The macroscopic Maxwell equations are a set of eight equations involving the components of the four fields \( \mathbf{E} \), \( \mathbf{B} \), \( \mathbf{D} \), and \( \mathbf{H} \). The four homogeneous equations can be solved formally by expressing \( \mathbf{E} \) and \( \mathbf{B} \) in terms of the scalar potential \( Φ \) and the vector potential \( \mathbf{Α} \), but the inhomogeneous equations cannot be solved until the derived fields \( \mathbf{D} \) and \( \mathbf{H} \) are known in terms of \( \mathbf{E} \) and \( \mathbf{B} \). These connections are known as constitutive relations:\( \mathbf{D} = \mathbf{D[E, B]} \)
\( \mathbf{H} = \mathbf{H[E, B]} \)
In addition, for conducting media there is the generalized Ohm’s law:
\( \mathbf{J} = \mathbf{J[E, B]} \)
The square brackets signify that the connections are not necessarily simple and may depend on past history (hysteresis), may be nonlinear, etc.
