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.

CC Image
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.

Note

(To be continued)