structural:reformatting
Conductive fluids

In a conductive fluid, such as a plasma, there is a similar effect. Consider a fluid moving with the velocity v {\displaystyle \mathbf {v} } in a magnetic field B {\displaystyle \mathbf {B} }. The relative motion induces an electric field E {\displaystyle \mathbf {E} } which exerts electric force on the charged particles giving rise to an electric current J {\displaystyle \mathbf {J} }. The equation of motion for the electron gas, with a number density n e {\displaystyle n_{e}}, is written as m e n e d v e d t = − n e e E + n e m e ν (v i − v e) − e n e v e × B, {\displaystyle m_{e}n_{e}{d\mathbf {v} _{e} \over dt}=-n_{e}e\mathbf {E} +n_{e}m_{e}\nu (\mathbf {v} _{i}-\mathbf {v} _{e})-en_{e}\mathbf {v} _{e}\times \mathbf {B},}

where e {\displaystyle e}, m e {\displaystyle m_{e}} and v e {\displaystyle \mathbf {v} _{e}} are the charge, mass and velocity of the electrons, respectively. Also, ν {\displaystyle \nu } is the frequency of collisions of the electrons with ions which have a velocity field v i {\displaystyle \mathbf {v} _{i}}. Since, the electron has a very small mass compared with that of ions, we can ignore the left hand side of the above equation to write σ (E + v × B) = J, {\displaystyle \sigma (\mathbf {E} +\mathbf {v} \times \mathbf {B})=\mathbf {J},}

where we have used the definition of the current density, and also put σ = n e e 2 ν m e {\displaystyle \sigma ={n_{e}e^{2} \over \nu m_{e}}} which is the electrical conductivity. This equation can also be equivalently written as E + v × B = ρ J, {\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =\rho \mathbf {J},} where ρ = σ − 1 {\displaystyle \rho =\sigma ^{-1}} is the electrical resistivity. It is also common to write η {\displaystyle \eta } instead of ρ {\displaystyle \rho } which can be confusing since it is the same notation used for the magnetic diffusivity defined as η = 1 / μ 0 σ {\displaystyle \eta =1/\mu _{0}\sigma }.