Electromagnetic waves

1
Electromagnetic waves

• Hecht, Chapter 2
• Monday October 21, 2002

2
Electromagnetic waves

• Consider propagation in a homogeneous medium (no
  absorption) characterized by a dielectric
  constant

?o permittivity of free space
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Electromagnetic waves
Maxwells equations are, in a region of no free
charges,
Gauss law electric field from a charge
distribution
No magnetic monopoles
Electromagnetic induction (time varying magnetic
field producing an electric field)
Magnetic fields being induced By currents and a
time-varying electric fields
µo permeability of free space (medium is
diamagnetic)
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Electromagnetic waves
For the electric field E,
or,
i.e. wave equation with v2 1/µo?
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Electromagnetic waves
Similarly for the magnetic field
i.e. wave equation with v2 1/µo?
In free space, ? ? ?o ?o
(? 1)
c 3.0 X 108 m/s
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Electromagnetic waves
In a dielectric medium, ? n2 and
? ? ?o n2 ?o
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Electromagnetic waves Phase relations
The solutions to the wave equations,
can be plane waves,
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Electromagnetic waves Phase relations

• Using plane wave solutions one finds that,

• Consequently

i.e. B is perpendicular to the Plane formed by k
and E !!
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Electromagnetic waves Phase relations

• Since also
• k ? E and k ? B ( transverse wave)
• Thus, k, E and B are mutually perpendicular
  vectors
• Moreover,

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Electromagnetic waves Phase relations

• Thus E and B are in phase since,
• requires that

E
k
B
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Irradiance (energy per unit volume)

• Energy density stored in an electric field
• Energy density stored in a magnetic field

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Energy density
Now if E Eosin(?tf) and ? is very large We
will see only a time average of E
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Intensity or Irradiance
In free space, wave propagates with speed c
c ?t
A
In time ?t, all energy in this volume passes
through A. Thus, the total energy passing through
A is,
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Intensity or Irradiance
Power passing through A is,
Define Intensity or Irradiance as the power per
unit area
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Intensity in a dielectric medium
In a dielectric medium,
Consequently, the irradiance or intensity is,
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Poynting vector
Define
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Poynting vector
For an isotropic media energy flows in the
direction of propagation, so both the magnitude
and direction of this flow is given by,
The corresponding intensity or irradiance is then,
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Example Lasers
?o 8.854 X 10-12 CV-1m-1 (SI units)
Laser Power 5mW

Spot diameter Intensity (W/m2) Electric field magnitude (V/m)
2 mm 1.6 X 103 1.1 X 103
20 µm (e.g. focus by lens of eye) 1.6 X 107 1.1 X 105
2 µm 1.6 X 109 1.1 X 106
Same as sunlight at earth
Near breakdown voltage in water
nb. Colossal dielectric constant material
CaCu3Ti4O12 , ? 10,000 at 300K
Subramanian et al. J. Solid State Chem. 151, 323
(2000)