A. F. Peterson:
Notes on Electromagnetic Fields & Waves
11/04
Fields & Waves Note #20
Energy and the Poynting Vector
Objectives:
Motivate and develop expressions for the instantaneous and
timeaverage Poynting vectors. Use the Poynting vectors to characterize the
power carried by an electromagnetic wave for several examples involving
lossless and lossy materials.
Just as a static electric or magnetic field acts as a store of potential energy, a timevarying
electromagnetic wave carries energy with it.
In this note, an expression for the power
carried by an electromagnetic wave is derived, and used to investigate several applications.
The Poynting Theorem
Suppose that the electric field and magnetic field are explicit functions of time, given by
exyzt
(,,,)
and
hxyzt
(,,, ), respectively, and that the medium of interest is characterized by
permittivity
e
, permeability
m
, and conductivity
s
. To investigate the power carried by an
electromagnetic wave, we begin with a mathematical derivation based on several equations.
The following relation is a vector identity, meaning that it is satisfied by any vector functions
e
and
h
:
—
∑
¥=
∑
—¥ 
∑
—¥
()
eh h
ee
h
(20.1)
However, the fields satisfy Faraday’s Law
—¥ =
∂
∂
e
h
t
(20.2)
and Ampere’s Law
=
∂
∂
+
h
e
t
e
es
(20.3)
which can be substituted into (20.1) to obtain
—
∑
∑

∂
∂
Ê
Ë
Á
ˆ
¯
˜

∑
∂
∂
+
Ê
Ë
Á
ˆ
¯
˜
=
∑
∂
∂

∑
∂
∂

∑
h
t
e
e
t
e
h
h
t
e
e
t
me
mes
(20.4)
The expression in (20.4) can be rewritten after recognizing that
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Notes on Electromagnetic Fields & Waves
11/04
1
2
1
2
1
2
d
dt
hh
h
dh
dt
dh
dt
dh
dt
()
∑
=
∑
+
∑
=
∑
(20.5)
to produce
—
∑
¥=

∂
∂
∑

∂
∂
∑

∑
=
∂
∂

∂
∂

eh
t
t
ee
t
h
t
me
s
22
2
(20.6)
We are concerned with the power stored by the field in some region of space and the power
flowing out of that region.
Therefore, we integrate both sides of (20.6) over a volume
V
:
—
∑

∂
∂

∂
∂

Ï
Ì
Ó
¸
˝
˛
ÚÚÚ
ÚÚÚ
d
v
t
h
t
d
v
VV
2
(20.7)
The lefthand side of this expression can be simplified using another vector identity known
as the
divergence theorem
:
—
∑
=
∑
ÚÚÚ
ÚÚ
Adv
A dS
VS
(20.8)
where
S
is the surface enclosing the volume
V
.
Equation (20.8) holds for any vector
A
, and
is essentially a generalization of the integrationbyparts formula to the vector situation.
Therefore, the lefthand side of (20.7) can be rewritten as
—
∑
¥
∑
ÚÚÚ
ÚÚ
d
v
ehd
S
(20.9)
By combining this with the righthand side of (20.7), we obtain
S
t
h
t
d
v
SV
¥
∑
∂
∂

∂
∂

Ï
Ì
Ó
¸
˝
˛
ÚÚ
ÚÚÚ
2
(20.10)
This result is known as the
Poynting Theorem
. It was developed in 1883 by J. H.
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 Spring '08
 CITRIN
 Energy, Electromagnet, Magnetic Field, Poynting vector, A. F. Peterson

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