by     Reginald O. Kapp


Appendix F - The Inverse Square Law

As is well known the inverse square law applies equally to electric charges, to magnet poles and to masses. It is often formulated more or less in these terms: 'The force exerted by an electric charge on another electric charge is inversely proportional to the square of the distance between the centres of gravity of the charges; this statement is equally true if one replaces the words "electric charge" by "magnet pole" or by "mass".' One would hardly expect so simple and familiar a law to allow scope for differences of opinion, and yet one meets various interpretations of its meaning and status. They are rarely stated explicitly, but are implicit in the way people talk about the law. I propose to ventilate this subject here because I believe that misconceptions about the inverse square law are symptomatic of a general misconception that is forming an obstacle to the unification of physical science. It is the misconception mentioned in Part I of this book, the notion that the laws of physics are specific and selective, that they are of the statute book kind. I select the inverse square law, not only because it is a typical example but also because I propose to show in Appendices G and H that a proper appreciation of this law offers a prospect of better insight into the relation between space and matter.

Many would, I feel sure, endorse every word of the following statement about the inverse square law:

'There are really three distinct inverse square laws. One of them defines the nature of electric charge, one the nature of magnet poles, and the third the nature of mass.

'Each of these three laws has become known by experiment and observation and could not have become known by any other means. These essential tools of the scientist had to be applied specifically to each of the three sources of force that are mentioned in the law. Thus the inverse square law for electric charges was discovered by experimenting with such charges; the inverse square law for magnet poles by experimenting with these; the inverse square law for masses by observing the movement of masses under the influence of the forces exerted between them. To deny that the three sets of distinct experiments were necessary is to forget that physics is essentially an experimental science. That the force was found to be the same function of distance in each of the sets of experiments is just one of those scientific facts that cannot, and need not, be explained.'

‘If there were a Cosmic Statute Book in which all the laws of physics were recorded, the inverse square law would have to occur there in three different sections. They would be those headed respectively "Behaviour of Electric Charges", "Behaviour of Magnet Poles" and "Behaviour of Masses". There would be no statutory means of ensuring that each of these natural phenomena obeyed the law, unless each of them was specifically mentioned in the formulation of the law.’

‘A terrestrial textbook on physics from which the student may learn about these three laws has to be compiled on similar lines. There is, for instance, no means of letting him know how the force between electric charges varies with distance without mentioning electric charges.’

'One could not replace the three laws by one single generalization in which electric charges, magnet poles and masses were not mentioned and that would, nevertheless, allow the student to infer how the force exerted by each varies with distance.’

'Each of these three laws is an independent discovery and none of them could have been found by purely deductive reasoning.'

This account of the inverse square law is in strict conformity with an approach to physics and its teaching that meets with wide approval. Nevertheless, I regard it as misleading and mischievous. By way of explaining why, I can usefully begin with a couple of further statements about the law that I have repeatedly heard quoted on good authority. They seem to have a fair measure of support and both are perfectly consistent with the account of the law that I have given above. I shall give in my own words what I think is a fair representation of the view that I am disputing.

The first statement is as follows:

'The inverse square law for mass has been obtained by observation of the movement of bodies within the solar system. It holds with great precision for the distances between these bodies, but it may, nevertheless, be only a first approximation to the truth. There is no reason why, for instance, a precise formulation should not include a second term, which might not be one of attraction but of repulsion. Instead of being an inverse function of distance, this term might be a direct function. For short distances it would then be negligible, but would predominate at those large distances at which the inverse term became very small. The assumption of such a correction to the inverse square law would provide an ad hoc explanation of some cosmological puzzles. One of them is the observed fact that the galaxies are moving away from each other. This may be because, at extra-galactic distances, the term in the corrected inverse square law that represents a repulsion between masses exceeds the term that represents an attraction.'

I have already shown in Chapter 24 that this interpretation of the recession of the galaxies is rendered untenable by simple mechanical considerations and that no satisfactory means have so far been found for avoiding the conclusion that space itself is expanding, originating. I shall show in a moment that the same conclusion results from a proper understanding of the inverse square law.

