where R = the rate of emmision in Watts per square meter
e = the emissivity of the surface. It varies between zero and one, depending on the nature of the surface.
sigma = the Stefan-Boltzmann constant
= 5.6696x10-8 watts per square meter Kelvin to the 4th power
T = Kelvin temperature [2]
If the object is in thermal equilibrium with its environment, it will absorb exactly the same amount of radiation as it emits. If an object is at a temperature T1 and surrounded by walls at a temperature T2, the net loss or gain of radiant energy is
This property of nature is used in many types of detection systems, such as night-vision goggles, spy satellites, and thermography.[3]
Objects which are good absorbers of radiant energy are also good emitters. An object which is highly reflective is a poor emitter of radiant energy. The best emitter would be a surface which also the best absorber. Objects which absorb all incident radiant energy would appear black (except for the case where the temperature of the object is high enough to produce visible light). Such objects are called blackbody radiators, or simply blackbodies. The emissivity e of an ideal blackbody is equal to one.
The study of blackbodies occcupied the minds of some of the most famous physicists of the late 19th and early 20th century. Wien found that the wavelength (in centimeters) of maximum radiation was related to the Kelvin temperature of the object by the simple relationship:
This is the mathematical statement of what is known as Wien's Displacement Law, which states that as the temperature of a blackbody is increased, the wavelength of maximum emission moves in the direction of shorter wavelengths in such a way that the product of this wavelength and the Kelvin temperature remains constant.
The culmination of blackbody research is Plank's Law of Radiation. Planck's law encompasses Stefan's Law, Wien's Law, as well as several other prior results of the study of ideal blackbodies. Plank's law shows that the bandwidth of thermal radiation is very large compared to other forms of electromagnetic radiation. The attached graph shows the spectrum of thermal radiation for several different temperatures. [4]
The general nature of the radiation from an ordinary lightbulb can be explained in terms of Planck's law and is easily demonstrated with a prism. The peak wavelength of emission is not in the visible region but rather in the infrared. Such bulbs are not very energy efficient.
In fact, ideal blackbody radiation is not a very efficient means of producing visible light. The maximum possible luminous efficiency of a blackbody is only 14% (See section on Photometry.)
Halogen bulbs are also incandescent, but operate at a higher temperature than ordinary tungsten bulbs. They are much more efficient producers of visible light.
where E = The energy lost in joules
h = Planck's constant
= 6.63x10-34 Joule-seconds
= 4.136x10-15 electron volt seconds
f = Frequency of the radiation
using
with E in electron-volts,
The simplest case to describe is what happens when an electron bound to a proton (a hydrogen atom) goes from one stable energy level to another. The stable energy levels are assigned an integer, n. In the level n = 1, the electron is bound to the proton with a binding energy of 13.6 eV ( 1eV = 1.6x10-19 joules). For level n = 2, the binding energy is 13.6/4 eV. For the nth level, the binding energy is 13.6/n2 eV. When an electron goes from one level to another that has more binding energy, it emits electromagnetic radiation. For hydrogen, transistions from lower binding energy levels to n = 2 result in visible and ultraviolet light.
The energy lost (i.e. radiated) in going from n = 3 to n = 2 is
The wavelength is
In similar fashion,
From n = 4 to 2 486.1 nm
5 to 2 434.1
6 to 2 410.2
7 to 2 397.1
8 to 2 389.0
9 to 2 383.6
10 to 2 379.8
11 to 2 377.1
infinity to 2 364.7
The SI unit of luminous intensity is the candela. "The luminous flux per steradian from a candela is equal to 1/60 th of the luminous flux per steradian of a blackbody radiator 1 cm2 in area at the solidification temperature of platinum (2042 K). The candela emits one lumen per steradian. ... The number of lumens in a source with a different spectral distribution from that of the standard should be defined as equalling the number of lumens in a standard source which appears "equally bright" to an average human eye." The eye response referred to here is that of the bright-adapted eye.
Two units of illumination are in common use. They both refer to the number of lumens passing through a portion of the surface of a sphere centered on the source. One lumen passing through a square-meter of such a surface is defined as a lux or a meter candle. One lumen passing through a square foot of such a surface is defined as a foodcandle (fc). ( 10.76 lux = 1 footcandle)
The steradian is in effect a unit of area measurement on the surface of a sphere equal in size to the square of the radius. Since the total surface area of a sphere is 4 PI R2, there are 4 PI steradians for a complete sphere. A hemisphere would have 2 PI steradians.
"Luminous efficiency, usually expressed in percent, is used to relate the effectiveness of a given light source in producing lumens compared to the effectiveness of a monochromatic source of the same power (watts) which radiates at the photopic eye's response peak." Photopic means bright-adapted, and the wavelength at its peak is 556 nm (Green-Yellow). A source of 100% efficiency would produce 680 lumens per watt. The figure below shows human eye response for both the bright and dark adapted case. [5]
The maximum luminous efficiency of a blackbody radiator is only 14% (95 lumens/watt) and occurs when the temperature of the object is 6600 K. The peak wavelength of such a source would be around 440 nm (Violet).
Since 1971, the Federal Trade Commission has required that light bulb packages be labelled in terms of lumens output as well as watts input. For example, a 100 watt soft white bulb is labelled 1710 lumens. The luminous efficiency would be 17.1 lm/watt or 2.5%. The table below is for General Electric Standard light bulbs.
Watts Lumens L/W Efficiency 40 505 12.6 1.86% 60 870 14.5 2.13% 75 1190 15.9 2.33% 100 1750 17.5 2.6 %
Note that one 100 watt bulb puts out more lumens than two 60 watt bulbs and more than three 40 watt bulbs.
[1] John Tyndall is also noted for: demonstrating the trapping of light in a stream of water; The bluish appearance of a soap bubble is called the Tyndall effect; Fifty years before Sir Alexander Fleming, he described the effect of penicillium mold in slowing down the rate of growth of bacteria.
[2] Kelvin temperature is the temperature of an object in degrees Celsius plus 273.16. Zero on this scale is known as absolute zero. Loosely speaking, this is the temperature at which random molecular motion ceases.
[3] In 1964, Arno Penzias and Robert Wilson, at Bell Laboratories, measured thermal radiation at a temperature of 3 K, which agrees with the big-bang model of the creation of the universe. They received the Nobel Prize for this work in 1978.
[4] This graph is a plot of Planck's law for four different temperatures. The equation is:
Where,
h = Planck constant = 6.63x10-34 Joule-seconds
c = speed of light in a vacuum = 3x108 meters per second
k = Boltzmann constant = 1.38x10-23 joules per Kelvin
T = Temperature in Kelvin
[5] Note that the response for the dark adapted eye shifts towards the violet end of the spectrum, compared to the response for the bright adapted eye. It takes only 2 or 3 minutes to go from dark adapted to bright adapted. BUT IT TAKES 30 MINUTES TO GO FROM BRIGHT ADAPTED TO DARK ADAPTED. On Navy ships at night, red lights are used so as not to destroy ones dark adaptation. Recently, these red night lights have been replaced by blue lights to minimize the contrast between the lighting and the CRT displays.