Continuity Equation

Conservation of electric charge is expressed by the continuity equation

$$\frac{\partial {\rho }_q}{\partial t}+\nabla \cdot \mathbf{J}=0,$$

where ${\rho }_q$ is the volume charge density and $\mathbf{J}$ the current density. Integrated over a control volume, it says that the net current leaving the volume equals the rate at which the enclosed charge decreases. It is the continuum version of Kirchhoff’s current law.

In Electrical Impedance Tomography no charge builds up inside the body and there are no interior sources, so in steady state

$$\nabla \cdot \mathbf{J}=0\text{in }\Omega .$$

All current enters and leaves through the boundary. With Ohm’s law $\mathbf{J}=-\gamma \nabla u$, this becomes the Conductivity Equation. The fact that the total injected current must be zero (see Neumann Problem) follows from integrating $\nabla \cdot \mathbf{J}=0$ over $\Omega$ with the divergence theorem.

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