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ElectromagnetismElectromagnetism

University of TwenteDepartment Applied Physics

First-year course on

Part III: Electromagnetic Waves : SlidesPart III: Electromagnetic Waves : Slides

©F.F.M. de Mul

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Electromagnetic WavesElectromagnetic Waves

• Gauss’ and Faraday’s Laws for E (D)-field• Gauss’ and Maxwell’s Laws for B (H)-field• Maxwell’s Equations and The Wave Equation• Harmonic Solution of the Wave Equation• Plane waves (1): orientation of field vectors• Plane waves (2): complex wave vector• Plane waves (3): the B - E correspondence• The Poynting Vector

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Gauss’ and Faraday’s Laws for Gauss’ and Faraday’s Laws for EE

0

E

div

B

c

dS

Faraday’s Law :

S S Sc

ind rotdt

dV dSEdlEdSB n

dt

drot

BE

E

S V dV VS V

dVdivdV .11

00

EdSE

Gauss’ Law :

Div = micro-flux perunit of volume

Rot = micro-circulationper unit of area

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Gauss’ and Maxwell’s Law for Gauss’ and Maxwell’s Law for BB

SV

dV

B

0. VS

dVdivBdSB

Gauss’ Law :

0Bdiv

j SSL dt

ddSEdSjdlB 000

S

L

Maxwell’s Fix for Ampere’s Law :

dt

drot

EjB 0 00

dt

drot f

DjH

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Maxwell’s Equations and the Maxwell’s Equations and the Wave EquationWave Equation

0

E

dt

dBE 0 B

dt

df

DjH

In vacuum ( = 0 and j = 0):

EEEE 22 )(

This is a 3-Dimensional Wave Equation 00

1

v

v = 2.99… x 108 m/s = light velocity

2

2

002

dt

d EE }2

2

00 dt

d

dt

d EBE

0 E

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Harmonic Solution of the Harmonic Solution of the Wave Equation: Plane WavesWave Equation: Plane Waves

2

2

002

dt

d EE

E may have 3 components: Ex Ey Ez

Choose x-axis // E Ey = Ez = 0(Polarization direction = x-axis).

Does a plane-wave expression for Ex satisfy the wave equation?

Ex = Ex0 exp{i (t-kz)} : Ex0 = amplitude + polarization vector +z-axis = direction of propagation

Insertion into wave equation: k2 = 0 0 2 = 2 / c2

k = / c = 2 / k = wave number ; = wavelength(in 3D-case: k = wave vector )

Analogously: By = By0 exp{i (t-kz)}

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-ik ez .E = 0

-ik ez .H = 0

-ik ez xE = -iH

-ik ez xH = E+iE

Plane waves (1): orientation of fieldsPlane waves (1): orientation of fieldsSuppose: E // x-axis; .. propagation // +z-axis: k // ez

.. Ex = Ex0 exp i (t-kz)

x

zy

EPropag-ation

(1) div E = 0

(2) div B = 0

(3) rot E = - dB/dt

(4) rot H = jf + dD/dtjf = E

Consequences:(1)+(2): E and H ez

(3)+(4): E H

If E chosen // x-axis, then H // y-axis

H

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Plane waves (2): complex wave vectorPlane waves (2): complex wave vector

(1) -ik ez .E = 0(2) -ik ez .H = 0(3) -ik ez xE = -iH(4) -ik ez xH = E+iE

(1)+(2): E and H ez

(3)+(4): E H

Suppose: E // x-axis; .. propagation // +z-axis: k // ez

.. Ex = Ex0 exp i (t-kz)

x

zy

EPropag-ation

H

).(

)(

i

kik

i

ik

EeeHeE zzz ez x ez x E = -E ik2=(+i).

Result: k complex: k = kRe+ ikIm

exp (-ikz) = exp (-ikRe z) . exp (kIm z)}harmonic

}

kIm<0 : absorption >0 : amplification (“laser”)

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Plane waves (3): Plane waves (3): B-EB-E correspondence correspondenceSuppose: E // x-axis; B // y-axis; .. propagation // +z-axis: k // ez

.. Ex = Ex0 exp i (t-kz)

.. By = By0 exp i (t-kz)

x

zy

EPropag-ation

BFaraday: rot E = - dB/dt

0

0

00y

x

dzd

dyd

dxd B

dt

d

E

zyx eeeyxydt

dxdz

d BkEBE 00

000 :Result cBBk

E

Maxwell (for j=0): rot H = dD/dt: similar result

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The Poynting vector The Poynting vector SSDefinition (for free space) : S = E x H

S = (EH) = H(E) - E(H) =

= H(-dB/dt) - E(jf+dD/dt) ft

E

t

BjE

2

20

2

2

0 22

1

Apply Divergence Theorem to integrate over wave surface A:

vv vA

dvdvEB

dt

ddvS )()( 2

00

2

21

fjEdAS

{Change in Electro- magnetic field energy

{

Joule heatinglosses [J/s]

{

Outflux of energy [J/s] = [W]

S = energy outflux per m2 = Intensity [W/m2]

Direction of S : // k

E

Hk

S