Physics equations/Faraday law

Electromagnetic induction

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

Faraday's law of induction makes use of the magnetic flux Φ_B through a hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface integral:

$\Phi _{B}=\iint \limits _{\Sigma (t)}\mathbf {B} (\mathbf {r} ,t)\cdot d\mathbf {A} \ ,$

where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B·dA is a vector dot product (the infinitesimal amount of magnetic flux). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.

When the flux changes—because B changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an EMF, ${\mathcal {E}}$, defined as the energy available from a unit charge that has travelled once around the wire loop. Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads. According to the Lorentz force law (in SI units),

$\mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$

the EMF on a wire loop is:

${\mathcal {E}}={\frac {1}{q}}\oint _{\mathrm {wire} }\mathbf {F} \cdot d{\boldsymbol {\ell }}=\oint _{\mathrm {wire} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\cdot d{\boldsymbol {\ell }}$

where E is the electric field, B is the magnetic field (aka magnetic flux density, magnetic induction), dℓ is an infinitesimal arc length along the wire, and the line integral is evaluated along the wire (along the curve the conincident with the shape of the wire).

The EMF is also given by the rate of change of the magnetic flux:

${\mathcal {E}}=-{{d\Phi _{B}} \over dt}\$,

where ${\mathcal {E}}$ is the electromotive force (EMF) in volts and Φ_B is the magnetic flux in webers. The direction of the electromotive force is given by Lenz's law.

For a tightly wound coil of wire, composed of N identical turns, each with the same Φ_B, Faraday's law of induction states that

${\mathcal {E}}=-N{{d\Phi _{B}} \over dt}$

where N is the number of turns of wire and Φ_B is the magnetic flux in webers through a single loop.

Problems and Examples

Faraday law example