The second statement is less crude. Its defect is not so much that it is downright incorrect as that it misses something essential. It is as follows:

'The inverse square law for electric charges has been obtained from experiments with charges separated by distances that are conveniently large for laboratory work. It holds with great precision for these distances, but there is no reason why it should also hold for distances too small to be measured in a laboratory. The assumption that the inverse square law fails to apply over the distances that span an atomic nucleus would provide an ad hoc explanation for a puzzle in nuclear physics. It is this:’

'According to the inverse square law the force of repulsion between the positive charges in the nucleus would be enormous. The nucleus would not cohere. But the coherence would be explained if the inverse square law does not hold for very short distances.'

The faint suggestion here that, when charges are very close together, Nature has granted them a dispensation so that they need not obey the inverse square law is not helpful. It is axiomatic that the laws of physics are valid everywhere, at all times, and in all circumstances. When appearances are against this, the reason is not that the law has failed to apply, but that it has been wrongly formulated. Many laws are special cases of wider generalizations and should then be so worded that this becomes evident. Let me apply these considerations to the inverse square law and seek statements about it that are not merely first approximations but must be absolutely true, and that do not admit exceptions but have universal validity.

During the search I shall try to reveal the weakness of the various statements that have appeared above between quotation marks. As an aid to clarity and cogency I shall employ an occasional touch of irony and I shall imagine a research worker who carries his faith in the paramount importance of experiment over every other means of gaining knowledge to absurd lengths.

An experimenter seeks to confirm the inverse square law for electric charges. He hopes that, by making his measurements sufficiently precise, he may discover a small correcting term, proving that the law as usually stated is only a first approximation.

Among various pieces of apparatus on his laboratory table there is a metal sphere. He places an electric charge on this. He then suspends a small metal disc in such a way that its deflection is a measure of the force exerted on it. He places a charge on the disc, and brings it, together with its suspension, to various distances from the charged sphere. He expects to find that the force varies inversely with the square of the distance, at least to a first approximation, for is this not what the textbooks say? But he is disappointed. The relation between force and distance is far from confirming the law or even seeming to obey any law at all.

A spring balance has been left on the table from a previous experiment. When this is moved, the force on the metal disc changes. For a moment our experimenter thinks that he has discovered a new law about spring balances. He is about to write in his notebook: 'My experiments show that a spring balance exerts a force on a metal disc in its vicinity.' But he is just saved from doing so when he finds that the force is influenced by many things: by the presence of the table, of the apparatus on it, of his own body. All these things, he then concludes rightly, distort the field.

Realizing this the experimenter decides that the law ought to be worded: 'In an undistorted field the force exerted by one electric charge on another is, at least to a first approximation, inversely proportional to the square of the distance between the centres of gravity of the charges'.

The experimenter appreciates, of course, that an experiment must be performed under controlled conditions. These require measures to prevent the field from being distorted. It would obviously not be good enough merely to remove the spring balance and other pieces of apparatus. In order to confirm the law as revised, and possibly also to find a correction factor, very careful steps must be taken to ensure an undistorted field. But what sort of a field should it be? Being most assiduous at his experimental work, the experimenter decides to try out a great variety of undistorted fields and to begin with the most uniform shape that he can think of. This is the field that occurs between charged parallel plates. By making them extensive he can eliminate disturbing factors and secure ideally controlled conditions.

He places two large copper plates parallel to each other and separated by a short distance. He puts an electric charge on the plates and moves a small test charge into various positions in the space between the plates. Again, he fails to confirm the inverse square law, even approximately. Instead of being inversely proportional to the square of the distance of the test charge from either plate, the force is found to be independent of the distance.

The experimenter does, however, discover something by this experiment. When he varies the charge on the plates he finds that the force varies in proportion. This proves to him that the force on the test charge is proportional to the flux density. As flux density is defined as the property of the electrostatic field by virtue of which the field exerts a force on an electric charge in it, this discovery is true by definition. It is one of those tautologies that are so helpful in scientific work. Being true, it is a discovery that one may expect also to confirm by experiment.

