Magnetic diffusion refers to the motion of magnetic fields, typically in the presence of a conducting solid or fluid such as a plasma. The motion of magnetic fields is described by the magnetic diffusion equation and is due primarily to induction and diffusion of magnetic fields through the material. The magnetic diffusion equation is a partial differential equation commonly used in physics. Understanding the phenomenon is essential to magnetohydrodynamics and has important consequences in astrophysics, geophysics, and electrical engineering.
Equation edit
The magnetic diffusion equation (also referred to as the induction equation) is
In the case of a non-uniform conductivity the magnetic diffusion equation is
Derivation edit
Starting from the generalized Ohm's law:[1][2]
Limiting Cases edit
In some cases it is possible to neglect one of the terms in the magnetic diffusion equation. This is done by estimating the magnetic Reynolds number
Physical Condition | Dominating Term | Magnetic Diffusion Equation | Examples | |
---|---|---|---|---|
Large electrical conductivity, large length scales or high plasma velocity. | The inductive term dominates in this case. The motion of magnetic fields is determined by the flow of the plasma. This is the case for most naturally occurring plasmas in the universe. | The Sun or the core of the earth | ||
Small electrical conductivity, small length scales or low plasma velocity. | The diffusive term dominates in this case. The motion of the magnetic field obeys the typical (nonconducting) fluid diffusion equation. | Solar flares or created in laboratories using mercury or other liquid metals. |
Relation to Skin Effect edit
At low frequencies, the skin depth for the penetration of an AC electromagnetic field into a conductor is:
Examples and Visualization edit
For the limit , the magnetic field lines become "frozen in" to the motion of the conducting fluid. A simple example illustrating this behavior has a sinusoidally-varying shear flow
For the limit , the magnetic diffusion equation is just a vector-valued form of the heat equation. For a localized initial magnetic field (e.g. Gaussian distribution) within a conducting material, the maxima and minima will asymptotically decay to a value consistent with Laplace's equation for the given boundary conditions. This behavior is illustrated in the figure below.
Diffusion Times for Stationary Conductors edit
For stationary conductors with simple geometries a time constant called magnetic diffusion time can be derived.[5] Different one-dimensional equations apply for conducting slabs and conducting cylinders with constant magnetic permeability. Also, different diffusion time equations can be derived for nonlinear saturable materials such as steel.
References edit
- ^ Holt, E. H.; Haskell, R. E. (1965). Foundations of Plasma Dynamics. New York: Macmillan. pp. 429-431.
- ^ Chen, Francis F. (2016). Introduction to Plasma Physics and Controlled Fusion (3rd ed.). Heidelberg: Springer. pp. 192–194. ISBN 978-3-319-22308-7.
- ^ Landau, L. D.; Lifshitz, E. M. (2013). The Classical Theory of Fields (4th revised ed.). New York: Elsevier. ISBN 9781483293288.
- ^ Longcope, Dana (2002). "Notes on Magnetohydrodynamics" (PDF). Montana State University - Department of Physics. Retrieved 30 April 2019.
- ^ Brauer, J. R. (2014). Magnetic Actuators and Sensors (2nd ed.). Hoboken NJ: Wiley IEEE Press. ISBN 978-1-118-50525-0.