Formulations in DC conduction

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QuickField is able to calculate the distribution of electric current in systems of conductors.
The problems of current distribution are described by the Poisson's equation for scalar electric potential U.

The equation for planar case is \[ \frac{\partial}{\partial x} \left( σ_x \frac{\partial U}{\partial x} \right) + \frac{\partial}{\partial y} \left( σ_y\frac{\partial U}{\partial y} \right) = 0, \]

and for axisymmetric case is \[ \frac{1}{r} \frac{\partial}{\partial r} \left( r σ_r \frac{\partial U}{\partial r} \right) + \frac{\partial}{\partial z} \left( σ_z \frac{\partial U}{\partial z} \right) = 0, \]
where components of electric conductivity tensor σ_x, σ_y or σ_z, σ_r are constant within each model block.

The electric current density j can be obtained from the equation
j = σ·grad U.