The logic of modern physics

by P. W. Bridgman

If the body is obviously not homogeneous, it is still a matter of experience that it can be divided into small pieces, each of which are by themselves sufficiently homogeneous, and the first law in its usual form may be applied to each of the pieces.

Finally, we emphasize a fact already implicitly mentioned, namely, that no physical significance can be directly given to flow of heat, and there are no operations for measuring it. All we can measure are temperature distributions and rates of rise of temperature. As at present defined, a heat current is a pure invention, without physical reality, for any determined heat flow may always be modified by the addition of a solenoidal vector, with change in no measurable quantity. If someone states that throughout all space there is a uniform heat current of 10⁶ cal./cm.² sec. in the direction of Polaris, no disproof can be given, for such a stream is solenoidal, and as much heat flows out of every closed surface in unit time as flows in. Such a solenoidal flow is meaningless in terms of operations; we could give meaning to such a flow only in terms of some slight modification of the solenoidal condition introduced by the measuring instrument. In all ordinary conditions the flow of heat given by the simple relation q = k Grad t corresponds exactly to what our atomic pictures lead to expect in those cases where the details of the picture can be worked out. But there may be cases where it is advantageous to supplement the ordinary heat flow (= k Grad t) by the pure fiction of a solenoidal flow, because in this way it may be possible to account for new phenomena which appear when the solenoidal conditions are slightly departed from. Thus if in a conductor at uniform temperature carrying a steady electric current we say that a heat current is also flowing proportional to the electric current and therefore solenoidal, we may provide the possibility for a simple correlation of phenomena found under those more complicated conditions when an electric current flows in a conductor of non-uniform temperature in a magnetic field. If it should turn out that the heat current is uniquely determined by considerations of this character, then we have taken the first step away from the pure formalism which this sort of thing otherwise is in the direction of giving physical reality to the invention of "heat current."

There are other interesting questions of a fundamental thermodynamic character, such for example, as whether the entropy concept has any general significance apart from the scale of our measuring instruments, and what is the operational significance of applying thermodynamic concepts to radiation, but we shall not consider these questions here.

ELECTRICAL CONCEPTS

We now set ourselves the problem of finding the meaning of the various concepts in terms of which we describe the behavior of electrical systems, assuming that we understand what we mean by "electrical." We start with the simplest electrical systems, namely, those in which we deal with static phenomena on a large scale. In such systems there are independent physical operations by which we may find the magnitude of any charge, provided that it is effectively concentrated in a geometrical point. The measurements involved in these operations are measurements of ordinary mechanical forces; we assume that our knowledge of mechanics has already taught us how to make such measurements. An electrically charged body experiences forces, which may be measured by tying a string to it and pulling on the string with a spring balance hard enough to keep the body in equilibrium. Three charges are numerically equal if when each is placed at unit distance from another, in the absence of the third (or other charge), the forces are always the same. If furthermore the forces are of unit magnitude, the charges are defined as unit charge. Having obtained unit charge, we define the magnitude of any other charge as equal to the force which it experiences when placed at unit distance from unit charge. This of course is all very trite; the important thing for us is merely that magnitude of charge, or quantity of electricity, is an independent physical concept, and that unique operations exist for determining it. These operations presuppose the ability to perform certain operations of mechanics. Having now learned how to measure electrical quantities, we discover experimentally the inverse square law of force, and later arrive at the concept of the electric field. As we have seen, the field is an invention; here we shall use this concept only for the purpose for which the invention was made, and shall not involve ourselves in any of the implications of ascribing physical reality to the field. Notice that as long as we deal only with point charges we do not have to define field strength in terms of the limiting procedure of making the exploring charge smaller, for the limiting small charge is necessary only to avoid the reaction of the exploring charge on the positions of the charges which generate the field. All this again is trite; the important point is that the operations by which the inverse square law and the concept of the field are established presuppose that the charge is given as an independent concept, since the operations involve a knowledge of charges. The operations also involve the measurement of forces by the ordinary static procedure of mechanics with spring balances. With the means now at our command we establish one very important property of electric charges, namely that the total amount of charge on an isolated body of finite size is conserved, no matter how the charge is forced to rearrange itself by the motion of charges on adjacent bodies.

By procedures exactly like those outlined above, we may treat all the corresponding magnetic quantities; there is formal parallelism between the two sets of phenomena, but there is the physical difference that we have to realize a single magnetic pole by the device of using a very long slender magnet.

