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 of length dS. The dot stands for the inner product between the magnetic induction B and dS.

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

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

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:

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

If C is a conductor, 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 B; this phenomenon is known as Lenz' law.

Note

(To be continued)