The rate at which a substance emits or takes up radiant energy depends upon the nature of its surface. A rough, black surface, such as may be obtained by holding an object in the smoke from burning camphor, radiates and absorbs heat with greater freedom than any other; whilst a polished, metallic surface, which acts as a reflector, is worst of all in these respects. Even a surface of finely divided soot, however, does not completely absorb all the radiations which fall upon it, but exhibits a small degree of reflection. An “absolute black surface,” if such could be found, would be totally devoid of reflecting power, and would absorb all the radiant energy incident upon it; and conversely would radiate all energy reaching it from its under side, without reflecting any back, or allowing any to pass through in the manner that light waves are transmitted through a transparent substance. No such perfect surface is known; but, as Kirchoff showed, it is possible to make a radiating arrangement which will give the same numerical result for the energy radiated as would be obtained by a perfect surface at the same temperature. Such an arrangement is termed a “black body,” and radiations from it are designated “black-body radiations.”
Fig. 42.—Black-Body Radiations.
Any enclosure, if opaque to radiant energy, and kept at a constant temperature, constitutes a black body, and radiations received from the interior through a small opening in the side are black-body radiations. [Fig. 42] represents such an enclosure; in which, to show the application to pyrometry, a body A is indicated opposite to an opening in the side, through which radiations escape from the surface of A. If this surface were “perfect,” all the waves falling upon it would be completely absorbed and completely radiated; but to prevent change of temperature the energy radiated must balance the energy received. If, on the other hand, the surface of A were a polished metal, the waves falling upon it from the sides of the enclosure would in the main be reflected; but here again the energy leaving the surface must equal the amount received if the temperature be constant. It follows, therefore, that if no alteration in temperature occur, the energy leaving the surface of A is independent of the nature of that surface; and the amount escaping through the opening will therefore be the same, whatever be the character of the surface opposite the opening. With a good radiating surface the rays from the enclosure will first be absorbed and then radiated through the opening; in the case of a poor radiating surface, the rays will be directly reflected through the opening; the total energy escaping being the same in either case. It will be seen later that radiation pyrometers are based upon black-body radiations; and it is important to note that the arrangement under discussion is realised in a furnace at a constant temperature, in which A might represent an object such as a block of steel. It happens, therefore, that the condition of perfect radiation is attained by the appliances in everyday use; and, moreover, black-body radiations can always be secured by placing a tube, closed at one end, in the heated space, and receiving the radiations through the open end; for this again represents an enclosure at a constant temperature. Similarly, radiations from a solid in the interior of the tube of the electric furnace shown in [fig. 29] will be of the same description, and we can therefore apply with accuracy any instrument based upon black-body radiations, knowing that the same may be readily realised in practice.
The law connecting the energy radiated by a substance, under given conditions, with its temperature, was variously stated by different observers until Stefan, in 1879, deduced the true relation from certain experimental data obtained by Tyndall. Stefan concluded that the figures given by Tyndall indicated that the energy radiated by a given solid varied as the fourth power of its absolute temperature. Numerous experiments, under different conditions, showed that the fourth-power law did not apply to all kinds of surfaces or circumstances; but a strong confirmation of its truth when applied to black-body radiations was forthcoming in 1884, when Boltzmann showed, from thermodynamic considerations, that the quantity of energy radiated in a given time from a perfect radiator must vary as the fourth power of its absolute thermodynamic temperature. Certain assumptions made by Boltzmann in this investigation were subsequently justified by experiment; and numerous tests under black-body conditions have since amply verified the law. It is upon the Stefan-Boltzmann law that radiation pyrometers are based; the energy received by radiation from the heated substance, under black-body conditions, being measured by the instrument, and translated into corresponding temperatures on its scale.
Expressed in symbols, the fourth-power law takes the form—
where E is the total energy radiated; T1 the absolute temperature of the black body; T2 the absolute temperature of the receiving substance, and K a constant depending upon the units chosen. If E be expressed as watts per square centimetre, the value of K is 5·6 × 10-12; if in calories per square centimetre per second, the value is 1·34 × 10-12. The introduction of the temperature of the receiving substance, T2, is rendered necessary by the fact, previously cited, that energy will be radiated back to the hot body, and the net loss of energy will evidently be the difference between that which leaves it and that which returns to it from the receiving substance. If T2 were absolute zero, the energy leaving the black body would be K(T1)4; whereas if T2 were equal to T1, the loss of energy would be nil, as a substance cannot cool by radiation to a lower temperature than its surroundings. The temperatures T1 and T2 refer to the thermodynamic scale (page 9), but as the gas scale is practically identical, Centigrade degrees may be used, measured from absolute zero, or -273°. An example is appended to illustrate the application of the law:—