maxwell’s equations & the electromagnetic wave...
TRANSCRIPT
SMR 1826 - 2
Preparatory School�� ���
Winter College on Fibre Optics, Fibre Lasers and Sensors
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Maxwell’s Equations&
The Electromagnetic Wave Equation
Imrana Ashraf Zahid������������ ����������
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5-02-2007 Preparatory School on Fiber
Optics, Fiber Lasers and Sensors
1
Maxwell’s Equations
&
The Electromagnetic Wave
Equation
Dr. Imrana Ashraf ZahidQuaid-i-Azam University, Islamabad
Pakistan
5-02-2007 Preparatory School on Fiber
Optics, Fiber Lasers and Sensors
2
Maxwell’s Equations
• Introduction
• Historical background
• Electrodynamics before Maxwell
• Maxwell’s correction to Ampere’s law
• General form of Maxwell’s equations
• Maxwell’s equations in vacuum
• Maxwell’s equations inside matter
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Optics, Fiber Lasers and Sensors
3
Introduction
• In electrodynamics Maxwell’s equations are a set of four equations, that describes the behavior of both the electric and magnetic fields as well as their interaction with matter
• Maxwell’s four equations express
– How electric charges produce electric field (Gauss’s law)
– The absence of magnetic monopoles
– How currents and changing electric fields produces magnetic fields (Ampere’s law)
– How changing magnetic fields produces electric fields (Faraday’s law of induction)
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Historical Background
• 1864 Maxwell in his paper “A Dynamical Theory of the Electromagnetic Field” collected all four equations
• 1884 Oliver Heaviside and Willard Gibbs gave the modern mathematical formulation using vector calculus.
• The change to vector notation produced a symmetric mathematical representation, that reinforced the perception of physical symmetries between the various fields.
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Nomenclature
• E = Electric field
• D = Electric displacement
• B = Magnetic flux density
• H = Auxiliary field
• = Charge density
• j = Current density
• 0 (permeability of free space) = 4 10-7
• 0 (permittivity of free space) = 8.854 10-12
• c (speed of light) = 2.99792458 108 m/s
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Electrodynamics Before Maxwell
JBiv
t
BEiii
Bii
Ei
o
o
)(
)(
0)(
)(
AB
t
AVE
Gauss’s Law
No name
Faraday’s Law
Ampere’s Law
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Electrodynamics Before Maxwell
(Cont’d)
Apply divergence to (iii)
.0becausezeroissidehandrightThe
zero.iscurlaofdivergencebecausezero,issidehandleftThe
B
Btt
BE
Apply divergence to (iv)
JB o
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• The left hand side is zero, because divergence of a curl is zero.
• The right hand side is zero for steady currents i.e.,
• In electrodynamics from conservation of charge
Electrodynamics Before Maxwell
(Cont’d)
0J
0t
tJ
is constant at any point in space which is
wrong.
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Optics, Fiber Lasers and Sensors
9
Maxwell’s Correction to Ampere’s Law
Consider Gauss’s Law
t
E
t
tE
t
E
o
o
o
t
E
t
Do Displacement current
This result along with Ampere’s law and the conservation of charge
equation suggest that there are actually two sources of magnetic field.
The current density and displacement current.
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Optics, Fiber Lasers and Sensors
10
Maxwell’s Correction to Ampere’s
Law (Cont’d)
Amperes law with Maxwell’s correction
t
EJB ooo
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General Form of Maxwell’s Equations
Differential Form Integral Form
S
ooenco
C
SC
S
VoS
SdEdt
dIldB
SdBdt
dldE
SdB
dVSdE
0
1
0
t
EJB
t
BE
B
E
ooo
o
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12
Maxwell’s Equations in vacuum
t
EB
t
BE
B
E
oo
0
0
• The vacuum is a linear, homogeneous, isotropic and dispersion less medium
• Since there is no current or electric charge is present in the vacuum, hence Maxwell’s equations reads as
• These equations have a simple solution interms of traveling sinusoidal waves, with the electric andmagnetic fields direction orthogonal to each other and the direction of travel
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13
Maxwell’s Equations Inside Matter
Maxwell’s equations are modified for polarized and
magnetized materials.
For linear materials the polarization P and magnetization M
is given by
HM
E
m
eo
And the D and B fields are related to E and H by
.andmaterialoflitysusceptibimagnetictheis
material,oflitysusceptibielectrictheisWhere
1
1
m
e
omo
oeo
HHMHB
EEPED
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Optics, Fiber Lasers and Sensors
14
Maxwell’s Equations Inside Matter
(Cont’d)
• For polarized materials we have bound charges in addition to free charges
P
nP
b
b
MJ
nMK
b
b
• For magnetized materials we have bound currents
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15
Maxwell’s Equations Inside Matter
(Cont’d)
• In electrodynamics any change in the electric
polarization involves a flow of bound charges
resulting in polarization current JP
Polarization current density is due
to linear motion of charge when the
Electric polarization changes t
PJ p
pbf
bft
JJJJ t
densitycurrentTotal
densitychargeTotal
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Optics, Fiber Lasers and Sensors
16
Maxwell’s Equations Inside Matter
(Cont’d)
• Maxwell’s equations inside matter are written as
t
EJJJB
t
BE
B
E
oobopofo
o
t
0
Dt
JH
PEt
JMB
t
EM
t
PJ
B
f
of
o
of
o
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17
Maxwell’s Equations Inside Matter
(Cont’d)
• In non-dispersive, isotropic media and µ are time-independent scalars, and Maxwell’s equations reduces to
t
EJH
t
HE
H
E
0
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Optics, Fiber Lasers and Sensors
18
Maxwell’s Equations Inside Matter
(Cont’d)
• In uniform (homogeneous) medium and µ areindependent of position, hence Maxwell’s equations reads as
Generally, and
µ can be rank-2
tensor (3X3
matrices)
describing
birefringent
anisotropic
materials.
