In magnetohydrodynamics, the induction equation is a partial differential equation that relates the magnetic field and velocity of an electrically conductive fluid such as a plasma. It can be derived from Maxwell's equations and Ohm's law, and plays a major role in plasma physics and astrophysics, especially in dynamo theory.

Maxwell's equations describing the Faraday's and Ampere's laws read: $${\displaystyle \nabla \times \mathbf {E} =-{\partial \mathbf {B} \over \partial t},}$$ and $${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} ,}$$ where:

The displacement current can be neglected in a plasma as it is negligible compared to the current carried by the free charges. The only exception to this is for exceptionally high frequency phenomena: for example, for a plasma with a typical electrical conductivity of 10⁷ mho/m, the displacement current is smaller than the free current by a factor of 10³ for frequencies below 2×10¹⁴ Hz.

The electric field can be related to the current density using the Ohm's law: $${\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =\mathbf {J} /\sigma }$$ where

Combining these three equations, eliminating ${\displaystyle \mathbf {E} }$ and ${\displaystyle \mathbf {J} }$, yields the induction equation for an electrically resistive fluid: $${\displaystyle {\partial \mathbf {B} \over \partial t}=\eta \nabla ^{2}\mathbf {B} +\nabla \times (\mathbf {v} \times \mathbf {B} ).}$$

Here ${\displaystyle \eta =1/\mu _{0}\sigma }$ is the magnetic diffusivity (in the literature, the electrical resistivity, defined as ${\displaystyle 1/\sigma }$, is often identified with the magnetic diffusivity).

If the fluid moves with a typical speed ${\displaystyle V}$ and a typical length scale ${\displaystyle L}$, then $${\displaystyle \eta \nabla ^{2}\mathbf {B} \sim {\eta B \over L^{2}},\nabla \times (\mathbf {v} \times \mathbf {B} )\sim {VB \over L}.}$$

The ratio of these quantities, which is a dimensionless parameter, is called the magnetic Reynolds number: $${\displaystyle R_{m}={LV \over \eta }.}$$