Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Induction equation
Summary
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:
and
where:
is the electric field.
is the magnetic field.
is the electric current density.
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 , the displacement current is smaller than the free current by a factor of for frequencies below .
The electric field can be related to the current density using the Ohm's law:
where
is the velocity field.
is the electric conductivity of the fluid.
Combining these three equations, eliminating and , yields the induction equation for an electrically resistive fluid:
Here is the magnetic diffusivity (in the literature, the electrical resistivity, defined as , is often identified with the magnetic diffusivity).
If the fluid moves with a typical speed and a typical length scale , then
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 in a typical star. In this case, the fluid can be called a perfect or ideal fluid. 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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.