Consider an electrically conducting fluid (i.e., a plasma). Ampère's law, ignoring the Maxwell displacement current term, states

where B is the magnetic field, is the permeability of free space, and J is the current density. Ohm's law, including the induction electric field, is

where is the electrical conductivity, E is the electric field, and v is the fluid velocity. Finally, Faraday's law states that

Taking the curl of (2) gives

Plugging in (1) and (3),

Now, if is constant, it can be pulled out, giving

Now define the magnetic diffusivity

and use the Maxwell equation gives

to obtain

Plugging (7) and (9) into (6) gives


This is the dynamo equation, also known as the hydromagnetic equation. From this equation, it can be shown that fluid motions cannot generate an exact dipole field or any other field with rotational symmetry, a result known as Cowling's theorem.
For incompressible plasmas (note that in reality, most plasmas are compressible), conservation of mass requires that the velocity be divergenceless, so

Using the vector identity

and plugging (12) and (13) into (11) then gives

so the resulting equation is

These terms physically correspond to



In order for the field to be nondecaying, the magnetic Reynolds number

must be larger than a critical value . The frozen flux approximation assumes that the magnetic field is not changing in space, so

and