CHAPTER V

FULL RADIATION

+The Full Radiator.+--We have assumed that a lamp-blacked surface is a perfect absorber, and consequently a full radiator, but although it is a very near approach to the ideal it is not absolutely perfect. No actual surface is a perfectly full radiator, but the exact equivalent of one has been obtained by an ingenious device. A hollow vessel which is blackened on the inside has a small aperture through which the radiation from the interior of the vessel can escape. If the vessel is heated up, therefore, the small aperture may act as a radiator. The radiation which emerges through the aperture from any small area on the interior of the vessel is made up of two parts, one part which it radiates itself, and the other part which it scatters back from the radiation which it receives from the other parts of the interior of the vessel. These two together are equal to the energy sent out by a full radiator, and therefore the small aperture acts as a full radiator: _e.g._ suppose the inner surface has an absorbing power of 90 per cent., then it radiates 90 per cent. of the full radiation and absorbs 90 per cent. of the radiation coming up to it therefore scattering back 10 per cent. We have therefore coming from the inner surface 90 per cent. {45} radiated and 10 per cent. scattered, and the radiated and scattered together make 100 per cent.

One form in which such radiators have been used is shown in section in Fig. 22. A double walled cylindrical vessel of brass has a small hole, _a_, in one end. Steam can be passed through the space between the double walls, thus keeping the temperature of the inner surface at 100° C. A screen with a hole in it just opposite to the hole in the vessel, or rather several such screens, are placed in front of the vessel in order to shield any measuring instrument from any radiation except that emerging through the hole.

+The Full Absorber.+--In an exactly similar way an aperture in a hollow vessel will act as a full absorber, for the fraction of the incident radiation which is scattered on the inner surface again impinges on another portion of the surface and so all is ultimately absorbed except a minute fraction which is scattered out again through the aperture.

The variation in the heat radiated by a full radiator at different temperatures forms a very important part of the study of radiation, and a very large number of experiments and theoretical investigations have been devoted to it. These investigations may be divided into two sections: those concerned with the total quantity of heat radiated at different temperatures and those concerned with the variation in the character of the spectrum with varying temperatures.

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The experiments in the first section have been carried out mainly in two ways. In the first, the rate of cooling of the full radiator has been determined, and from the rate of cooling at any temperature the rate at which heat was lost by radiation was immediately calculated. Newton was the first to investigate in this way by observing the rate at which a thermometer bulb cooled down when it was surrounded by an enclosure which was kept at a uniform temperature. He found that the rate of cooling, and therefore the rate at which heat was lost by the thermometer, was proportional to the difference of temperature between the thermometer and its surroundings. This rule is known as Newton's Law of Cooling, and is still used when it is desired to correct for the heat lost during an experiment where the temperature differences are small. It is only true, however, for very small differences of temperature between the thermometer and its surroundings, and as early as 1740 Martine had found that it was only true for a very limited range of temperature.

+Prévost's Theory of Exchanges.+--In 1792, Prévost of Geneva, when endeavouring to explain the supposed radiation of cold, introduced the line of thought, that any body is not to be regarded as radiating heat only when its temperature is falling, or absorbing heat only when its temperature is rising, but that both processes are continually and simultaneously going on. The amount of heat radiated will depend on the temperature and character of the body itself, while the amount absorbed will depend upon the condition of the surroundings as well as upon the nature {47} of the body. If the amount of heat radiated is greater than the amount absorbed the body will fall in temperature, and _vice-versa_. This view of Prévost's is called the Theory of Exchanges, and we can see that it is a necessary consequence of our ideas as to the production of heat and light waves by the agitation of electrons in the radiating body.

If the rate of cooling of a body at a certain temperature is measured when it is placed in an enclosure at a lower temperature, it must be borne in mind that the rate of loss of heat is equal to the rate at which heat is radiated minus the rate at which it is absorbed from the enclosure.

A second way in which the heat lost by a body has been measured at different temperatures is by heating a conductor such as a thin platinum strip by means of an electric current, and measuring the temperature to which the conductor has attained. When its temperature is steady, all the energy given to it by the current must be lost as heat, and therefore the electrical energy, which can very easily be calculated, must be equal to the heat radiated by the body minus the heat received from the enclosure.

So many attempts have been made to establish, by one or other of these two methods, the relation between the quantity of heat radiated and the temperature, that it is impossible to give even a passing reference to most of them. Unfortunately, the results do not show the agreement with one another which we would like, but probably the most correct result is that stated by Stefan in 1878, after a close inspection of the experimental results of Dulong and {48} Petit. He stated that the quantity of heat radiated per second by a full radiator is proportional to the fourth power of its absolute temperature.[1] Thus the quantity of heat radiated by one square centimetre of the surface of a full radiator whose absolute temperature is T, is equal to ET(sup)4, where E is some constant multiplier which must be determined by experiment and which is called the radiation constant. If the absolute temperature of the enclosure in which the surface is placed is T, then the rate at which the surface is losing heat will be E(T(sup)4-T(sub)1(sup)4), for it will receive heat at the rate ET(sub)1(sup)4 and will radiate it at the rate ET(sup)4.

