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

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

  is the energy density of the field,  
 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  
\[\mathbf{\nabla} \times \mathbf{H} = \epsilon \frac{\partial \mathbf{E}}{\partial t} + \mathbf{J}\]
\[ \mathbf{J} = \mathbf{\nabla} \times \mathbf{H} - \epsilon \frac{\partial \mathbf{H}}{\partial t} \]
  and take tha dot product of  
  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
\[ \mathbf{E} \cdot \mathbf{J} =\mathbf{E} \cdot (\mathbf{\nabla} \times \mathbf{H}) - \epsilon \frac{1}{2} \frac{\partial; }{\partial t} ( \mathbf{E} \cdot \mathbf{E}) \]

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

Using another of the Maxwell equations  
\[\mathbf{\nabla} \times \mathbf{E} =- \frac{\partial \mathbf{H}}{\partial t} \]
\[\mathbf{H} \cdot (\mathbf{\nabla} \times \mathbf{E})= -\mathbf{H} \cdot \frac {\partial \mathbf{H}}{\partial t} = -\frac{1}{2} \frac{\partial}{\partial t} (\mathbf{H} \cdot \mathbf{H} ) \]

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

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