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Mathematics for Physics

61 Induction

A process in which a changing magnetic flux through a loop produces an electromotive force that is used to generate electric current from magnetic change, where magnetic flux is the surface integral of \(\mathbf{B}\).

Faraday’s law in integral form. The electromotive force around a loop equals minus the rate of change of magnetic flux through any surface bounded by the loop. This principle is used to compute the induced emf from a known flux history.

Faraday’s law in integral form is

\[ \oint_{\partial S} \mathbf{E}\cdot d\mathbf{r} = -\dfrac{\partial}{\partial t}\displaystyle\iint_{S} \mathbf{B}\cdot\hat{\mathbf{n}}\, dA \]

where

  • \(S\) is a surface spanning the loop \(\partial S\).
  • \(\mathbf{E}\) is the electric field along the loop.
  • \(\mathbf{B}\) is the magnetic field.
  • \(\hat{\mathbf{n}}\) is the unit normal to \(S\).
  • \(t\) is time.

Faraday’s law in differential form. In differential form a changing magnetic field produces a circulating electric field. This principle is used to write Faraday’s law as a local Maxwell equation.

Faraday’s law in differential form is

\[ \nabla \times \mathbf{E} = -\dfrac{\partial\mathbf{B}}{\partial t} \]

where

  • \(\nabla\times\) is the curl.
  • \(\mathbf{E}\) is the electric field.
  • \(\mathbf{B}\) is the magnetic field.
  • \(t\) is time.

The emf of a fixed loop. For a loop of fixed area in a uniform field the emf is minus the area times the rate of change of \(B\). This principle is used to compute the emf of a loop in a ramping laboratory field.

The emf of a fixed loop in a uniform field is

\[ \mathcal{E} = -A\dfrac{dB}{dt} \]

where

  • \(\mathcal{E}\) is the induced emf.
  • \(A\) is the area of the loop.
  • \(B\) is the magnetic field through the loop.
  • \(t\) is time.

Note: Also called electromagnetic induction. Also called Faraday induction.