Lecture 1: Maxwell, Kelvin and the Inverse Square Law of Electrostatics
Dr Isobel Falconer
Dr Falconer is a historian of mathematics and physics at St Andrews University. She is especially interested in the relations between maths and physics in the nineteenth century, and the interaction of both with the cultural context.
My notes from the lecture (if they don’t make sense then it is entirely my fault)
In 1877 James Clerk Maxwell and his student Donald McAlister refined Henry Cavendish’s 1773 null experiment demonstrating the absence of electric charge inside a charged conductor. Such absence of charge was a mathematical prediction of the inverse square law, and both Cavendish and Maxwell took the experiment as verifying the law. The experiment was aptly completed at the Cavendish Laboratory.
James Clerk Maxwell FRS FRSE (13 June 1831 – 5 November 1879) was a Scottish scientist in the field of mathematical physics. His most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as different manifestations of the same phenomenon. Maxwell’s equations for electromagnetism have been called the “second great unification in physics” after the first one realised by Isaac Newton.
Sir Donald MacAlister, 1st Baronet of Tarbet KCB FRSE (17 May 1854 – 15 January 1934) was a Scottish physician who was Principal and Vice-Chancellor and, later, Chancellor of the University of Glasgow. He was a member of the Cambridge Apostles intellectual secret society, from 1876. From 1904 to 1931 he was President of the General Medical Council.
Henry Cavendish FRS (10 October 1731 – 24 February 1810) was an English natural philosopher, scientist, and an important experimental and theoretical chemist and physicist.
Cavendish wrote papers on electrical topics for the Royal Society but the bulk of his electrical experiments did not become known until they were collected and published by James Clerk Maxwell a century later, in 1879, long after other scientists had been credited with the same results.
No actual measurement … was required: Maxwell and Cavendish’s null method for the inverse square law of electrostatics https://arxiv.org/ftp/arxiv/papers/1608/1608.01520.pdf
Null method is a method of measurement using an electrical device, as a Wheatstone bridge, in which the quantity to be measured is balanced by an opposing known quantity that is varied until the resultant of the two is zero.
Why did Maxwell repeat the experiment?
At the time Maxwell repeated the experiment the inverse square law had been open to question, and its experimental base was not very robust. He also wanted to close the door on alternative electrical theories by constructing an experimental tradition of null tests of the law that would be less open to critical scrutiny than Coulomb’s original experiments.
The experiment allowed him to promote the use of null methods and the electrical programme, which was putting great emphasis on the value of precision measurement.
His use of a sophisticated instrument such as Thomson’s quadrant electrometer in the null experiment protected the inverse square law by increasing the accuracy of the results in relation to the mathematics of the theory.
Maxwell stressed several times that null methods required only “detection” rather than “measurement”.
Maxwell, J. C. (1873). A treatise on electricity and magnetism. Vol. 1. Oxford: Clarendon Press.
Maxwell, J. C. (ed.) (1879). The electrical researches of Henry Cavendish. Cambridge University Press.
Maxwell, J. C. (1881). A treatise on electricity and magnetism. 2nd ed. by W. Garnett, Vol.1. Oxford: Clarendon.
Maxwell improved Cavendish’s experiment in two major ways.
First, where Cavendish had devised a hinged apparatus (AbcDCB in the diagram below left) to remove the outer hemispheres before grounding them, thus allowing access to the inner globe for testing, Maxwell left the outer hemispheres in place throughout. This shielded the inner globe from possible electrical disturbances. He argued that this also removed one of the chief sources of error in Cavendish’s experiment, charge leakage from the inner globe to the bench, as the globe now rested on insulating supports on the inside of the sphere. Thus “the potentials of the globe and sphere remained sensibly equal,” so any charge would not be attracted away from the globe (Maxwell, 1879 p417). He failed to point out that this argument assumed the inverse square law prediction of constant potential within the outer sphere.
Above right: diagram of Maxwell and McAlister’s apparatus, showing the inner globe supported on insulating blocks within the outer sphere, which remains closed throughout except for a small capped hole. The experiment involved opening the hole, charging the inner globe, disconnecting the charging mechanism, closing the hole, then opening the hole to measure the charge with an electrometer.
The experiment involved arranging an insulated conducting globe inside a concentric hollow conducting sphere comprised of two separable hemispheres. Initially a wire connected the globe and sphere. The apparatus was electrified from a Leyden jar, then the connecting wire was removed and the outer sphere discharged to earth. The charge on the inner globe was tested with an electrometer and found to be nil.
Maxwell arranged for the small access hole to be covered while the conductors were charged, providing further evidence for his recognition of the mathematical importance of a closed shell.
