handbook of modern sensors

3.12 Heat Transfer 105
Fig. 3.42. Spectral flux density for three
temperatures for the ideal radiator emanating
toward infinitely cold space.
energy in the visible range, and the power radiated by the cooler objects is concentrated
in the infrared and far-infrared portion of the spectrum.
Equation (3.132) is quite complex and cannot be solved analytically for any particular
bandwidth.Asolution can be found either numerically or by an approximation.
An approximation for a broad bandwidth (when λ 1 and λ 2 embrace well over 50% of
the total radiated power) is a fourth-order parabola known as the Stefan–Boltzmann
law:
b0 = Aεσ T 4 . (3.133)
Here σ = 5.67 × 10 −8 W/m 2 K 4 (Stefan–Boltzmann constant), A is the geometry
factor, and ε is assumed to be wavelength independent [37].
Whereas wavelengths of the radiated light are temperature dependent, the magnitude
of radiation is also a function of the surface property called emissivity, ε.
Emissivity is measured on a scale from 0 to 1. It is a ratio of flux which is emanated
from a surface to that emanated from the ideal emitter having the same temperature.
There is a fundamental equation which relates emissivity ε, transparency γ , and
reflectivity ρ:
ε + γ + ρ = 1. (3.134)
In 1860, Kirchhoff had found that emissivity and absorptivity, α, is the same thing.
As a result, for an opaque object (γ = 0), reflectivity ρ and emissivity ε are connected
by a simple relationship: ρ = 1 − ε.
The Stefan–Boltzmann law specifies radiant power (flux) which would be emanated
from a surface of temperature T toward an infinitely cold space (at absolute
zero). When thermal radiation is detected by a thermal sensor, 19 the opposite radiation
19 Here, we discuss the so-called thermal sensors as opposed to quantum sensors, which are
described in Chapter 13.