  We generalise the relativistic expression of Ohm's law by studying a
multi-fluid system of charged species using the 1+3 covariant formulation of
general relativistic electrodynamics. This is done by providing a fully
relativistic, fully nonlinear propagation equation for the spatial component of
the electric 4-current. Our analysis proceeds along the lines of the
non-relativistic studies and extends previous relativistic work on cold
plasmas. Exploiting the compactness and transparency of the covariant
formalism, we provide a direct comparison with the standard Newtonian versions
of Ohm's law and identify the relativistic corrections in an unambiguous way.
The generalised expression of Ohm's law is initially given relative to an
arbitrary observer and for a multi-component relativistic charged medium. Then,
the law is written with respect to the Eckart frame and for a hot two-fluid
plasma with zero total charge. Finally, we apply our analysis to a cold
proton-electron plasma and recover the well known magnetohydrodynamic
expressions. In every step, we discuss the approximations made and identify
familiar effects, like the Biermann-battery and the Hall effect.
