4  Maxwell’s Equations

The EM fields 2.1 are solutions of Maxwell’s equations

\[ \curl \mathbf h - \partial_t \mathbf d = \mathbf j \tag{4.1}\]

\[ \curl \mathbf e + \partial_t \mathbf b = \mathbf 0 \tag{4.2}\]

\[ \divergence \mathbf b = 0 \tag{4.3}\]

\[ \divergence \mathbf d = \rho_E \tag{4.4}\]

In the form presented here, Maxwell’s equations are an uncoupled set of ordinary differential equations.

4.1 Constitutive equations

The goal is to couple these equations. This can be achieved with the use of the constitutive equations, which are

\[ \begin{align} \mathbf d & = \varepsilon \mathbf e \\ \mathbf b & = \mu \mathbf h \end{align} \tag{4.5}\]

Generally, the linear parameters \(\varepsilon, \mu\) are rank-2 tensors represented as 3-by-3 matrices.

4.2 Ohm’s law

In an electrically conductive medium, any electric field gives rise to an electric current. This current is expressed by its current density as

\[ \mathbf j = \sigma \mathbf e, \tag{4.6}\]

where \(\sigma\) is a rank-2 tensor.

This tensor can be represented in matrix form as

\[ \sigma = \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{pmatrix}. \]

Tensors like introduced here cause anisotropy, i.e., the material properties have different values across different spatial directions.

A typical observation would be the deviation of the induced current density from the direction of the driving electric field.

4.2.1 Remarks about anisotropy

The rank-2 tensor of electrical conductivity \(\tilde\sigma\) may be represented in matrix form as

\[ \hat\sigma = \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{pmatrix}. \]

Any real symmetric (n-by-n) matrix \(A\) can be diagonalized (principal axis theorem), such that

\[ D_A = S^\top A S \]

is a diagonal matrix, and \(S\) is an orthogonal matrix.

Interpreting the matrix \(A\) as a linear map in \(\mathbb {R} ^3\), then the matrix \(S\) can be thought of as a transformation to the new basis. Between the old and new coordinates there is the relation \(\mathbf {x}=S \boldsymbol {\xi }\). The action of the matrix \(A\) in the new coordinate system is taken over by the diagonal matrix \(D_{A}\).

After transformation of the tensor in diagonal form, we have \[ \tilde\sigma = \begin{pmatrix} \sigma_{xx} & 0 & 0 \\ 0 & \sigma_{yy} & 0 \\ 0 & 0 & \sigma_{zz} \end{pmatrix}. \]

If \(\sigma_{xx} = \sigma_{yy} = \sigma_{zz} = \sigma\), then \[ \tilde\sigma = \sigma \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \sigma. \] In this case, the conductivity does not depend on the spatial direction and hence is labelled isotropic.