The induction equation, one of the magnetohydrodynamic equations, is a partial differential equation that relates the magnetic field and velocity of an electrically conductive fluid such as a plasma. It can be derived from Maxwell's equations and Ohm's law, and plays a major role in plasma physics and astrophysics, especially in dynamo theory.
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Mathematical statement
Maxwell's equations describing the Faraday's and Ampere's laws read
and
where the displacement current has been neglected as it usually has small effects in astrophysical applications as well as in most of laboratory plasmas. Here,
Here,
If the fluid moves with a typical speed
The ratio of these quantities, which is a dimensionless parameter, is called the magnetic Reynolds number:
Perfectly conducting limit
For a fluid with infinite electric conductivity,
This is taken to be a good approximation in dynamo theory, used to explain the magnetic field evolution in the astrophysical environments such as stars, galaxies and accretion discs.
Diffusive limit
For very small magnetic Reynolds numbers, the diffusive term overcomes the convective term. For example, in an electrically resistive fluid with large values of
It is common to define a dissipation time scale