Joule heating for a fluid Say you have a conductive liquid with a changing magnetic field going right through it, causing an electric current.  How exactly does the electric current travel and how could you calculate the effect of Joule heating on the liquid?
 A: You are looking for something called the generalized Ohm's law, which is given by:
$$
\mathbf{E} + \mathbf{v} \times \mathbf{B} \approx \frac{ \mathbf{j} \times \mathbf{B} }{ n \ e } - \frac{ \nabla}{ n \ e } \cdot \left( \mathcal{P}_{e} + \frac{ m_{e} }{ m_{i} } \mathcal{P}_{i} \right) + \eta \ \mathbf{j} + \frac{ m_{e} }{ n \ e^{2} } \frac{ d \mathbf{j} }{ d t } \tag{1}
$$
where $\mathbf{j}$ is the total current density, $n$ is the total number density (assuming quasi-neutrality, i.e., $n_{e} = n_{i}$), $e$ is the fundamental charge, $\mathcal{P}_{s}$ is the pressure tensor of species $s$, $m_{s}$ is the mass of species $s$ ($s$ can be $e$ for electron or $i$ for ion), and $\eta$ is the scalar electrical resistivity.
The Joule heating term is the $\eta \ \mathbf{j}$.  In linear circuits, one often assumes that Equation 1 reduces down to something akin to:
$$
\mathbf{E} \approx \eta \ \mathbf{j} \tag{2}
$$
and then one can use a relationship from Poynting's theorem which relates the rate of change of electromagnetic energy per unit volume into mechanical energy per unit volume (e.g., heat and/or particle kinetic energy).  This term is given by $\mathbf{E} \cdot \mathbf{j}$ or approximated as $\eta \ j^{2}$.
In your specific example, Equation 2 would probably include the Hall term as well, i.e., the $\mathbf{j} \times \mathbf{B}$ term.
