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.
[Illustration: FIG. 22.]
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.
{46}
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.
{49}
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.
[Illustration: FIG. 23.]
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.
{50}
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.
{52}
+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.
{56}
+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°.
{57}