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Adapted from notes by Prof. Stuart A. Long 1 Notes 5 Poynting’s Theorem ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018
Adapted from notes by Prof. Stuart A. Long
1
Notes 5 Poynting’s Theorem
ECE 3317 Applied Electromagnetic Waves
Prof. David R. Jackson Fall 2018
Poynting Theorem
The Poynting theorem is one on the most important in EM theory. It tells us the power flowing in an electromagnetic field.
John Henry Poynting (1852-1914)
John Henry Poynting was an English physicist. He was a professor of physics at Mason Science College (now the University of Birmingham) from 1880 until his death. He was the developer and eponym of the Poynting vector, which describes the direction and magnitude of electromagnetic energy flow and is used in the Poynting theorem, a statement about energy conservation for electric and magnetic fields. This work was first published in 1884. He performed a measurement of Newton's gravitational constant by innovative means during 1893. In 1903 he was the first to realize that the Sun's radiation can draw in small particles towards it. This was later coined the Poynting-Robertson effect. In the year 1884 he analyzed the futures exchange prices of commodities using statistical mathematics.
(Wikipedia) 2
Poynting Theorem (cont.)
t
t
∂∇× = −
∂∂
∇× = +∂
BE
DH J
From these we obtain:
( )
( )t
t
∂⋅ ∇× = − ⋅
∂∂
⋅ ∇× = ⋅ + ⋅∂
BH E H
DE H J E E
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Subtract, and use the following vector identity:
( ) ( ) ( )⋅ ∇× − ⋅ ∇× = ∇ ⋅ ×H E E H E H
( )t t
∂ ∂∇ ⋅ × = − ⋅ − ⋅ − ⋅
∂ ∂B D
E H J E H E
We then have:
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( )
( )t
t
∂⋅ ∇× = − ⋅
∂∂
⋅ ∇× = ⋅ + ⋅∂
BH E H
DE H J E E
Poynting Theorem (cont.)
σ=J ENext, assume that Ohm's law applies for the electric current:
( ) ( )t t
σ ∂ ∂∇ ⋅ × = − ⋅ − ⋅ − ⋅
∂ ∂B D
E H E E H E
or
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( )t t
∂ ∂∇ ⋅ × = − ⋅ − ⋅ − ⋅
∂ ∂B D
E H J E H E
( ) 2
t tσ ∂ ∂
∇ ⋅ × = − − ⋅ − ⋅∂ ∂B D
E H E H E
Poynting Theorem (cont.)
From calculus (chain rule), we have that
( )
( )
12
12
t t t
t t t
ε ε
µ µ
∂ ∂ ∂ ⋅ = ⋅ = ⋅ ∂ ∂ ∂ ∂ ∂ ∂ ⋅ = ⋅ = ⋅ ∂ ∂ ∂
D EE E E E
B HH H H H
Hence we have
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Poynting Theorem (cont.)
( ) 2
t tσ ∂ ∂
∇ ⋅ × = − − ⋅ − ⋅∂ ∂B D
E H E H E
( ) ( ) ( )2 1 12 2t t
σ µ ε∂ ∂∇ ⋅ × = − − ⋅ − ⋅
∂ ∂E H E H H E E
This may be written as
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Poynting Theorem (cont.)
( ) 2 2 21 12 2t t
σ µ ε∂ ∂ ∇ ⋅ × = − − ∂ ∂ E H E H E
( ) ( ) ( )2 1 12 2t t
σ µ ε∂ ∂∇ ⋅ × = − − ⋅ − ⋅
∂ ∂E H E H H E E
Final differential (point) form of Poynting’s theorem:
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( ) 2 2 21 12 2t t
σ µ ε∂ ∂ ∇ ⋅ × = − − ∂ ∂ E H E H E
Poynting Theorem (cont.)
Volume (integral) form
Integrate both sides over a volume and then apply the divergence theorem:
( ) 2 2 21 12 2V V V V
dV dV dV dVt t
σ µ ε∂ ∂ ∇ ⋅ × = − − − ∂ ∂ ∫ ∫ ∫ ∫E H E H E
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( ) 2 2 21 12 2t t
σ µ ε∂ ∂ ∇ ⋅ × = − − ∂ ∂ E H E H E
( ) 2 2 21 1ˆ2 2S V V V
n dS dV dV dVt t
σ µ ε∂ ∂ × ⋅ = − − − ∂ ∂ ∫ ∫ ∫ ∫E H E H E
Poynting Theorem (cont.)