[1] See page 56.

Stefan's fourth power law has been verified by a number of good experiments, notably those of Lummer and Pringsheim (_Congrés International de Physique_, Vol. II. p. 78), so that although some experiments do not agree with it, we are probably justified in taking it as correct.

In 1884 Boltzmann added still further evidence in support of this law by deriving it theoretically. He applied to a space containing the waves of full radiation the two known laws which govern the transformation of energy, by imagining the space to be taken through a cycle of compressions and expansions in just the same way as a gas is compressed and expanded in what is known as Carnot's cycle.

+Variation of Spectrum with Temperature.+--The variation of the character of the spectrum of a full radiator has been determined mainly by the use of Langley's bolometer, but the general nature of the change may be readily observed by the eye.

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As the temperature of a full radiator rises it first gives out only invisible heat waves; as soon as its temperature exceeds about 500° C. it begins to emit some of the longest visible rays; and as the temperature rises further, more and more of the visible rays in the spectrum are emitted until, when the radiator is white hot, the whole of the visible spectrum. is produced. Thus the higher the temperature of the radiator the more of the shorter waves are produced.

By means of Langley's bolometer the distribution of energy in the spectrum has been measured accurately, with the results of confirming and amplifying the general results just stated. The energy in the spectrum of even the hottest of terrestrial radiators is mostly in the longer waves of the infra-red, but the position of the maximum of energy moves to shorter and shorter wave-lengths as the temperature rises, and so more of the shorter waves make their appearance. The sun is not a full radiator, but is nearly so, and its temperature is so high that the maximum of energy in its spectrum is in the visible part near to the red end.

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Fig. 23 shows the results obtained by Lummer and Pringsheim, and brings out clearly the shift of the maximum with rising temperature and also the position of the greatest part of the energy in the infrared region.

+Wien's Laws.+--Examination of the results also shows that the wave-length at which the maximum energy occurs is inversely proportional to the absolute temperature and that the actual energy at the maximum point is proportional to the fifth power of the absolute temperature. These two results have both been derived theoretically by Wien[2] in a similar way to that in which Boltzmann derived Stefan's fourth power law, _i.e._ by imagining a space filled with the radiation to be taken through a cycle of compressions and rarefactions.

[2] _Wied. Ann._, 46, p. 633; 52, p. 132.

Wien derived an amplification of the last result by showing that if a wave-length in the spectrum of a full radiator at one temperature and another wave-length in the spectrum at another temperature are so related as to be inversely proportional to the two absolute temperatures, they may be said to correspond to each other, and the energy in corresponding wave-lengths at different temperatures is proportional to the fifth power of the absolute temperature.

We see therefore that if the distribution of energy in the spectrum of the full radiator be known at any one temperature it may be calculated for any other temperature by applying these two laws of corresponding wave-lengths and the energy in them.

Neither of them give us any information, however, {51} about the actual distribution of energy at any one temperature from which we may calculate that at any other temperature. For that, some relation must be found between the energy and the wave-length. Planck, by reasoning founded on the electromagnetic character of the waves, derived such a relation, but both his reasoning and his results are a little too complicated to be introduced here. His results have been confirmed in the most striking manner by experiments carried out by Rubens and Kurlbaum (_Ann. der Physik_, 4, p. 649, 1901). They measured the energy in a particular wave-length (.0051 cms., _i.e._ nearly 100 times the wave-length of red light) in the radiation of a full radiator from a temperature of 85° up to 1773° absolute, and their results are given in the following table:

Absolute Temperature. Observed Energy. Energy calculated from Planck's Formula.

85 -20.6 -21.9 193 -11.8 -12 293 0 0 523 +31 +30.4 773 64.6 63.8 1023 98.1 97.2 1273 132 132 1523 164 160 1773 196 200

We have therefore the means of calculating both the total quantity and the kind of radiation given out by any full radiator at any temperature, and a number of very interesting problems may be solved by means of the results.

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+Efficiency in Lighting.+--One very simple problem is concerned with efficiency in lighting. We see by reference to Fig. 16, that in the radiation from the electric arc very little of the energy is in the visible part of the spectrum even though the temperature in the arc is the highest yet obtained on the earth, whereas the energy in the visible part of the spectrum from a gas flame is almost wholly negligible. The problem of efficient lighting is to get as big a proportion as possible of the energy into the visible part of the spectrum, and therefore the higher the temperature the greater the efficiency. This is the reason of the greater efficiency of the incandescent gas mantle over the ordinary gas burner, for the introduction of the air into the gas allows the combustion to be much more complete, and therefore the temperature of the mantle becomes very much higher than that of the carbon particles in the ordinary flame. The modern metallic filament electric lamps have filaments made of metals whose melting point is extremely high, and they may therefore be raised to a much higher temperature than the older carbon filaments. The arc is even more efficient than the metallic filament lamps, because its temperature is higher still; and we must assume that the temperature of the sun is very much higher even than the arc, since its maximum of energy lies in the visible spectrum.

