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Mathematics for Physics

59 Induced Emf

A potential difference generated in a conducting loop by changing magnetic flux that is used to measure the work per unit charge done by the induced electric field, where electromotive force is that work per unit charge around the loop.

The definition of emf. The induced emf is the line integral of the electric field around the loop. This principle is used to convert Faraday’s law into a circuit quantity.

The definition of emf is

\[ \mathcal{E} = \oint\mathbf{E}\cdot d\mathbf{l} \]

where

  • \(\mathcal{E}\) is the electromotive force.
  • \(\mathbf{E}\) is the electric field.
  • \(d\mathbf{l}\) is a displacement along the loop.

Faraday’s flux rule. Faraday’s law equates that emf to minus the rate of change of magnetic flux. This principle is used to compute the induced emf from \(\Phi_{B}\).

Faraday’s flux rule is

\[ \mathcal{E} = -\dfrac{d\Phi_{B}}{dt} \]

where

  • \(\mathcal{E}\) is the induced emf.
  • \(\Phi_{B}\) is the magnetic flux through the loop.
  • \(t\) is time.

Motional emf. A conductor moving in a magnetic field develops a motional emf. This principle is used to compute the emf of a sliding bar on rails.

The motional emf is

\[ \mathcal{E} = \oint\bigl(\mathbf{v}\times\mathbf{B}\bigr)\cdot d\mathbf{l} \]

where

  • \(\mathbf{v}\) is the velocity of a length element of the conductor.
  • \(\mathbf{B}\) is the magnetic field.
  • \(d\mathbf{l}\) is a displacement along the conductor.