}$

where c is the speed of light. If C is a conducting loop, then under influence of the EMF a current i_ind will run through it. The minus sign in Faraday's law has the consequence that the magnetic field generated by i_ind opposes the change in Φ this phenomenon is known as Lenz' law. If the surface S is constant the change in Φ is solely due to a change in B.

Faraday's law forms the theoretical basis of the electric motor, the dynamo, and the electric generator.

Connection to Lorentz force

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Conducting bar moves to the right on conducting rails in homogeneous field B. Surface grows with v. Lorentz force is downward and gives an electric current i_ind in clockwise direction around the closed circuit.

It is instructive to derive Faraday's law from the Lorentz force for a special case. To that end an experimental setup is depicted in the figure on the right. A conducting bar of length a moves in positive x direction over two parallel pairs of conducting rails. The speed v of the bar is constant. The whole setup is placed in a homogeneous magnetic field B = μ₀ H (we are using SI units) that points toward the reader, i.e., in positive z-direction. Homogeneity means that everywhere in the plane of drawing a magnetic field of the same strength points toward the reader.

The magnetic field is constant, but the surface S increases linearly in time

$S=xa\Longrightarrow {\frac {dS}{dt}}={\frac {dx}{dt}}a\equiv va.$

The vector B points in positive z-direction and so does the normal to S,

${\begin{aligned}{\frac {d}{dt}}\iint _{S}\mathbf {B} \cdot d\mathbf {S} &={\frac {d}{dt}}\iint _{S}BdS\\&=B{\frac {d}{dt}}\iint _{S}dS=B{\frac {dS}{dt}}=Bva.\end{aligned}}$

The time-dependence of the growing flux is solely due to the time-dependence of the growth of the surface. The time derivative of the magnetic flux in the figure is positive and equal to

${\frac {d\Phi }{dt}}=Bva.\qquad \qquad \qquad \qquad (1)$

The Lorentz force (in SI units) is the cross product,

$\mathbf {F} _{\mathrm {Lor} }=\mathbf {v} \times \mathbf {B} =-\mathbf {e} _{y}vB.$

This constant force acts over the length a of the moving bar and gives the electromotive force (EMF) equal to force times path length,

${\mathcal {E}}=-vBa\qquad \qquad \qquad \qquad (2).$

From equations (1) and (2) follows

${\mathcal {E}}=-{\frac {d\Phi }{dt}},$

which indeed is Faraday's law of magnetic induction.

The EMF gives a current i_ind in the direction of the Lorentz force, i.e., in a clockwise direction in the plane of drawing (the x-y plane). By Biot-Savart's law a current gives a circular magnetic field. The current i_ind has a tangent vector in the x-y plane that is in negative z-direction. Hence B_ind opposes B, in accordance with Lenz' law.