Fundamentals of Electromagnetics - RF/Microwave Interaction with Biological Tissues - page 32

< e

() >=

t 4 RkTB when < hf < kT (1.41)

2

where the brackets mean the average value. This is the well-known expression
for the Johnson-Nyquist noise emitted by a resistor. This expression depends
upon the value of the resistor and its range of application is limited because
of the limitations of the Rayleigh-Jeans approximation.
It is interesting to calculate the blackbody radiation and compare it with
other radiation sources evaluated earlier in this section. Equation (1.40) can
be used in most cases. It shows that the noise power of a blackbody source at
a physical absolute temperature of 300 K and measured in a bandwidth of
10 MHz is 41.4 ¥ 10 -15 W. If the source is at 3000 K and the receiver band-
width is 100 MHz, the power is 41.4 ¥ 10 -13 W. It appears that the blackbody
radiation is at a power level much smaller than most of the physical and in-
dustrial sources.

Stefan-Boltzmann Law Planck's law predicts the brightness curves presented
in Figure 1.6 as a function of frequency and temperature. Integrating the
Planck radiation law over all frequencies, which is summing the area of the
Planck radiation law curve for that temperature, yields the total brightness B t
for a blackbody radiator:

2 h f

3

= Ê
Ë

ˆ
¯

•

Ú

B df W sr m (1.42)

- 1 - 2

t

c

2

e

hf kT

- 1

0

The integration yields the Stefan-Boltzmann relation [15]

BT
t

= s W sr m (1.43)

4 - 1 - 2

where B t is the total brightness, s a constant equal to 5.67 ¥ 10 -8 Wm -2 K -4 , and
T the absolute blackbody temperature in kelvin. It must be observed that the
total brightness of a blackbody increases as the fourth power of its tempera-
ture. As an example, the total brightness of a blackbody at 1000 K should be
16 times that of at 500 K. This temperature dependence is valid only for the
total brightness and not for the brightness (1.33) or (1.35).

Wien Displacement Law Planck's law represented in Figure 1.6 shows that the
peak brightness shifts to higher frequencies with an increase in temperature.
Maximizing Eqn. (1.33) by differentiating with respect to frequency f , that is,
in terms of unit frequency, and setting the result equal to zero yield a quanti-
tative expression for this displacement at a specific frequency f P . Noting that
the peak occurs at high values of the ratio hf P / kT and simplifying accordingly
yield the expression of the wavelength l P at which B is a maximum [15]:

hc
==

l p T 0 0048 . m K (1.44)

3

This relation, in which the product of wavelength and temperature is a con-
stant, is called the Wien displacement law : The wavelength of the maximum of