Above shows a photograph of Maxwell and McAlister’s apparatus . The additional ball hanging from the retort stand has traditionally been considered part of the apparatus, but does not correspond to the brass ball for testing sensitivity of Maxwell’s description. Photograph courtesy of the Cavendish Laboratory, Cambridge.
Second, Maxwell benefited from Thomson’s recently invented quadrant electrometer to test for charge, which was far more sensitive than the pith ball electroscope available to Cavendish.
Cavendish’s original experiment had used a one-fluid model of electricity. He assumed an exact inverse square law, and showed the charge on a closed spherical conductor must reside in the shallowest possible depth on the surface, with no charge inside (1771 pp592-4).
Cavendish, H. (1771). An attempt to explain some of the principal phæaenomena of electricity by means of an elastic fluid. Philosophical Transactions of the Royal Society, 61, 584-677, repr. in Maxwell (1879), 3-63.
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them. It is called an inverse problem because it starts with the results and then calculates the causes. This is the inverse of a forward problem, which starts with the causes and then calculates the results.
Maxwell was well aware of this problem of reverse inference, at least in general terms, and his draft account of Cavendish’s methodology raised it. “It is obvious that the mere agreement between experiment and the deductions from a hypothesis cannot prove that hypothesis to be true unless it can also be shown that no other hypothesis will agree with the experiments …. we must form such an exhaustive classification of hypotheses as to enable us from the result of an experiment to conclude that all hypotheses except those belonging to a certain class must be wrong. By means of other experiments we may narrow the boundary of this class ….” (Harman, 2002 pp541-542).
Maxwell Treatise pp 75-76
Maxwell took both a mathematical and an experimental approach to ‘narrowing the class’ and demonstrating that no other hypothesis would agree with the null result. Already, in 1873, he countered mathematically any who still believed the law might vary in different circumstances by stating that an inverse square was, “the only law of force which satisfies the condition that the potential within a uniform spherical shell is zero,” (1873 p76).
Maxwell’s detailed demonstration was due to his friend, Peter Guthrie Tait, outlined in a postcard dated 5 June 1871 (Harman, 1995 p650).
Tait’s demonstrations suffered from the same fundamental flaw if they were to be realised experimentally: they ignored the quantifiers at a crucial step in the argument. This is very clearly seen in Maxwell’s 1873 reproduction of Tait’s demonstration.
The lack of reported care in conducting the experiment suggests that Maxwell’s aim in repeating the experiment was not purely metrological, to push it to the limits of precision.
It was well known that electricity resides on the surface
“The fact that electricity resides only on the external surface of a conductor, combined with the fact that there is no electrical force in the space enclosed by this surface, affords a rigorous proof of the law of inverse squares…. Now it admits of proof, and is well known to mathematicians, that a uniform spherical shell exerts no attraction at any point of the interior space, if the law of attraction be that of inverse squares, and that the internal attraction does not vanish for any other law,” (Everett, 1872 p521-22,).
J.D. Everett 1872 Elementary Treatise Nat. Phil. P523-4
Everett J. D. (1872). Elementary Treatise on Natural Philosophy. London : Blackie.
Everett, J. D. (1878). Elementary Text-Book of Physics. London: Blackie.
When a conductor is at equilibrium, the electric field inside it is constrained to be zero. Since the electric field is equal to the rate of change of potential, this implies that the voltage inside a conductor at equilibrium is constrained to be constant at the value it reaches at the surface of the conductor. A good example is a charged conducting sphere, but the principle applies to all conductors at equilibrium.
TEM Vol. 1 Pg 76
By 1879 Maxwell had repeated Cavendish’s experiment having dropped Tait’s demonstration. Instead he asserted the result as fact, calling on the authority of Laplace who, “gave the first direct demonstration that no function of the distance except the inverse square satisfies the condition that a uniform spherical shell exerts no force on a particle within it,” (1879 p422; 1881 p82). Here Maxwell was probably drawing on Isaac Todhunter’s account of Laplace, published in 1873. Todhunter’s demonstration, and Laplace’s original, are similar to Maxwell’s, but differ crucially in stating the quantifiers (Laplace, 1829 pp298-299; Todhunter, 1873 pp181-182). However, they do not spell out the experimental implications. Jon Dorling has described Maxwell as “disingenuous” for glossing over the essence of Laplace’s mathematical demonstration – every possible spherical distribution, i.e. tested with shells of all possible radii, would need to produce a “no force” result in order to infer an inverse square law (1974 p336).
Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French scholar whose work was important to the development of engineering, mathematics, statistics, physics and astronomy.
The power of the mathematical methods based on the inverse square law, and its role in securing their electrical programme, had raised the stakes for Maxwell and Thomson in protecting it.
Maxwell positioned his repetition of the experiment at the head of both a mathematical tradition, from Pratt or Laplace, and the experimental one that he, Thomson and Everett, had constructed.
