It is useful, in passing, to note some properties of the temperature field which are special, as well as some properties of our law to describe it, which are quite general. Obviously, the law is in some sense intrinsically expressed in terms of an approximation. We said, right from the beginning, that the whole point of the field description is that neither the air nor the temperature has any obvious features, hence neither specifies any obvious size scales. Yet we use some particular sphere to compute averages and estimate the rate of change. The actual rate of change, of course, is some number that an observer can stand someplace and measure, so it must not depend on the choice of sphere we use to predict it. On the other hand, it is also clear that if we use a large enough sphere (say, one that contains the entire stove/ice combination) we will miss most of the structure of the temperature field, and do not expect the answer to be very good. Having said this, it will not be surprising that the correct statement of the law for temperatures involves the use of a sufficiently small sphere to compute averages. The reason the law works is that, once we start using spheres that are much smaller than any of the features of the room, like the stove or the ice, our answer becomes increasingly independent of which sphere we choose. There seems to be some clearly identifyable limiting behavior when we use arbitrarily small spheres, that not only ignores which sphere is used, but also approaches the correct answer, the rate an observer actually measures when we check the law against experiments.
This kind of behavior is typical of field laws. Fields are useful as the clear counterpoint to objects precisely because they don't have obvious features or size scales. Yet people are objects, and most of the experiments they perform eventually make use of objects (like measuring machines), so the laws that relate the field behavior to the outcomes of experiments are often phrased in terms of many measurements of the effects on such objects, taken at many places. The trick, and the reason the laws work, is that with care we can choose sufficiently many individual measurements, in terms of sufficiently fine objects, that we can predict behaviors for the field which do not reflect artifacts of the way we measure them. As we learn more, we will be able to improve this by using a description that gets away from the language of objects. That is the subject of the second section of the chapter.
Another, particularly interesting thing to note about the temperature field is that the law of approaching the average, as long as it involves only one field, can never overshoot. We put this in the following context. A spring, recall, typically has some preferred, unstretched length. When it is stretched, it pulls back to try to approach that preferred length. Yet a mass, whether still or set in motion, wants to continue doing whatever it has been doing. Thus the interacting system of the spring and mass possesses very interesting properties. If the spring is stretched, it pulls back, and when released, it can set the mass in motion. By the time the mass has been pulled back to the spring's preferred length, the spring is no longer pulling, but the mass has aquired a significant momentum. Not being affected by the spring, it sails through the preferred position and stretches the spring the other way, after which the spring pulls back until it stops the mass and starts to bring it back, and so on. This is how oscillations are possible. The point is that the law for temperatures has no property analogous to the inertia of the mass, so there is nothing to cause an overshoot. Therefore, no matter how we heat or cool the air, unless we oscillate the sources by hand, the temperature field does not oscillate of its own accord.
There is one last, very important, thing that we should note
about the temperature field and its affects. It involves a new
concept that is `familiar' but that we have not introduced
carefully. That is the concept of heat. Heat is different from
temperature, and the difference is important, which is why we
have always spoken of a temperature field, but never of a `heat
field'. Without pursuing a long tangent on the subject, we note
that what we think of as heat
is a particular kind of
energy. A better way to say it is that what we feel as
heat describes a certain way that we can interact with the
energy that other things can have.
We have said that it has been recognized for a long time that air is made of small, very hard atoms and molecules, of things like Nitrogen, Oxygen (in molecules) and Carbon Dioxide, or Helium, Argon, Radon, etc. (as atoms). In air these atoms are not packed very closely together, which is why air is not very dense, compared to substances like water, metals or rocks, in which the atoms are. The thing that holds atoms of air widely separated, and keeps the ocean of air from sinking to the surface of the earth like its ocean of water has, is that the molecules are in violent motion. Like so many marbles in a shaken box, they collide with one another and their bouncing keeps them from being compacted more than a certain amount by any given force on the walls of the box. For air, the box is the surface of the earth, the force is the force of gravity making things try to fall through that surface. The shaking of the marbles in the box is provided by the light from the sun. (It happens that the difference between the behavior of water, which does condense into a dense liquid, and air, which does not, is a chemical one caused by the different qualities of the molecules and atoms themselves, because both sets of molecules are at similar temperatures and in similarly violent states of motion. In the particular case of Helium, in fact, the atoms are so light that the force of the earth's gravity is not enough to hold them down when sunlight shakes them. If Helium were not constantly seeping up from places where it is trapped in the ground or the ocean, all of it that is presently here would eventually `boil off' from the earth and escape into space.)
Anyway, the property that we call heat is simply the energy of motion of all of the atoms and molecules, which do not hit us like individual golf balls, but rather in enormous numbers of very tiny impacts. It is this heat energy that can be transferred to ice, and which causes it to melt. While, for a given composition of air, higher temperature is associated with the presence of more heat energy, the two are not the same thing. In particular, the importance of temperature is that heat flows from whatever has a higher temperature to whatever has a lower one, if the two are put in contact. Thus air can be either hotter or colder than ice, and respectively either heat or cool it. However, temperature alone does not control how much heat is available in a certain substance. This is apparent as follows. An airport worker can stand on a taxiway in Arizona in the summer, where the air temperatures routinely reach 140 degrees Fahrenheit for hours at a time. Yet various chemical processes like sweating can carry off enough heat, enough faster than the air can impart it, that it does not kill him. On the other hand, a person immersed in water of the same temperature would suffer brain damage and death in a few tens of minutes, and in Mercury of the same temerature, in several minutes. This is because these other substances, having so many more molecules in a given volume, contain much more kinetic energy to impart at a given temperature, and also because their thermal contact with the person differs.
The point to be understood is that it is heat that is being conveyed from the stove to the ice in the system considered here, yet the particular measureable quantity that is the relevant field is the temperature. This point would seem pedantic and silly, except that when we relate light to properties of the electromagnetic field in a later chapter, and talk about the way that light can convey energy and momentum, though the field itself is not a `field of energy or momentum', it will be helpful to remember the precedent that what a field transports may not look anything like what measurable quantity the field ``is''.