35 Cavity radiation and thermal noise
Cavity radiance:
Energy den
sity
E
(
f
)
=
N
(
f
)
V
W
(
f
)
=
8
π
f
2
c
3
hf
e
hf/KT

1
J/m
3
Hz
.
in a 3D cavity in thermal equi
librium resides by equal amounts
in the traveling wave components
of the cavity modes arriving with
speed
c
from the boundaries of
the cavity subtending
4
π
sterads.
Multiplying
E
(
f
)
by
c/
4
π
we ob
tain
L
(
f
) =
2
f
2
c
2
hf
e
hf/KT

1
W/m
2
/ster
Hz
,
which is called
radiance
and rep
resents the
power density per unit
solid angle
of the waves traveling
within the cavity.
Radiance
L
(
f
)
also represents the
spectrum of power
radiated
per
unit solid angle by a unit area of
a
blackbody surface
at temper
ature
T
(since nonreflective walls
of a cavity will produce the same
E
(
f
)
as partialreflecting walls as
mentioned earlier).
•
In a 1D cavity of some length
L
— e.g. a TL with
shorts
at both ends
as discussed in ECE 329 notes — the resonant frequencies are
f
m
=
c
λ
m
=
c
2
L/m
=
c
2
L
m,
where
m
= 1
,
2
,
3
,
· · ·
which indicates that the mode density in
f
is
N
(
f
) =
2
L
c
modes
Hz
.
Therefore the energy density in a 1D cavity in thermal equilibrium will
be
E
(
f
) =
N
(
f
)
L
W
(
f
)
=
2
c
hf
e
hf/KT

1
J/m
Hz
in analogy with the energy density of 3D cavities. This energy density
will reside by equal amounts in the traveling wave components of the
1D resonant modes arriving with speed
c
from the opposite ends of the
1D resonator. Power spectral content
P
(
f
)
of each of these traveling
wave components can thus be calculated as
c/
2
times
1
E
(
f
)
, i.e.,
P
(
f
) =
hf
e
hf/KT

1
W
Hz
.
1
Note that per TEM plane wave,
c
(
1
4
o
˜

E

2
+
1
4
μ
o
˜

H

2
) =
˜

E

2
4
η
o
+
η
o
˜

H

2
4
=
˜

E

2
2
η
o
,
which confirms that the timeaveraged stored energy density times
c
is indeed the timeaveraged power
transported per unit area.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Thermodynamics, Electromagnet, Resistor, Energy density, Electrical impedance, Thévenin's theorem

Click to edit the document details