# Steady State Equations

## Direct Current Resistivity

A DC resistivity survey is ultimately an electromagnetic phenomenon and is therefore governed by Maxwell’s equations. However, the fact that the ground is energized with a time invariant direct current allows us to use a much simpler model.

### Deriving the DCR Equations

We can start from time domain differential form of the Ampere-Maxwell equation equation (equation (5) on Ampere-Maxwell)

where \(\mathbf{h}\) is the magnetic field, \(\mathbf{j}_{total}\) is the total current in the system, and \(\mathbf{d}\) is the electric displacement. We can divide up \(\mathbf{j}_{total}\) as follows

which states that the total current densisty can be divided into the current in the ground (\(\sigma\mathbf{e}\) due to Ohm’s law) and the current in the source wires (\(\mathbf{j}_{source}\)). Since we are in the steady state case, \(\frac{\partial \mathbf{d}}{\partial t}=0\). Using that assumption and substituting (327) into (326) we obtain

Faraday’s law is also simplified in steady state. It becomes

In other words it is a potential field. In particular, this means that

where \(\phi\) is the electric potential. This allows us to write (328) as

Taking the divergence of both sides of (330) gives the governing equation for DC resistivity