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The magnetic diffusion equation describes how non-uniformities in a magnetic field will be ironed out. The equation is derived by combining Ohm's, Ampere's and Faraday's laws. These equations deal with the current density J, the electric field E and the magnetic field B. The permeability μ and the electrical conductivity σ also appear.

Ohm's Law  JE
Ampere's Law  curl.BJ

We may proceed by substituting the expression for J in Ampere's law into Ohm's law.

(1/μ)curl.BE

This provides an expression for the electric field which can now be inserted into Faraday's Law.

(1/σμ)curl(curl.B)=-δB/δt
The curl of the curl of B may now be restated using a vector identity and remembering that the divergence of the magnetic field is zero. The result is
(1/σμ)div2B=-δB/δt
It is apparent that this equation is a diffusion equation since it states that the second spatial derivative is proportional to the first time derivative of some physical vector quantity (i.e. it is a parabolic differential equation). The magnetic diffusivity Dm is given by (1/σμ). The time taken for the irregularity in the magnetic field to diffuse away is known as the resistive diffusion time, tdiff. It is given by L2/Dm where L is the dimension of the field irregulartiy. Rewriting the magnetic diffusion equation in one-dimension-
Dmδ2B/δx2=-δB/δt

Applications
A sunspot has a radius of the order 107 m and has a magnetic diffusivity of about 103 m2s-1. Thus the time it will take for this field to diffuse away is 1011 seconds or 3,000 years.

The magnetic diffusion equation is also used in plasma physics in order to determine how the magnetic field in a plasma will evolve.

Note: This derivation assumed a uniform conductivity.

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