Joseph Everett 1872 Pg. 521-322
Thomson, W. (1872). The Mathematical Theory of Electricity in Equilibrium. Papers on electrostatics and magnetism. London: Macmillan. Pg. 15-27.
Thomson, W. (1876). Electrical Measurement, 238–9ff. Conferences Held in Connection with the Special Loan Collection of Scientific Apparatus. London: Chapman and Hall. Pg. 238-250
Inventing the deduction of the electrostatics inverse square law:
1) Coulomb’s law, or Coulomb’s inverse-square law, is a law of physics for quantifying the amount of force with which stationary electrically charged particles repel or attract each other. In its scalar form, the law is F = kq1q2/r2 where k is coulomb’s constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the centres of the charges.
The law was first published in 1785 by French physicist Charles-Augustin de Coulomb and was essential to the development of the theory of electromagnetism.
The torsion balance apparatus contains several parts but is relatively easy to replicate in order to reproduce Coulomb’s experiment. The top of the balance is a suspension head which is attached to a fibre that hangs through a column and hangs inside a lower glass cylinder. On this cylinder a paper scale marks the angular position. One of the charges is placed on a small conducting sphere that is able to transfer its charge through an opening at the top of the lower glass cylinder. The second charge is on a second conducting sphere placed inside the lower glass cylinder and is suspended from the fibre at a balanced crossbar.
The inverse square law of electrostatics wasn’t really known until Maxwell wrote about it in the 1870s.
A personal comment: I have never been able to replicate the experiment in the classroom and apparently Coulomb only took three sets of results.
Cavendish took a different approach. In his experimental set up, Cavendish did not measure electrostatic force; he simply confirmed the lack of it. That is, he confirmed the following phenomena, which is now very well known:
A charged conductor produces no electric field in its interior.
As described earlier a conducting sphere is suspended, using the support of insulating frames, within another conducting sphere, which may be divided into two hemispheres by hinges. Both conductors are initially uncharged. A conducting wire is placed connecting the inner and outer spheres. The outer sphere is then charged, and the wire was cut by means of a silk thread. After the outer sphere was disassembled, the inner sphere’s charge was measured by an electrometer.
If the outer sphere produced interior electric fields, then charge would naturally migrate to the inner sphere. However, the electrometer failed to show any significant charge on the inner sphere, confirming the hypothesis that electric conductors cannot produce interior electric fields.
3) For the sake of historical interest, Joseph Priestly, the discoverer of oxygen, first showed in 1767 that the absence of internal electric fields in conductors gives rise to the inverse-square law.
Joseph Priestley FRS (24 March [O.S. 13 March] 1733 – 6 February 1804) was an 18th-century English Separatist theologian, natural philosopher, chemist, innovative grammarian, multi-subject educator, and liberal political theorist who published over 150 works. He has historically been credited with the discovery of oxygen, having isolated it in its gaseous state, although Carl Wilhelm Scheele and Antoine Lavoisier also have strong claims to the discovery.
4) Poisson’s potential theory
In mathematics, Poisson’s equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace’s equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson.
One of the cornerstones of electrostatics is setting up and solving problems described by the Poisson equation. Solving the Poisson equation amounts to finding the electric potential φ for a given charge rf
Baron Siméon Denis Poisson FRS FRSE (21 June 1781 – 25 April 1840) was a French mathematician, engineer, and physicist, who made several scientific advances.
Faraday’s ice pail experiment is a simple electrostatics experiment performed in 1843 by British scientist Michael Faraday that demonstrates the effect of electrostatic induction on a conducting container.
Apparatus Faraday used in the experiment: a metal pail (A) is supported on a wooden stool (B) to insulate it from the ground. A metal ball (C) charged with static electricity can be lowered into the pail on a nonconducting silk thread. A gold-leaf electroscope (E), a sensitive detector of electric charge, is attached by a wire to the outside of the pail. When the charged ball is lowered into the pail without touching it, the electroscope registers a charge, indicating that the ball induces charge in the metal container by electrostatic induction. An opposite charge is induced on the inside surface of the pail.
The ice pail experiment was the first precise quantitative experiment on electrostatic charge.
Michael Faraday FRS (22 September 1791 – 25 August 1867) was a British scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis.
6) William Thomson, ‘On the mathematical theory of electricity in equilibrium, 1845)
William Thomson, 1st Baron Kelvin, OM, GCVO, PC, FRS, FRSE (26 June 1824 – 17 December 1907) was a Scots-Irish mathematical physicist and engineer who was born in Belfast in 1824.