Final volume form of Poynting theorem:
For a stationary surface:
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( ) 2 2 21 1ˆ2 2S V V V
n dS dV dV dVt t
σ µ ε∂ ∂ × ⋅ = − − − ∂ ∂ ∫ ∫ ∫ ∫E H E H E
( ) 2 221 1ˆ2 2S V V V
n dS dV dV dVt t
σ µ ε∂ ∂ × ⋅ = − − − ∂ ∂ ∫ ∫ ∫ ∫E H E H E
Poynting Theorem (cont.)
Physical interpretation:
Power dissipation as heat (Joule's law)
Rate of change of stored magnetic energy
Rate of change of stored electric energy
Right-hand side = power flowing into the region V.
(Assume that S is stationary.)
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( ) 2 221 1ˆ2 2S V V V
n dS dV dV dVt t
σ µ ε∂ ∂ − × ⋅ = + + ∂ ∂ ∫ ∫ ∫ ∫
E H E H E
Poynting Theorem (cont.)
( ) ˆS
n dS− × ⋅ =∫ E H power flowing into the region
( ) ˆS
n dS× ⋅ =∫ E H
power flowing out of the region
Or, we can say that
Define the Poynting vector: ≡ ×S E H
Hence
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ˆS
n dS⋅ =∫ S
power flowing out of the region
Poynting Theorem (cont.)
ˆS
n dS⋅ =∫ S
power flowing out of the region
Analogy:
ˆS
n dS⋅ =∫ J
current flowing out of the region
J = current density vector
S = power flow vector
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Poynting Theorem (cont.)
Direction of power flow
The units of S are [W/m2].
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Poynting Theorem (cont.)
= ×S E H
SE
H
Power Flow
The power P flowing through the surface S (from left to right) is:
( ) ˆS
t n dS= ⋅∫P S
Surface S
n̂
= ×S E H
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Time-Average Poynting Vector
( ) ( ) ( ) ( )*1 Re2
t t t E H= × = ×S E H
Assume sinusoidal (time-harmonic) fields)
( ) ( ){ }, , , Re , , j tx y z t E x y z e ω=E
( ) ( ){ }, , , Re , , j tx y z t H x y z e ω=H
From our previous discussion (notes 2) about time averages, we know that
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Complex Poynting Vector
Define the complex Poynting vector:
We then have that
( )*12
S E H≡ ×
( ) ( )( ), , , = Re , ,x y z t S x y zS
Note: The imaginary part of the complex Poynting vector corresponds to the
VARS flowing in space.
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The units of S are [VA/m2].
Complex Power Flow
The complex power P flowing through the surface S (from left to right) is:
Surface S
n̂
ˆS
P S n dS= ⋅∫
*12
S E H= ×
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( )( )
Re P
Im P
=
=
Watts
Vars
Complex Poynting Vector (cont.) What does VARS mean?
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( )
2 21 1VARS 24 4
2V
m e
H E dVω µ ε
ω
= −
= −
∫W W
Equation for VARS flowing into a region (derivation omitted):
The VARS flowing into the region V is equal to the difference in the time-average magnetic and electric stored energies inside the region (times a factor of 2ω).
E
H
VARS consumed Power (watts) consumed
σ
Watts + VARS flowing into V
V
S
Note on Circuit Theory
Although the Poynting vector can always be used to calculate power flow, at low frequency circuit theory can be used, and this is usually easier.
*1 ˆ2S
P E H z dS = × ⋅ ∫
Example (low frequency circuit):
*12
P VI=
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The second form is much easier to calculate!
(low frequency fields)
PI
LZ
S
Vz
+−
Example: Parallel-Plate Transmission Line
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At z = 0:
( ) 0jkzV z V e−=
( ) 0jkzI z I e−=
( ) 00V V=
( ) 00I I=
The voltage and current have the form of waves that travel
along the line in the z direction.
EH
x
y
h
w
+ - x
y
z
h
w
V
I
,ε µ
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0ˆ( , ,0) VE x y yh
= −
( ) 0ˆ, ,0 IH x y xw
=
At z = 0:
( )*12
S E H= ×
*0 01 ˆ
2V IS zh w
=
Example (cont.)
(from ECE 3318)
+ - x
y
z
h
w
V
I
,ε µ
EH
x
y
h
w
Example (cont.)
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*0 01 ˆ
2V IS zh w
=
0 0
ˆh w
fP S z dx dy= ⋅∫ ∫
( )*
0 012f
V IP whh w
=
*0 0
12fP V I=
Hence + - x
y
z
h
w
V
I
,ε µ
EH
x
y
h
w
= 0fP z z= complex power flowing in the direction at
