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 **J**=σ**E**

Ampere's Law curl.**B**=μ**J**

Faraday's Law curl.**E**= -δ**B**/δt

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

(1/μ)curl.**B**=σ**E**

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

(1/σμ)curl(curl.**B**)=-δ**B**/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)div^{2}**B**=-δ**B**/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 D

_{m} 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, t

_{diff}. It is given by L

^{2}/D

_{m} where L is the dimension of the field irregulartiy. Rewriting the magnetic
diffusion equation in one-dimension-

D_{m}δ^{2}**B**/δx^{2}=-δ**B**/δt

**Applications**

A sunspot has a radius of the order 10^{7} m and has a magnetic diffusivity of about 10^{3} m^{2}s^{-1}. Thus the time it will take
for this field to diffuse away is 10^{11} 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.