Download Electromagnetic Waves

Lecture, Summer Term 2015
Physics of the Atmosphere 2
Radiation and Energy Balance
Ulrich Foelsche
Institute of Physics, Institute for Geophysics, Astrophysics, and Meteorology (IGAM)
University of Graz
und
Wegener Center for Climate and Global Change
[email protected]
http://www.uni-graz.at/~foelsche/
Atmo II 01
Textbooks
C. Donald Ahrens, Meteorology Today: An Introduction to Weather, Climate,
and the Environment, Brooks/Cole, 9. Ed., ISBN: 0495555746 (also
paperback)
UB-Semesterhandapparat, IGAM-Library
K.N. Liou, (Ed.), An Introduction to Atmospheric Radiation, Academic Press,
2nd Ed., ISBN: 978-0-12-451451-5, 2002
<http://books.google.at/books?id=mQ1DiDpX34UC> (partial)
Murry L. Salby, Physics of the Atmosphere and Climate, Cambridge Univ.
Press, 2nd Ed., ISBN: 978-0-521-76718-7, 2012
<http://books.google.at/books?id=CeMdwj7J48QC> (partial)
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Lehrbücher
Helmut Kraus, Die Atmosphäre der Erde - Eine Einführung in die
Meteorologie, Springer, Berlin, 3. Auflage, ISBN: 978-3-540-20656-9 (auch
paperback)
UB-Semesterhandapparat, IGAM-Bibliothek
Gösta H. Liljequist & Konrad Cehak, Allgemeine Meteorologie, Springer,
Berlin, 3. Auflage ISBN: 3540415653 (nützliches deutsch-englisches
Register)
UB-Semesterhandapparat, IGAM-Bibliothek
Ludwig Bergmann & Clemens Schaefer, Lehrbuch der
Experimentalphysik, Band 7, Erde und Planeten, (Kapitel 3 – Meteorologie,
Kapitel 4 – Klimatologie), de Gruyter, Berlin, ISBN: 978-3-11-016837-2
UB-Semesterhandapparat, IGAM-Bibliothek
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Exams
No, it will be the other way round –
you will be forced to answer
questions – in my office & IGAM
Exam dates and registration via
UNIGRAZonline: online.uni-graz.at
Picture credit: Gary Larson
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Different Aspects of
Atmospheric Radiation
UF
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Physics of the Atmosphere II
(1) Electromagnetic Waves
NASA
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The Electromagnetic Field
Basic Properties of the Electromagnetic Field
Within the framework of classical electrodynamic theory, it is
represented by the vector fields:
Electric field E [V/m]
Magnetic field B [Vs/m2] = [T] (Tesla)
To describe the effect of the field on material objects, it is necessary to
introduce a second set of vectors: the
Electric current density
j [A/m2]
Electric displacement field
D [As/m2]
Magnetizing field
H [A/m]
The space and time derivatives of the vectors field are related by
Maxwell's equations – we will focus on the differential form.
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The Electromagnetic Field
The electric field E and the electric displacement field D are related by
D  ε0E  P
where ε0 is the electric constant 8.854 187 817 · 10-12 AsV-1m-1 (exact)
[NIST Reference: http://physics.nist.gov/cuu/Constants/index.html], and
P is the electric polarization – the mean electric dipole moment per
volume.
The magnetic field B and the magnetizing field H are related by
B  μ0H  M
where µ0 is the magnetic constant 4π·10-7 VsA-1m-1 (exact), and
M is the magnetic polarization – the mean magnetic dipole moment
per volume.
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Maxwell's Equations in Matter
The First Maxwell Equation, also known as Gauss’s Law:
  D  ρf ree
relates the divergence of the displacement field to the (scalar) free
charge density: Positive electric charges are sources of the
displacement field (negative electric charges are sinks). Closed field
lines can be caused by induction.
The Second Maxwell Equation or Gauss’s Law for Magnetism:
 B  0
states that there are no magnetic charges (magnetic monopoles).
The magnetic field has no sources or sinks – its field lines can only
form closed loops.
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Maxwell's Equations in Matter
The Third Maxwell Equation, or Faraday’s Law of Induction:
B
E  
t
describes how a time-varying magnetic field causes an electric field
(induction).
The Fourth Maxwell Equation:
D
H  j
t
shows that magnetic (magnetizing) fields can be caused by electric
currents (Ampère’s Law), but also by changing electric (displacement)
fields (Maxwell’s Correction – which is very important, since it “allows”
for electromagnetic waves – also in vacuum).
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Maxwell's Equations in Gas
The previous formulations are known as Maxwell’s Macroscopic
Equations or Maxwell’s Equations in Matter. Under specific conditions
the relations on slide 07 can be simplified.
The Earth's atmosphere is a linear medium – the induced polarization
P is a linear function of the imposed electric field E. The Earth’s
atmosphere is also an isotropic medium – P is parallel to E:
P  ε0  e E
The electric susceptibility χe degenerates to a simple scalar (in
general it would be a tensor of second rank) and we get:
D  ε0 1   e E  ε 0εr E  εE
D  εE
where ε is the permittivity (or dielectric constant in a homogenous
medium) and εr = 1 + χe is the dimensionless relative permittivity,
which depends on the material and is unity for vacuum.
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Maxwell's Equations in Gas
Similar considerations for M and H yield:
M  μ0  m H
B  μ0 1   m H  μ0 μr H  μ H
where χm is the (scalar) magnetic susceptibility (in general it would be
again a tensor of second rank), µ is the permeability and µr = 1 + χm is
the dimensionless relative permeability (which is also unity in vacuum).
The electric current density j is related to the electric field E via the
electric conductivity σ [Ω-1m-1] (a scalar for isotropic media, but in
general again a tensor) through the differential form of Ohm’s Law:
jσE
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Maxwell's Equations in Neutral Gas
The lower atmosphere (troposphere and stratosphere, at least up to
~ 50 km) is a neutral (ρfree = 0), and isotropic medium, and has a
negligible electric conductivity (σ = 0) yielding j = 0.
Maxwell’s equations can therefore be written as:
 E  0
 B  0
B
E  
t
E
  B  εμ
t
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Electromagnetic Waves
Maxwell's equations relate the vector fields by means of simultaneous
differential equations. On elimination we can obtain differential
equations, which each of the vectors must separately satisfy. Applying
the curl operator on Faraday’s law, interchanging the order of
differentiation with respect to space and time (which can be done for a
slowly varying medium like the atmosphere is one at frequencies of
practical interest) and inserting the fourth Maxwell equation yields:
with
we get

