\[- \frac{\partial u}{\partial t} + \mathbf{\nabla} \mathbf{S} + \mathbf{E} \cdot \mathbf{J}\]
Where
\[u\]
is the energy density of the field, \[\mathbf{S}\]
is the energy flux , and \[\mathbf{E}, \mathbf{J}\]
are the electric field and current density respectively.This equation can be expressed entirely in terms of the electric and magnetic fields
\[\mathbf{E}, \mathbf{H}\]
.
Write Maxwell's equation \[\mathbf{\nabla} \times \mathbf{H} = \epsilon \frac{\partial \mathbf{E}}{\partial t} + \mathbf{J}\]
as \[ \mathbf{J} = \mathbf{\nabla} \times \mathbf{H} - \epsilon \frac{\partial \mathbf{H}}{\partial t} \]
and take tha dot product of \[\mathbf{E}\]
with both sides.\[ \mathbf{E} \cdot \mathbf{J} =\mathbf{E} \cdot (\mathbf{\nabla} \times \mathbf{H}) - \epsilon \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} \]
Use in this equation that
\[\mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{2} \frac{\partial; }{\partial t} ( \mathbf{E} \cdot \mathbf{E})\]
, obtaining\[ \mathbf{E} \cdot \mathbf{J} =\mathbf{E} \cdot (\mathbf{\nabla} \times \mathbf{H}) - \epsilon \frac{1}{2} \frac{\partial; }{\partial t} ( \mathbf{E} \cdot \mathbf{E}) \]
Use the identity
\[\mathbf{\nabla} \cdot (\mathbf{H} \times \mathbf{E}) = \mathbf{E} \cdot (\mathbf{\nabla} \times \mathbf{H}) - \mathbf{H} \cdot (\mathbf{\nabla} \times \mathbf{E})\]
rearranged as
\[\mathbf{E} \cdot (\mathbf{\nabla} \times \mathbf{H}) = \mathbf{\nabla} \cdot (\mathbf{H} \times \mathbf{E}) + \mathbf{H} \cdot (\mathbf{\nabla} \times \mathbf{E})\]
so that
\[ \mathbf{E} \cdot \mathbf{J} =\mathbf{E} \cdot (\mathbf{\nabla} \times \mathbf{H}) - \epsilon \frac{1}{2} \frac{\partial; }{\partial t} ( \mathbf{E} \cdot \mathbf{E}) \]
becomes
\[ \mathbf{E} \cdot \mathbf{J} =\mathbf{\nabla} \cdot (\mathbf{H} \times \mathbf{E}) + \mathbf{H} \cdot (\mathbf{\nabla} \times \mathbf{E}) - \epsilon \frac{1}{2} \frac{\partial; }{\partial t} ( \mathbf{E} \cdot \mathbf{E}) \]
Using another of the Maxwell equations
\[\mathbf{\nabla} \times \mathbf{E} =- \frac{\partial \mathbf{H}}{\partial t} \]
gives\[\mathbf{H} \cdot (\mathbf{\nabla} \times \mathbf{E})= -\mathbf{H} \cdot \frac {\partial \mathbf{H}}{\partial t} = -\frac{1}{2} \frac{\partial}{\partial t} (\mathbf{H} \cdot \mathbf{H} ) \]
Hence
\[ \mathbf{E} \cdot \mathbf{J} =\mathbf{\nabla} \cdot (\mathbf{H} \times \mathbf{E}) -\frac{\partial}{\partial t} ( \frac{1}{2} ( \mathbf{H} \cdot \mathbf{H}) + \epsilon \frac{1}{2} \frac{\partial }{\partial t} ( \mathbf{E} \cdot \mathbf{E}) \]