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)