00
C
C
S
S
S
encff
S
encff
SdDdt
dIldH
t
EJH
SdHdt
dldE
t
HE
SdHH
QSdDD
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19
The Electromagnetic Wave Equation
(EM Wave)
• The EM wave from Maxwell’s Equation
• Solution of EM wave in vacuum
• EM plane wave
• Polarization
• Energy and momentum of EM wave
• Inhomogeneous wave equation
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The Electromagnetic Wave from
Maxwell’s Equations
Take curl of
][t
BE
t
BE
Change the order of differentiation on the R.H.S
][ Bt
E
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21
The Electromagnetic Wave from
Maxwell’s Equations (cont’d)
As
t
EB oo
Substituting for we haveB
2
2
][
][][][
t
EE
t
E
tE
t
BE
oo
oo
•Assuming that µo and o are constant in time
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Optics, Fiber Lasers and Sensors
22
The Electromagnetic Wave from
Maxwell’s Equations (cont’d)
2
2
t
EE
2
22)(
t
EEE oo
0E
Using the vector identity
becomes,
In free space
And we are left with the wave equation
02
22
t
EE oo
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Optics, Fiber Lasers and Sensors
23
The Electromagnetic Wave
from Maxwell’s Equations (cont’d)
Similarly the wave equation for magnetic field
02
22
t
BB oo
oo
c1where,
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Optics, Fiber Lasers and Sensors
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Solution of Electromagnetic Waves
in Vacuum
02
22
t
BB oo
The solutions to the wave equations, where there is no
source charge is present
02
22
t
EE oo
can be plane waves, obtained by method
of separation of variables
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Optics, Fiber Lasers and Sensors
25
Solution of Electromagnetic Waves
in Vacuum (Cont’d)
ti
o
ti
o
eBB
eEE
rk
rk
Where Eo and Bo are the complex amplitudes of electric
and magnetic fields and related to each other by
relation
)ˆ(1
oo Ekc
B
Where is a propagation vector.k̂
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Optics, Fiber Lasers and Sensors
26
Electromagnetic Plane waves
• Plane electromagnetic waves can be expressed as
Ec
neEc
B
neEE
ti
o
ti
o
kkrk
rk
ˆ1ˆˆ1
ˆ
Where is the polarization vector.n̂
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Electromagnetic Plane waves
The real electric and magnetic fields in a
monochromatic plane wave with propagation vectorkˆ and polarization nˆ are therefore
nktrkEc
trB
ntrkEtrE
o
o
ˆ)cos(1
,
ˆ)cos(,
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Polarization
•The polarization is specified by the orientation of the
electromagnetic field.
•The plane containing the electric field is called the
plane of polarization.
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Polarization (Cont’d)
• Can be horizontal, vertical, circular, or elliptical
x
y
zE
Horizontal Polarization
Electric Field
Magnetic Field
Electromagnetic
Wave
x
y
z
E
Vertical Polarization
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Energy and Momentum of
Electromagnetic Waves
The energy per unit volume stored in electromagnetic field is
22 1
2
1BEU
o
o
In the case of monochromatic plane wave
)(cos
1
222
22
2
2
tkxEEU
EEc
B
ooo
oo
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Energy and Momentum of
Electromagnetic Waves (Cont’d)
• As the wave propagates, it carries this energy along with it.
The energy flux density (energy per unit area per unit time)
transported by the field is given by the poynting vector
BESo
1
For monochromatic plane waves
icUitkxEcS ooˆˆcos2
2
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Homogenous Wave Equations Inside
Matter
MatterVacuum
2
22
2
22
2
22
2
22
11
11
t
BB
t
BB
t
EE
t
EE
oo
oo
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Homogenous Wave Equations Inside
Matter (cont..)
rrrr
v1111
0000
n
c
v
=c = n
Permittivity: = r o ( r is dielectric constant)
Permeability: µ=µrµo (µr is relative permeability 1
n=Refractive Index
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Inhomogeneous Electromagnetic
Wave Equation
Inside linear dielectric medium with no free charge present,
Maxwell’s equations reads as
(iv)
(iii)
(ii)0
(i)0
t
DH
t
BE
B
D
Where, HBPED oo and
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36
Inhomogeneous Electromagnetic
Wave Equation (Cont’d)
Taking curl of (iii)
][][][ Ht
Bt
E o
Using (iv)
2
2
22
2
2
2
2
2
22
2
2
2
2
2
2
22
2
22
11
11
][
t
P
ct
E
cE
t
P
ct
E
cE
t
P
t
EE
t
DEE
o
o
ooo
o
Source term
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Solution of Inhomogeneous
Electromagnetic Wave Equation
2
2
22
2
2
2 11
t
P
ct
E
cE
o
Inhomogeneous wave equation can be solved
with the help of Green’s Theorem