The use of Physical Analogies

64.] In many cases the relations of the phenomena in two different physical questions have a certain similarity which enables us, when we have solved one of these questions, to make use of our solution in answering the other. The similarity which constitutes the analogy is not between the phenomena themselves, but between the relations of these phenomena.

To begin with a case of extreme simplicity;—a person slow at arithmetic having to find the price of 52 yards of cotton at 7 pence a yard, if he hap- pened to remember that there are 52 weeks and a day in a year of 365 days, might at once give the answer, 364 pence, without performing the calculation.

Here there is no resemblance whatever between the quantities themselves— the weeks and the yards of cotton,—the sole resemblance is between the arith- metical relations of these quantities to others in the same question.

The analogy between electrostatic phenomena and those of the uniform conduction of heat in solid bodies was first pointed out by Sir W. Thomson in a paper ‘On the Uniform Motion of Heat in Homogeneous Solid Bodies, and its connection with the Mathematical Theory of Electricity,’ published in the Cambridge Mathematical Journal, Feb. 1842; reprinted in the Phil. Mag. 1854, and in the reprint of Thomson’s papers on Electrostatics and Mag- netism. The analogy is of the following nature:—ANALOGIES BETWEEN ELECTROSTATICS AND HEAT. Electrostatics.

The electric field. A dielectric medium. The electric potential at different points of the field. The electromotive force which tends to move positively electrified bodies from places of higher to places of lower po- tential. A conducting body. The positively electrified surface of a con- ductor. The negatively electrified surface of a con- ductor. A positively electrified body. A negatively electrified body. An equipotential surface. A line or tube of induction. 57 Heat.

An unequally heated body. A body which conducts heat. The temperature at different points in the body. The flow of heat by conduction from places of higher to places of lower temperature. A perfect conductor of heat. A surface through which heat flows into the body. A surface through which heat escapes from the body. A source of heat. A sink of heat, that is, a place at which heat disappears from the body. An isothermal surface. A line or tube of flow of heat.

By a judicious use of this analogy and other analogies of the same nature the progress of physical science has been greatly assisted. In order to avoid the dangers of crude hypotheses we must study the true nature of analogies of this kind. We must not conclude from the partial similarity of some of the relations of the phenomena of heat and electricity that there is any real physical similarity between the causes of these phenomena.

The similarity is a similarity between relations, not a similarity between the things related. This similarity is so complete as far as it goes that any result we may have obtained either about electricity or about the conduction of heat may be at once translated out of the language of the one science into that of the other without fear of error; and in pursuing our investigations in either subject we are at liberty to make use of the ideas belonging to the other, if by so doing we are enabled to see more clearly the connection between one step and another of the reasoning.

We must bear in mind that at the time when Sir W. Thomson pointed out the analogy between electrostatic and thermal phenomena men of science were as firmly convinced that electric attraction was a direct action between

LIMITATION TO THE USE OF ANALOGIES.

distant bodies as that the conduction of heat was the continuous flow of a material fluid through a solid body. The dissimilarity, therefore, between the things themselves appeared far greater to the men of that time than to the readers of this book, who, unless they have been previously instructed, have not yet learned either that heat is a fluid or that electricity acts at a distance. 65.] But we must now consider the limits of the analogy—the points be- yond which we must not push it.

In the first place, it is only a particular class of cases of the conduction of heat that have analogous cases in electrostatics. In general, when heat is flowing through a body it causes the temperature of some parts of the body to rise and that of others to fall, and the flow of heat, which depends on the relation of these temperatures, is therefore variable. If the supply of heat is maintained uniform, the temperatures of the different parts of the body tend to adjust themselves to a state in which they remain constant. The quantity of heat which enters any given portion of the body is then exactly equal to that which leaves it during the same time. Under these circumstances the flow of heat is said to be steady.

Now the analogy with electric phenomena applies to the steady flow of heat only. The more general case, that of variable flow of heat, has nothing in electrostatics analogous to it. Even the restricted case of steady flow of heat differs in a most important element from the electrostatic analogue. The steady flow of heat must be kept up by the continual supply of heat at a con- stant rate and the continual withdrawal of heat at an equal rate. This involves a continual expenditure of energy to maintain the flow of heat in a constant state, so that though the state of the body remains constant and independent of time, the element of time enters into the calculation of the amount of heat required.

The element of time does not enter into the corresponding case in electro- statics. So far as we know, a set of electrified bodies placed in a perfectly insulating medium might remain electrified for ever without a supply of any- thing from external sources. There is nothing in this case to which we can apply the term ‘flow,’ which we apply to the case of the transmission of heat with the same propriety that we apply it to the case of a current of electricity, of water, or of time itself.

FARADAY’S CUBE.

66.] Another limitation to the analogy is that the temperature of a body cannot be altered without altering its physical state. The density, conductivity, electric properties, &c. all vary when the temperature rises. The electrical potential, however, which is the analogue of temperature is a mere scientific concept. We have no reason to regard it as denoting a physical state. If a number of bodies are placed within a hollow metallic vessel which completely surrounds them, we may charge the outer surface of the vessel and discharge it as we please without producing any physical effect whatever on the bodies within. But we know that the electric potential of the enclosed bodies rises and falls with that of the vessel. This may be proved by passing a conductor connected to the earth through a hole in the vessel. The relation of the enclosed bodies to this conductor will be altered by charging and dis- charging the vessel. But if the conductor be removed, the simultaneous rise and fall of the potentials of the bodies in the vessel is not attended with any physical effect whatever.

67.] Faraday∗ proved this by constructing a hollow cube, twelve feet in the side, covered with good conducting materials, insulated from the ground and highly electrified by a powerful machine. ‘I went into this cube,’ he says, ‘and lived in it, but though I used lighted candles, electrometers, and all other tests of electrical states, I could not find the least influence upon them, or indication of anything particular given by them, though all the time the outside of the cube was powerfully charged and large sparks and brushes were starting off from every part of its outer surface.’

It appears, therefore, that the most sudden changes of potential produce no physical effects on matter, live or dead, provided these changes take place simultaneously on all the bodies in the field. If Faraday, instead of raising his cube to a high electric potential, had raised it to a high temperature, the result, as we know, would have been very different.