 2E
    E     B   εμ 2
t
t
    E    E  E   E  0
 2E
E  εμ 2
t
 2B
B  εμ 2
t
  2
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Electromagnetic Waves
These partial differential equations are standard wave equations.
Considering plane waves, the solutions have the form:
E(r, t )  E0 expi k  r  ωt

B(r, t )  B0 expi k  r  ωt 
where k is the wave number vector, pointing in the direction of wave
propagation. The angular frequency, ω [rad/s] and the angular wave
number, k [rad/m], are defined as (with k = |k|):
ω  2
and
k
2

where ν is the frequency (Hz) and λ is the wavelength [m]. Inserting the
above solutions into Maxwell’s equations yields:
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Electromagnetic Waves
i k E  0
i k B  0
i k  E  i B
i k  B   i E
which shows that the field vectors E and B are perpendicular to each
other and that both are perpendicular to k. Electromagnetic waves are
thus transverse waves.
micro.magnet.fsu.edu
wikimedia
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Electromagnetic Waves
Inserting the first equation of slide 14 in to the wave equation using the
vector identity:
k  k  E  k  E k  k  k  E
and the orthogonality
k E  0
yields
k 2  2
With the definition of the phase velocity:

c
k
we see that monochromatic electromagnetic waves propagate in a
medium with the phase velocity:
c
1


1
 0  0 r  r
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Electromagnetic Waves
And in vacuum we get:
c0 
1
 0 0
C0 = 299 792 458 m s-1 which is nothing else
than the speed of light in vacuum
In geometric optics the refractive index of a medium (n) is defined as
the ratio of the speed of light in vacuum (c0) to that in the medium (c):
c0
n
  r r
c
which is known as the Maxwell Relation.
In the Earth’s atmosphere the relative permeability is almost exactly = 1,
thus we get (in a general, frequency-dependent form):
n( )   r ( )
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Electromagnetic Waves
James N. Imamura, Univ. Oregon
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Electromagnetic Waves
ESA
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Gamma Rays
A gamma-ray blast 12.8 billion light years away,
Supernova Cassiopeia A, Cygnus region.
NASA/DOE/Swift
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X Rays
The Sun and the Earth’s northern Aurora-Oval in X-Rays.
JAXA/NASA/POLAR
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Ultraviolet
The Sun in UV and the “Ozone Hole” above Antarctica.
NASA/SDO
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Visible
(US) National Optical Astronomy Observatory
The visible part of the solar spectrum – including Fraunhofer lines
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Visible
Color temperatures of stars and spectral signatures on Earth.
Jenny Mottar SOHO/Jeannie Allen
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Near–Infrared
Vegetation and different Soil types from reflected near–infrared.
Jeff Carns/NASA
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Infrared
Saturn’s strange aurora and forest fires in California.
NASA/Jeff Schmaltz MODIS
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Microwaves
Hurricane Katrina.
Corresponding Wavelengths: 1m to 1 mm
NASA
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Radio Waves
Michael L. Kaiser, Ian Sutton, Farhad Yusef-Zedeh NASA
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Radio Waves
In German:
LW – Langwelle
MW – Mittelwelle
KW – Kurzwelle
UKW – Ultrakurzwelle
Microwaves – 1m to 1 mm
Physics Hypertextbook
Military Radar Nomenclature:
L (1 – 2 GHz), S, C, X (8 – 12 GHz),
Ku, K (18 – 27 GHz) and Ka bands
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Solar EM Waves
Wiki
For processes in the lower atmosphere wavelengths from
0.2 to 100 µm are most important