We now give our electrical system more freedom, in that we allow the charges to be in motion with respect to each other. Perhaps the most immediate question which we now have to ask is whether charge continues to be conserved when set in motion, or whether the total charge on an isolated body is a function of its velocity? To answer this question we must generalize the procedure by which we assigned a numerical value to a stationary charge. Perhaps the simplest way is to allow two unit charges each to move with constant velocity, remaining at unit distance apart, and measure with a spring balance the force required to keep them at constant distance apart. Now we immediately find that the force is altered under these conditions, so that our first impulse is to say that the charge is a function of the velocity. But as we experiment further, we find that the state of affairs is very complicated; the force between the two charges at any moment of their motion depends not only on the charges, their distance apart, and their velocities, but also on the angle between the line joining them and the direction of motion in the lines. Further experiment of other kinds yields other information; it requires a force to maintain a charge in uniform motion in a magnetic field, or to maintain a magnetic pole in motion in an electric field. A moving electric charge exerts a force on a stationary magnetic pole, so that by definition the moving charge is surrounded by a magnetic field, and similarly a moving magnetic pole is surrounded by an electric field. Returning to our two moving electric charges, we are impelled to ask whether, if all these complications are possible, the numerical constant (unity for static charges) in the inverse square law of force is a function of velocity as well as the magnitude of the charges themselves? If we broaden the question in this way, as we apparently must, our problem becomes indeterminate, for we are trying to answer two different questions with a single kind of measurement, namely of the force between moving charges. I have had no better luck on trying other methods of measurement. Apparently the operations do not exist by which unique meaning can be given to the question of whether the magnitude of a charge is a function of its velocity. On realizing this situation, we are at first embarrassed to know how to proceed, but we reflect that the embarrassment is not of our own making, but corresponds to a physical fact. The concept of charge as a unique and independent thing essentially pertains only to static systems. We may extend the concept to moving systems if we wish, as a matter of convenience to ourselves, but must recognize that such an extension is an invention of ours and not a reality of nature. Now we do make such an extension, and we make it in the simplest possible way, that is, we define the charge on an isolated body in motion as that which we should find on it if we reduced it to rest and made measurements according to the regular static procedure. That this is a convenient thing to do depends on the experimental result that the charge so found is independent of the way in which velocity is imparted to or removed from the body; in other words, whenever the body is reduced to rest, the same charge is always found on it.

Although this is pure definition on our part, it turns out to have a most simple and convenient connection with experimental facts which were discovered after the decision to treat a moving charge in this way was made; the discovery is of the atomic structure of electricity. If then we agree to call each elementary charge a constant independent of the velocity, the total charge on a body becomes merely proportional to the count of the total number of atomic charges on the body, which is certainly highly convenient and suggestive.

Having now fixed what we mean by the magnitude of a moving charge, we are ready to turn to the general problem of the behavior of any system of charged bodies in motion. For the present we consider only phenomena of the scale of everyday experience. The most general problem that has meaning here is to determine all measurable properties of the system in terms of those data which experiment shows can be arbitrarily specified. Now we have already emphasized that the electromagnetic field itself is an invention, and is never subject to direct observation. What we observe are material bodies, with or without charges (including eventually in this category electrons), their positions, motions, and the forces to which they are subject. The forces are to be measured according to definition in mechanical terms, either by the strains in members of a framework if the system is in equilibrium, or in terms of accelerations and masses if it is not in equilibrium. The electromagnetic field as such is not the final object of our calculations, but the calculation of it is only an intermediate auxiliary step, convenient to make because our mathematical formulation gives so simple a connection between electromagnetic field, charges, and mechanical action that the latter can be calculated at once in terms of the former. In fact the connection is so simple that in many cases we have come to regard our problem as solved if we can compute the electromagnetic field, overlooking the fact that the field has no immediate meaning in terms of experience.

Electromagnetic theory now presents us with a solution of the general problem; this solution is contained in the four-field equations of Maxwell, the constitutive equations, and those additional equations (quite often lost sight of) which give the forces exerted by the field on electric charges, or currents, or dielectrics. Let us inquire how we may set about testing the physical correctness of these equations. We may begin with one of the simplest possible tests, and inquire whether the equations are correct in stating that the force acting on a charge moving in an electric field is simply the product of the charge and the field strength. This, on the face of it, is a surprising statement. The field itself is affected by the motion of the charges which generate it, and it is natural to expect a converse effect. If, furthermore, we have sympathy with the medium point of view, it is easy to think that whatever it is in the medium that gets hold of a charge and exerts a force on it will find it harder to take hold when the charge is in motion.