In magnetohydrodynamics, the induction equation 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.
Maxwell's equations describing the Faraday's and Ampere's laws read:
where: The displacement current can be neglected in a plasma as it is negligible compared to the current carried by the free charges.
The only exception to this is for exceptionally high frequency phenomena: for example, for a plasma with a typical electrical conductivity of 107 mho/m, the displacement current is smaller than the free current by a factor of 103 for frequencies below 2×1014 Hz.
The electric field can be related to the current density using the Ohm's law:
, yields the induction equation for an electrically resistive fluid:
is the magnetic diffusivity (in the literature, the electrical resistivity, defined as
, is often identified with the magnetic diffusivity).
[1] If the fluid moves with a typical speed
and a typical length scale
The ratio of these quantities, which is a dimensionless parameter, is called the magnetic Reynolds number:
For a fluid with infinite electric conductivity,
, the first term in the induction equation vanishes.
This is equivalent to a very large magnetic Reynolds number.
For example, it can be of order 109 in a typical star.
So, the induction equation for an ideal conductive fluid such as most astrophysical plasmas is
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.
More generally, the equation for the perfectly-conducting limit applies in regions of large spatial scale rather than infinite electric conductivity, (i.e.,
), as this also makes the magnetic Reynolds number very large such that the diffusion term can be neglected.
For very small magnetic Reynolds numbers, the diffusive term overcomes the convective term.
For example, in an electrically resistive fluid with large values of
, the magnetic field is diffused away very fast, and the Alfvén's Theorem cannot be applied.
It is common to define a dissipation time scale