In the first approach, the magnetic induction equation is derived from Ohm’s law and Maxwell’s equation. The equation provides the coupling between the flow field and the magnetic field.

In general, Ohm’s law that defines the current density is given by:

(21–7)

where is the electrical conductivity of the media. For fluid velocity field in a magnetic field , Ohm’s law takes the form:

(21–8)

From Ohm’s law and Maxwell’s equation, the induction equation can be derived as:

(21–9)

From the solved magnetic field , the current density can be calculated using Ampere’s relation as:

(21–10)

Generally, the magnetic field in a MHD problem can be decomposed into the externally imposed field and the induced field due to fluid motion. Only the induced field must be solved.

From Maxwell’s equations, the imposed field satisfies the following equation:

(21–11)

where is the electrical conductivity of the media in which field is generated. Two cases need to be considered.