Ohm's Law in Continuum Form

For a resistor, Ohm’s law $U=RI$ relates the voltage $U$ to the current $I$ through the resistance $R$. Inside a continuous medium it takes a local form. It links the current density $\mathbf{J}(x)$ (current per unit area perpendicular to the flow, in A/m²) to the electric field $\mathbf{E}(x)$:

$$\mathbf{J}=\gamma \mathbf{E},$$

where $\gamma (x)>0$ is the conductivity (S/m) and $\rho =1/\gamma$ is the resistivity (Ω·m).

Under the Quasi-Static Approximation the electric field is curl-free, so it is the negative gradient of a scalar electric potential $u$ (the voltage):

$$\mathbf{E}=-\nabla u,\mathbf{J}=-\gamma \nabla u.$$

The minus sign says that current flows from high to low potential. For anisotropic media such as muscle fibres, $\gamma$ is a symmetric positive definite matrix field instead of a scalar (see Anisotropic Conductivities).

Combined with the Continuity Equation, this gives the Conductivity Equation.

References

1. J. D. Jackson (1999). Classical Electrodynamics, 3rd ed., Wiley. ISBN 978-0-471-30932-1
2. M. Cheney, D. Isaacson, J. C. Newell (1999). Electrical Impedance Tomography. SIAM Review 41(1), 85–101. doi:10.1137/S0036144598333613