Failure with the parallel plates suggests to the experimenter that the field between these is probably too uniform. He experiments with many other shapes of undistorted field. But the long story of trial and error need not concern us. Among the shapes of field on our experimenter's list is a uniformly divergent one, and eventually he comes to it. He finds it a little difficult to construct such a field, but eventually hits on a good way. He obtains two concentric spheres, each of a conducting material. He decides to place a charge on one of them, which will induce an equal and opposite charge on the other. The Faraday tubes of force between the surfaces of the spheres will then be straight lines all diverging from the centre of the spheres, which is another way of saying that the field will be uniformly divergent.

The experimenter arranges to introduce a small test charge into this field and devises ingenious means of moving this about and measuring the force on it. Provided the supports of the outer sphere, the device for moving the test charge and the device for measuring the force on it are not too bulky, the field will then not be greatly distorted.

Having spent many laborious weeks constructing the apparatus, the experimenter takes the unusual step of retiring to his study to think.

It occurs to him that every tube of unit flux that originates on the inner sphere ends on the outer one, and that every one of these tubes would pass through the surface of any intermediate sphere that was concentric with the two conducting ones. The area of each such sphere would be proporttional to the square of its radius,r, and so the flux density at a distance, r, from the centre would be inversely proportional to r2. The force on the test charge that he proposes to insert will, he knows from his experiment with the parallel plates, as well as from the definition of 'flux', be proportional to the flux density, and so this force, too, will be inversely proportional to the square of the distance r, from the centre.

Having at long last reached this conclusion, our experimenter decides to abandon the experiment. He now knows what the result will be. The inverse square law will be confirmed all right. Five minutes of thought in his study have taught him more than he could learn from five weeks of trial and error in his laboratory.

Let us list the things that thought has taught him:
(1) A correct statement of the inverse square law must strictly limit its application. If it does not do so, it is misleading. It is necessary to begin the formulation of the law with the words ‘In a uniformly divergent field, and only in such a field . . . .'
(2) It is incorrect to say that the law defines the nature of electric charge, magnet poles, mass, or any other source of a field of force. It defines the nature of one thing only, a sphere. Hence it is also incorrect to say that there are really three laws. The proper place of the law is not in electricity, magnetism, astronomy, mechanics or in any other such science. It is in geometry.
(3) It is incorrect to say that the law can only be known by experiment and observation. It can also be known by taking thought.
(4) It is incorrect to say that the law is just one of those scientific facts that cannot, and need not, be explained. It can be explained with the help of simple geometry.
(5) It is incorrect to say that one could not replace the three inverse square laws by one single generalization in which the source of a force was not mentioned, and that would, nevertheless, enable the student to infer how the force varies with distance. The generalization that the area of a sphere is proportional to the square of its radius would suffice. The student could do the rest by the use of deductive reasoning. This reasoning would be based on the definition of force and the fact implicit in the definition that action and action are equal and opposite.
(6) To say that the inverse square law is only a first approximation to the truth is the same as to say that the area of a sphere only to a first approximation is proportional to the square of its radius.

Therewith one may say good-bye to all hope of explaining the recession of the nebulae or any other large-scale cosmic events by inventing the ad hoc hypothesis that the inverse square law needs the addition of a further term. But I do not think that one ought to dismiss this law from one's mind when seeking to explain the cohesion of the atomic nucleus. Let us examine statement (5) above. Is it necessarily true that the area of a sphere is proportional to the square of its radius?

Undoubtedly, one is inclined to answer hurriedly: ‘It follows from Euclid.' This is true enough; but do the spatial relations within the nucleus necessarily follow from Euclid? Is space within the nucleus Euclidean?

What I am suggesting is this. Instead of saying that charges within the nucleus do not obey the inverse square law, it would be both more precise and more fruitful to say that space within the nucleus is non-Euclidean. Such a mode of expression would lead to exploration of the various ways in which the space contained by the nucleus could be curved, and might give insight into the nature of the nucleus and of the forces that hold it together. I suggest this theme as likely to prove rewarding to a research worker. If the curvature is very intense forces must, for geometrical reasons alone, be very far from proportional to the inverse square of the distance between the bodies in question.

It may, for all I know, be impossible to postulate curvature of such a kind that a force of repulsion in flat space would operate as a force of attraction in curved space. But this is not the only way in which the puzzle of nuclear coherence might be solved. I shall venture to suggest a different approach in Appendix H. It is one that seems to follow from the new theory of gravitation.

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