+Temperature of the Sun.+--The actual temperature of the sun may be calculated approximately by means of Stefan's fourth power law. We will first assume that the earth and the sun are both full radiators, and {53} that the earth is a good conductor, so that its temperature is the same all over. The first assumption is very nearly true, and we will make a correction for the small error it introduces; and the second, although far from true, makes very little difference to the final result, for it is found that the values obtained on the opposite assumption that the earth is an absolute non-conductor differ by less than 2 per cent. from those calculated on the first assumption. We will further assume that the heat radiated out by the earth is exactly equal to the heat which it receives from the sun. This is scarcely an assumption, but rather an experimental fact, for experiment shows that heat is conducted from the interior of the earth to the exterior, and so is radiated, but at such a small rate that it is perfectly negligible compared with the rate at which the earth is receiving heat from the sun.

The sun occupies just about one 94,000th part of the hemisphere of the heavens or one 188,000th part of the whole sphere. If the whole sphere surrounding the earth were of sun brightness, the earth would be in an enclosure at the temperature of the sun, and would therefore be at that temperature itself. The sphere would be sending heat at 188,000 times the rate at which the sun is sending it, and the earth would be radiating it at 188,000 times its present rate. But the rate at which it radiates is proportional to the fourth power of its absolute temperature, and therefore its temperature would be the fourth root of 188,000 times its present temperature, _i.e._ 20.8 times. If the radiating or absorbing power of the earth's surface be taken as 9/10, which is somewhere near the mark, {54} the calculation gives the number 21.5 instead of 20.8. The average temperature of the earth's surface is probably about 17° C. or 290° absolute, and therefore the temperature of the sun is 290 x 21.5, _i.e._ about 6200° absolute.

It is easy to see that if we had known the temperature of the sun and not of the earth, we could have calculated that of the earth by reversing the process.

By this means we can estimate the temperatures of the other planets, at any rate of those for which we may make the same assumptions as for the earth. Probably those planets which are very much larger than the earth are still radiating a considerable amount of heat of their own, and therefore to them the calculation will not apply; but the smaller planets Mercury, Venus and Mars have probably already radiated nearly all their own heat and are now radiating only such heat as they receive from the sun. The temperatures calculated in this way are--

Average Absolute Temperature

Mercury . . . . . . . . . 467° Venus . . . . . . . . . . 342° Earth . . . . . . . . . . 290° Mars . . . . . . . . . . 235°

Since the freezing point of water is 273° absolute, we see that the average temperature of Mars is 38° C. below freezing, and it is almost certain that no part of Mars ever gets above freezing point.

In a very similar way we may find the temperature to which a non-conducting surface reaches when it is exposed to full sunlight by equating the heat absorbed to the heat radiated, and the result comes {55} to 412° absolute, _i.e._ 139° C., or considerably above boiling point. This would be the upper limit to the temperature of the surface of the moon at a point where the sun is at its zenith.

On the surface of the earth the sunlight has had to pass through the atmosphere, and in perfectly bright sunshine it is estimated that only three-fifths of the heat is transmitted. Any surface is also radiating out into surroundings which are at about 300° absolute. Taking into account these two facts, we find that the upper limit to a non-conducting surface in full sunshine on the earth is about 365° absolute, or only a few degrees less than the boiling point of water.

+Effective Temperature of Space.+--The last problem we will attack by means of the fourth power law is the estimation of the effective temperature of space, _i.e._ the temperature of a full absorber shielded from the sun and far away from any planet.

It is estimated by experiment that zenith sun radiation is five million times the radiation from the stars. This estimate is only very rough, as the radiation from the stars is so minute. As the sun only occupies one 94,000th part of the heavens, the radiation from a sunbright hemisphere would be five million times 94,000 times starlight, _i.e._ 470,000,000,000 times. The temperature of the sun is therefore the fourth root of this quantity times the effective temperature of space, _i.e._ about 700 times. Since the temperature of the sun is about 6200°, the temperature of space is a little under 10° absolute; _i.e._ lower than -263° C.

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+Note on Absolute Temperature.+--It is found that, if a gas such as air has its temperature raised or lowered while its pressure is kept uniform, for every one degree centigrade rise or fall its volume is increased or decreased by one two hundred and seventy-third of its volume at freezing point, _i.e._ at 0° centigrade. If therefore it continued in the same way right down to -273° centigrade, its volume would be reduced to zero at this temperature. This temperature is therefore called the absolute zero of temperature, and temperatures reckoned from it are called absolute temperatures. To get absolute temperatures from centigrade temperatures we evidently need to add 273°.

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