Ch 34 Electromagnetic Waves

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```					               Ch 34 Electromagnetic
Waves

Eunil Won
Department of Physics
Korea University

Fundamentals of Physics by Eunil Won, Korea University                           1
Maxwell’s Rainbow

Present understanding of all electromagnetic waves
: very wide spectrum (Maxwell’s Rainbow)
Fundamentals of Physics by Eunil Won, Korea University         2
The traveling electromagnetic
wave: qualitatively
Generation of macroscopic (~1m) electromagnetic wave:

Feature:
1) E and B are always perpendicular to the wave (transverse wave)
2) E and B are perpendicular each other
3) The ﬁelds vary sinusoidally in phase     E = Em sin (kx − ωt)

B = Bm sin (kx − ωt)
Fundamentals of Physics by Eunil Won, Korea University                          3
The traveling electromagnetic
wave: qualitatively
wave speed: speed of light                          1
c= √
µ0   0

A most curious wave
: no medium is required for
light wave

c = 299792458 m/s
(deﬁned as 9192631779 times the
oscillation period of the hyper-ﬁne
splitting of Cs133)

Fundamentals of Physics by Eunil Won, Korea University                                             4
The traveling electromagnetic
wave: quantitatively

Decreasing B induces E and dE
: counterclockwise loop (Lenz’s law)
dΦB
E · ds    = −
dt
= (E + dE)h − Eh = hdE
dΦB           dB
= hdx
dt           dt
Combining two equations above,
dE           dB
= −
dx           dt   (we used partial derivatives as E and B are
∂E           ∂B   functions of x and t)
= −
∂x           ∂t
Fundamentals of Physics by Eunil Won, Korea University                                                          5
The traveling electromagnetic
wave: quantitatively
E = Em sin (kx − ωt)
Since E and B have the forms of
B = Bm sin (kx − ωt)

we get                   ∂E
= kEm cos (kx − ωt)
∂x
∂B
= −ωBm cos(kx − ωt)
∂t
ω                  Em
= c=            (amplitude ratio)
k                  Bm
Fundamentals of Physics by Eunil Won, Korea University                                       6
The traveling electromagnetic
wave: quantitatively
Applying the same reasoning:
decreasing E induces B
Using Maxwell’s law of induction,
dΦE
B · ds   = µ0 0
dt
= −(B + dB)h + Bh = −hdB
dΦE           dE
= hdx
dt            dt
Combining two                      ∂B           ∂E
= µ0
equation above, we get:
−            0
∂x           ∂t
−kBm cos (kx − ωt) = −µ0 0 ωEm cos (kx − ωt)
Using the wave                             Em          1         1
equation:                                  Bm
=
µ0 0 (ω/k)
=
µ0 0 c
1
c = √
µ0 0
Fundamentals of Physics by Eunil Won, Korea University                                                  7
Energy transport and the
Poynting vector
The rate of energy transport per unit area                                        1
: Poynting vector
S=    E×B
µ0
energy/time                             power
The SI unit is W/m2
S=                                                   =
area                                 area
inst               inst

1       1            1 2
S=    EB =    E(E/c) =     E                                        (instantaneous energy ﬂow rate)
µ0      µ0          cµ0
Most of sensors of
electromagnetic wave detect
electric component only

Practically time averaged I = S = 1 [E 2 ] = 1 [E 2 sin2 (kx − ωt)] = 1 E 2
value of S (I: intensity) is
avg        avg                      avg
cµ0        cµ0 m                      cµ0 rms
more useful                                                      E
Em
= √                               rms
Fundamentals of Physics by Eunil Won, Korea University                                                           2   8
Energy transport and the
Poynting vector
Energy density of the electromagnetic wave:
1       1         1     1          B2
uE = 0 E 2 = 0 (cB)2 = 0 (      )B 2 =     = uB
2       2         2    0 µ0        2µ0
The energy associated with electric ﬁeld is same is with the magnetic ﬁeld

Intensity with a point source (Ps: power of the source) : I(r) = ?

The energy has to be conserved:
2
4πr I        = Ps
Ps
I   =
Fundamentals of Physics by Eunil Won, Korea University
4πr2                   9
A beam of electromagnetic radiation on an object (free to move)
for a given amount of time:
∆U
1) Radiation is entirely absorbed: momentum change becomes ∆p =
c
∆U
2) Radiation is totally reﬂected                           ∆p = 2
c

From Newton’s 2nd law: F =
∆p       The energy intercepted
by area A:
∆U = IA∆t
∆t

1 ∆U   IA                                                   1 ∆U    IA
Combining two above: F =      =    for 1)                                     F =2        =2    for 2)
c ∆t    c                                                   c ∆t     c

pr = F/A =       for 1)    pr = F/A = 2       for 2)
becomes:                                                    c                            c

Fundamentals of Physics by Eunil Won, Korea University                                                           10
Polarization

Polarized light: E oscillates within x-y plane              Unpolarized light: E oscillates without
(ex: TV signal)                                             direction (ex: light from Sun)

Fundamentals of Physics by Eunil Won, Korea University                                                 11
Reﬂection and Refraction
n: index of refraction (n=c/v)

Law of Reﬂection
θ1 = θ1
Law of Refraction
n2 sin θ2 = n1 sin θ1
(Known as Snell’s law. We will derive
this in Ch. 36)

Medium         Index         Medium         Index
Vacuum            1         Diamond          2.42
Air        1.00029      Water (20oC)      1.33
Ethyl alcohol     1.36         Sapphire        1.77
Fundamentals of Physics by Eunil Won, Korea University                                                           12
Chromatic Dispersion
Index of refraction is a function of the wavelength associate with.
So a light beam consisting of rays of different wavelengths will be
refracted at different angles by a surface (n is greater for shorter
wavelength, in general)

Triangular prism

Fundamentals of Physics by Eunil Won, Korea University                                 13
Total Internal Reﬂection
The angle of incidence that gives
no refracted light in the upper
medium : critical angle θ
c
n1 sin θc   = n2 sin 90o
−1 n2
θc    = sin
n1

Application of total
internal reﬂection:
optical ﬁbers

Fundamentals of Physics by Eunil Won, Korea University                                       14
Polarization by Reﬂection
In general reﬂected light is
partially polarized. At a particular
angle, the reﬂected light has only
perpendicular component
(Brewster’s angle) θB

Experimental observation:
θB + θr = 90o

n1 sin θB    = n2 sin θr
Using Snell’s law:                            n1 sin θB    = n2 sin (900 − θB ) = n2 cos θB
−1 n2
θB     = tan
n1
Fundamentals of Physics by Eunil Won, Korea University                                                      15
Summary
E = Em sin (kx − ωt)                     1
c= √
µ0   0
B = Bm sin (kx − ωt)
1
S=    E×B
µ0

Critical angle

Brewster’s angle

Fundamentals of Physics by Eunil Won, Korea University                       16

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