08/28/2013 PHY 530 -- Lecture 01 1
Light is electromagnetic radiation!
• = Electric Field
• = Magnetic Field
• Assume linear, isotropic, homogeneous media.
),( txE
),( txB
08/28/2013 PHY 530 -- Lecture 01 2
Maxwell’s Equations
• Published by J.C. Maxwell in 1861 in the paper “On Physical Lines of Force”.
• Unite classical electricity and magnetism.• Predict the propagation of electromagnetic energy
away from time varying sources (current and charge) in the form of waves.
08/28/2013 PHY 530 -- Lecture 01 3
Maxwell’s Equations
• Four partial differential equations involving E, B that govern ALL electromagnetic phenomena.
• Gauss’s Law (elec, mag)
• Faraday’s Law, Ampere’s Law
08/28/2013 PHY 530 -- Lecture 01 4
Gauss’s Law (elec)
E
q
Charge density
Electric permittivityconstant of medium
Total chargeenclosed
Electric charges give rise to electric fields.
dS – outward normal
E dS = q
08/28/2013 PHY 530 -- Lecture 01 5
Gauss’s Law (mag)
0 B
No Magnetic Monopoles!
B dS = 0
08/28/2013 PHY 530 -- Lecture 01 6
Faraday’s Law
= mag. flux
A changing B field gives rise to an E field E field lines close on themselves (form loops)
where:
08/28/2013 PHY 530 -- Lecture 01 7
Ampere’s Law
t
E
jΒ
If E const in time:
Where:
Magneticpermeabilityof medium
Electric currents give rise to B fields.
i Electric current
j = current density
08/28/2013 PHY 530 -- Lecture 01 8
What Maxwell’s Equations Imply
In the absence of sources, all components of E, B satisfy the same (homogeneous) equation:
02
22
t
EE
02
22
t
BB
The properties of an e.m. wave (direction of propagation, velocity of propagation, wavelength, frequency) can be determined by examining the solutions to the wave equation.
08/28/2013 PHY 530 -- Lecture 01 9
What does it mean to satisfy the wave equation?
),( txImagine a disturbance traveling along thex coordinate (1-dim case).
0t 0tt ),( tx
x
08/28/2013 PHY 530 -- Lecture 01 10
What does a wave look like mathematically?
)()(),( vtxgvtxftx
General expression for waves traveling in +ve, -ve directions:
Argument affects the translation of wave shape.
is the velocity of propagation.v
08/28/2013 PHY 530 -- Lecture 01 11
Waves satisfy the wave equation
• Try it for f! Use the chain rule, differentiate:
• This is the (homogeneous) 1-dim wave equation.
01
2
2
22
2
t
f
vx
f
08/28/2013 PHY 530 -- Lecture 01 12
E, B satisfy the 3-dim wave equation!!
2
1
v
01
2
2
22
tv
can be zyxzyx BBBEEE or...,,,,,
and
08/28/2013 PHY 530 -- Lecture 01 13
Index of Refraction (1)
1
vOkay, Velocity of light in a medium dependent on medium’s electric,magnetic properties.
In free space:
m/s103.00
H/m1026.1
F/m1085.8
8
60
120
cv
08/28/2013 PHY 530 -- Lecture 01 14
Index of Refraction (2)
For any l.i.h. medium, define index of refraction as:
00
vcn
NOTE: dimensionless.
08/28/2013 PHY 530 -- Lecture 01 15
Index of Refraction (3)
medium n
vacuum 1, by definition
Air 1.0003
Water 1.33
Flint glass 1.5
Crown glass 1.7
Diamond 2.417
08/28/2013 PHY 530 -- Lecture 01 16
Plane waves
Back to the 3-dim wave equation, but assume
),( tz
z
x
y has constant valueon planes:
08/28/2013 PHY 530 -- Lecture 01 17
Seek solution to wave eqn
Solving PDEs is hard, so assume solution of the form:
)()(),( tTzZtz
(so-called “separable” solution…)
Now, Becomes:01
2
2
22
tv
08/28/2013 PHY 530 -- Lecture 01 18
Voilà! Two ordinary differential equations!
22
2
22
2
2
2
22
2
const)(
)(
1)(
)(
1
)()(
1)()(
kdt
tTd
tTvdz
zZd
zZ
dt
tTdzZ
vdz
zZdtT
022
2
Zkdz
Zd022
2
2
Tkvdt
Tdand
Note!
08/28/2013 PHY 530 -- Lecture 01 19
We know the solutions to these...
tik
tikk
ikzk
ikzkk
tT
zZ
ee)(
ee)(
where222 kv .
(Sines and cosines!)
08/28/2013 PHY 530 -- Lecture 01 20
How to build a wave
Choose w positive, +ve z dir, then have
)(1
)(0 ee),( tkzitkzitz
Any linear combination of solutions of this form isalso a solution.
Start with sines and cosines, make whatever shapelike.
08/28/2013 PHY 530 -- Lecture 01 21
Let’s get physical
Sufficient to study )(0 e tkzi
2
2
k wavelength
frequency
Harmonic wave
kz-ωt - phase (radians) ω - angular frequencyk – propagation number/vector
08/28/2013 PHY 530 -- Lecture 01 22
3-D wave equation
)(0 e),( tit xkx
Solution:
Reduces to the 1-D case when
kzkzk ˆˆ||
08/28/2013 PHY 530 -- Lecture 01 23
Back to Plane Waves
Assume we have)(
0 e),( tkzit ExE(plane waves in the z-direction, E0 a constant vector)
0
yx
EEE
Eik
z
Similar equations for B.
,
EE it
08/28/2013 PHY 530 -- Lecture 01 24
Electromagnetic Waves are Transverse
Differentiate first equation of previous slide,can show then using Maxwell’s equations that:
0zkE
0zkB Byx )ˆˆ( xy EEk
Eyx2
)ˆˆ(v
BBk xy
Try it!
08/28/2013 PHY 530 -- Lecture 01 25
EM Waves are Transverse (2)
This implies:
0Ek
0Bk BEk
EBk2v
Fields must be perpendicular to the propagation direction!
08/28/2013 PHY 530 -- Lecture 01 26
EM Waves are Transverse (3)
Also, fields are in phase in the absence of sources and E is perpendicular to B since
BEk ˆ
k
E
B
08/28/2013 PHY 530 -- Lecture 01 27
What light looks like close up
Magnetic Field Waves
Electric Field Waves
The Electric and Magneticcomponents of light are perpendicular (in vacuum).
+
Movingcharge(s)
Waves propagate withspeed 3x108 m/s.
08/28/2013 PHY 530 -- Lecture 01 28
The Poynting Vector
S is parallel to the propagation direction. In free space,
S gives us the energy transport of waveform. Energy/time/area
I = <|S|>time=1/2(c ε0) E02 - irradiance (time average of the
magnitude of the Poynting vector)
08/28/2013 PHY 530 -- Lecture 01 29
The Electromagnetic Spectrum
08/28/2013 PHY 530 -- Lecture 01 30
The Electromagnetic Spectrum