## Conservation of Energy for Electromagnetic Fields in Terms of E and H

We can write the conservation of energy for an electromagnetic field as
$- \frac{\partial u}{\partial t} + \mathbf{\nabla} \mathbf{S} + \mathbf{E} \cdot \mathbf{J}$

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
$\mathbf{E} \cdot \mathbf{J} =\mathbf{\nabla} \cdot (\mathbf{H} \times \mathbf{E}) -\frac{\partial}{\partial t} ( \frac{1}{2} ( \mathbf{H} \cdot \mathbf{H}) + \epsilon \frac{1}{2} \frac{\partial }{\partial t} ( \mathbf{E} \cdot \mathbf{E})$

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