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
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
'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
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
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
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 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
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
(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
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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