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PHYS809 Class 35 Notes
Energy density and energy flux
In looking for solutions to Maxwell’s equation, we found it convenient to consider physical quantities as
the real parts of complex expressions, e.g.
( )
(
)
0
,
Re
,
i
t
t
e
ω
⋅ 
=
k x
E x
E
(35.1)
where the complex vector
E
0
contains amplitude and phase information. This approach works because
Maxwell’s equations are linear in the fields. However quantities such as the energy density and the
Poynting vector are quadratic in the fields and some care is need in evaluating them. A simplification
arises because we are usually interested in the time averages of such quantities. Consider a product
such as
( ) ( )
,
,
.
t
t
⋅
E x
D x
Since the real part of a complex number
z
is
( )
*
2,
z
z
+
we have
( ) ( )
(
)
(
)
(
)
(
)
(
)
(
)
*
*
0
0
0
0
2
2
*
*
*
*
0
0
0
0
0
0
0
0
1
,
,
4
1
.
4
i
t
i
t
i
t
i
t
i
t
i
t
t
t
e
e
e
e
e
e
⋅ 

⋅ 
⋅ 

⋅ 
⋅ 

⋅ 
⋅
=
+
⋅
+
=
⋅
+
⋅
+
⋅
+
⋅
k x
k x
k x
k x
k x
k x
E x
D x
E
E
D
D
E D
E
D
E D
E
D
(35.2)
On timeaveraging of a product, the harmonically varying terms go to zero, and we have
( ) ( )
( )
*
*
*
0
0
0
0
0
0
1
1
,
,
Re
.
4
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 Fall '11
 MacDonald
 Energy

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