electromagnetic power density introduction – …eie.polyu.edu.hk/~em/em06pdf/poynting vector.pdf1...

1

Electromagnetic Power Density

Introduction– Power is transmitted by an electromagnetic wave in the

direction of propagation.

– Poynting vector

• not only applied to uniform plane waves• can be used for any electromagnetic waves

)/(),,,(),,,(),,,( 2mWtxyxtzyxtzyx HES ×=

mV / mA /

2

Poynting Vector from Maxwell’s Equations

It can be derived from Maxwell’s equation in the time domain that:

dvHEt

dvdVVS ∫∫∫

+

∂∂

+⋅=⋅×−22

22 µεJESHE

power loss

the rate of increase of energies stored in the electric and magnetic fields

V

Snet power supplied to the enclosed surface

3

Poynting Vector

Net power supplied to the volume:Power density of the wave:

Therefore, Poynting vector P (in W/m2) represents the direction and density of power flow at a point.

Example: If a plane wave propagating in a direction that makes an angle θ with the normal vector of an aperture

HEP

SHE

×=

⋅×= ∫Sout dW

θcos

ˆ

SA

dAnPS

=

⋅= ∫S

k̂n̂

4

Example

Find the Poynting vector on the surface of a long straight conducting wire (of radius b and conductivity σ) that carries a direct current I.

Solution:Since we have a d-c situation, the current in the wire is uniformly distributed over its cross-sectional area.

L2ˆbIz

π=J

2ˆbIz

σπσ==

JE

5

Example

On the surface of the wire,

Integrating the Poynting vector over the wall of the wire

bIπ

φ2

ˆ=H

32

2

32

2

2ˆ

2ˆˆ

bIr

bIz

σπσπφ −=×=×= HEP

RIbLI

bLb

Idsrdss

22

2

32

2

22

ˆ

=

=

=×=⋅−=⋅− ∫∫

σπ

πσπ

HEPsP

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Average Power Density

In time-varying fields, it is more important to find the average power. We define the average Poynting vector for periodic signals as Pavg

It can be shown that the average power density can be expressed as

∫=Tdt

T 0

1 PPavg

( )*),,(),,(Re21 zyxzyx HEPavg ×=

Complex conjugate

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Example: Plane Wave in a lossless Medium

The general expression for the phasor electric field of a uniform plane wave with arbitrary polarization, traveling in the +z direction, is given by

The magnetic field is

( ) zjyoxo

yx

eEyEx

zEyzExzβ−+=

+=

ˆˆ

)(ˆ)(ˆ)(E

( ) zjyoxo eExEy

zzz

β

η

η

−−−=

×=

ˆˆ1

)(ˆ1)( EH

( )22

21ˆ yoxoav EEzP +=∴η

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Example: Solar Power

9

Example: Solar Power

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Example: Solar Power

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Example: Solar Power

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Example: Plane wave in a lossy medium

The general expression for the phasor electric field of a uniform plane wave with arbitrary polarization, traveling in the +z direction, is given by

and magnetic field is

( ) zjzyoxo

yx

eeEyEx

zEyzExzβα −−+=

+=

ˆˆ

)(ˆ)(ˆ)(E

( ) zjzyoxo eeExEy

zzz

βα

η

η

−−−−=

×=

ˆˆ1

)(ˆ1)( EH

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Example: Plane wave in a lossy medium

The average power density is

Demonstration: D7.6

( )

( )( ) ηθ

ηα

α

ηηθη

η

jzyoxo

zyoxo

eeEEz

eEEz

zyxzyx

=+=

+=

×=

−

−

ifcos21ˆ

1Re21ˆ

*),,(),,(Re21

222

*222

HEPavg

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Reflection of Plane Waves at Normal Incidence

x

z

Medium 2 (σ2,ε2,µ2)Medium 1(σ1,ε1,µ1)

(incident wave)

(reflected wave)

(transmitted wave)

Ei

ak

ak

ak

Et

Er

Hi

Hr

Ht

ak – propagation direction

y direction

-y direction

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Incident wave:

yzio

yz

ioi

xz

ioi

eEeH

eE

aaH

aE

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1

1

γγ

γ

η−−

−

==

=

Reflected wave:

yzro

yz

ror

xz

ror

eEeH

eE

aaH

aE

11

1

1

)( γγ

γ

η−=−=

=

Transmitted wave:

yzto

yz

tot

xz

tot

eEeH

eE

aaH

aE

22

2

2

γγ

γ

η−−

−

==

=

Reflection of Plane Waves at Normal Incidence

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Boundary conditions at the interface (z=0):

Tangential components of E and H are continuous in the absence of current sources at the interface, so that

( )21

1ηηto

roio

toroio

toroio

EEE

HHHEEE

=−

=+=+

Reflection coefficient12

12

ηηηη

+−

==Γio

ro

EE

Transmission coefficient21

22ηη

ητ+

==io

to

EE

Reflection of Plane Waves at Normal Incidence

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Note that Γ and τ may be complex, and

101

≤Γ≤

=Γ+ τ

Similar expressions may be derived for the magnetic field.

In medium 1, a standing wave is formed due to the superimposition of the incident and reflected waves. Standing wave ratio can be defined as in transmission lines.

Reflection of Plane Waves at Normal Incidence

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Example: Normal Incidence on a good conductor

For good conductor,

12

12

ηηηη

+−

=Γ21

22ηη

ητ+

=

∞→σ

02 →+

=ωεσ

ωµηj

j

1−=Γ

0=τ