Magnetic Flux - Magnetic Flux Through An Open Surface

Magnetic Flux Through An Open Surface

While the magnetic flux through a closed surface is always zero, the magnetic flux through an open surface need not be zero and is an important quantity in electromagnetism. For example, a change in the magnetic flux passing through a loop of conductive wire will cause an electromotive force, and therefore an electric current, in the loop. The relationship is given by Faraday's law:

where

is the EMF,

Φ_B is the magnetic flux through the open surface Σ,

∂Σ is the boundary of the open surface Σ; note that the surface, in general, may be in motion and deforming, and so is generally a function of time. The electromotive force is induced along this boundary.

dℓ is an infinitesimal vector element of the contour ∂Σ,

v is the velocity of the boundary ∂Σ,

E is the electric field,

B is the magnetic field.

The two equations for the EMF are, firstly, the work per unit charge done against the Lorentz force in moving a test charge around the (possibly moving) surface boundary ∂Σ and, secondly, as the change of magnetic flux through the open surface Σ. This equation is the principle behind an electrical generator.