Thomson did more than any other electrician up to his time in introducing accurate methods and apparatus for measuring electricity. As early as 1845 he pointed out that the experimental results of William Snow Harris were in accordance with the laws of Coulomb. In the Memoirs of the Roman Academy of Sciences for 1857 he published a description of his new divided ring electrometer, based on the old electroscope of Johann Gottlieb Friedrich von Bohnenberger and he introduced a chain or series of effective instruments, including the quadrant electrometer, which covered the entire field of electrostatic measurement. He invented the current balance, also known as the Kelvin balance or Ampere balance (SiC), for the precise specification of the ampere, the standard unit of electric current. From around 1880 he was aided by the electrical engineer Magnus Maclean FRSE in his electrical experiments.
‘It was left for Faraday to make … the … experiment which crowned Cavendish’s theory. Faraday found by the most thoroughly searching investigation that the electrical force in the circumstances supposed was zero…. Therefore the law of force varies with the inverse square of the distance. This result was obtained with far less searching accuracy by Coulomb and Robinson, because their method did not admit of the same searching accuracy. On this law is founded the whole system of electrostatic measurement in absolute measure’. (William Thomson, 1876, ‘Electrical Measurement’, Lecture at South Kensington)
Frederick C. Webb was one of the first cable engineers, and also a writer and lecturer on the subject. Articles from the London trade paper The Electrician describe his life and career.
Frederick Charles Webb was born on the 1st of October, 1828, at No. 14, Surrey-square, Old Kent-road, London.
“Although the attractive force may vary within certain limits … as some particular power of the distance, if increased beyond these limits, the ratio will begin to vary, and ultimately the attractive force will vary as some other power of the distance” (Frederick Charles Webb, in The Electrician, 1862)
Sir William Snow Harris (1 April 1791 – 22 January 1867) was a British physician and electrical researcher, nicknamed Thunder-and-Lightning Harris, and noted for his invention of a successful system of lightning conductors for ships. It took many years of campaigning, research and successful testing before the British Royal Navy changed to Harris’s conductors from their previous less effective system. One of the successful test vessels was HMS Beagle which survived lightning strikes unharmed on her famous voyage with Charles Darwin.
“If we examine the physical data upon which … the laws of these forces rest, we do not find the experimental investigations to be extensive, nor are they always satisfactory … It is easy to see that the limit of the law of electrical or magnetic force is a simple inverse ratio of the distance…It is only, then, between certain limits, and under certain free states or conditions of magnetic or electrical change, that we obtain the law arrived at by Coulombe” [sic]. (Sir William Snow Harris, Treatise on Frictional Electricity, 1868)
9) Maxwell on Faraday in 1864
‘before I began the study of electricity I resolved to read no mathematics on the subject till I had first read through Faraday’s Experimental Researches in Electricity’.
‘As I proceeded with the study of Faraday, I perceived that his method of conceiving of phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols… I found, also, that several of the most fertile methods of research discovered by the mathematicians could be expressed much better in terms of ideas derived from Faraday than in their original form.’
‘Faraday’s null experiment is far more conclusive than any measurements of electrical forces can be…’ (Maxwell, 1873, Treatise on Electricity an Magnetism, 1st edn)
Maxwell’s drawing of Faraday’s apparatus (1871 Treatise on Electricity and Magnetism)
Maxwell put a lid on Faraday’s apparatus
What was good about the null experiment?
It was superior because it was intrinsically more accurate than Coulomb’s method. Precision measurement, and associated claims of accuracy, went hand in hand with mathematical analysis in Thomson and Maxwell’s electrical programme.
Maxwell, J. C. (1890). Introductory lecture on experimental physics. In W. D. Niven, ed. The Scientific Papers of James Clerk Maxwell, pp241-255. Cambridge University Press.
Remarks on the Mathematical Classification of Physical Quantities http://www.clerkmaxwellfoundation.org/MathematicalClassificationofPhysicalQuantities_Maxwell.pdf
‘All quantities may be classed together in one respect, that they may be defined by means of two factors, the first of which is a numerical quantity, and the second is a standard quantity of the same kind with that to be defined’.
Mathematical methods raised the stakes in validity but there was a cavalier approach to experiments. Some of the experiments had logical flaws. The null method elevated mathematical theory over experimental care.
Null method was discussed. The term “null method” appears to have been popularised, only a few years before Maxwell’s Treatise, by Robert Sabine. In his 1867 book The Electric Telegraph, Sabine described a number of these methods, for example de Sauty’s use of a Wheatstone bridge for comparing the capacities of Leyden jars, “one of the most elegant of the many applications of the null methods” (p379; de Sauty, 1865). Follow-up of the works picked out by the Ngram viewer suggests they come almost entirely from the field of telegraphy and electrical engineering. There is a strong positive correlation between the rise in the relative importance of the term “null method” and the growth of these fields.
Answers to questions
1) Maxwell recognised that Coulomb’s torsion experiment was very difficult. Many A level physics teachers would agree with him.
2) Charges have to be static for symmetry and the charges must be single.