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Lecture 11
Poynting TheoremChapter 8, Page 346, Griffith
Lecture :Electromagnetic Power Flow
Lecture 11
Statement: Poynting’s Theorem“Conservation of Energy”
“The work done on the charges by the electromagnetic force is equal to the decrease in energy stored in the field, less the energy that flowed out through the surface”
SVV
i sdHEdvHEt
dvJE 22
2
1
2
1
Lecture 11
Suppose we have some charge and current configuration, produces fields E and B at time t.
Work done (by applying Electromagnetic forces ) on a charge q isgiven by
Rate of work done on charges and current available in the system
VVV
dJEdvEvEddt
dW
JvanddtqNow
.).(.)(
,
Total energy stored in Electromagnetic fields is
V
em dvHEU 22
2
1
2
1
vdtqEvdtBvEqdlF .).(.
Lecture 11
• Maxwell’s curl equations in differential form
t
DJHand
t
BE
,,
HEEHHE
t
BHEH
t
DEJEHE
• Recall a vector identity
• Furthermore,
Can we obtain V
JdE . Using maxwell’s equations ?
Lecture 11
Derivation of Poynting’s Theorem in the Time Domain (Cont’d)
t
DEJE
t
BH
HEEHHE
• Integrating over a volume V bounded by a closed surface S, we have
VVV
dvHEdvt
BH
t
DEdvJE
• Using the divergence theorem, we obtain the general form of Poynting’s theorem
SVV
sdHEdvt
BH
t
DEdvJE
Lecture 11
Derivation of Poynting’s Theorem in the Time Domain (Cont’d)
• Note that 22
1A
tt
AA
t
AA
• Hence, we have the form of Poynting’s theorem valid in simple, lossless media:
SVV
sdHEdvHEt
dvJE 22
2
1
2
1
Lecture 11
Physical Interpretation of the Terms in Poynting’s Theorem
• Hence, the terms
represent the total electromagnetic energy stored in the
volume V.
V
dvHE 22
2
1
2
1
• The term
represents the flow of instantaneous power out of the volume V through the surface S.
S
sdHE
• The term
represents the total electromagnetic energy generated (Rate of work done) by the sources in the volume V.
V
dvJE
Lecture 11
V
dvJE
S
sdHE
V
dvHE 22
2
1
2
1
System of q and I
Rate of work in the system
Applied Lorentz forcePower flow
Rate of decrease in stored energy
Lecture 11
Differential form of Poynting’s Theorem
SVV
sdHEdvHEt
dvJE 22
2
1
2
1
]:[.)(
,
TheoremGaussApplyingNotedvSdvUUt
VectorPoyntingisSwheresdSdvUt
dvt
U
VV
EMmec
SV
EM
V
mec
)(. EMmec UUt
S
Shows Conservation of Energy
Lecture 11
Poynting Vector in the Time Domain
• We define a new vector called the (instantaneous) Poynting vector as
• The Poynting vector has the same direction as the direction of propagation.
• The Poynting vector at a point is equivalent to the power density of the wave at that point.
HES • The Poynting vector has units of W/m2.
Lecture 11
Boundary conditions, Page 333, Ch. 7
• If there is no free charge or free current at the interface of two medium, then
2
//2
1
//1
//2
//1
21
2211
.4
.3
.2
.1
BB
EE
BB
EE
Incident
Reflected
Refracted
Medium-1(ε1 , µ1)
EE//
E┴
Medium-2(ε2 , µ2)