### Cavendish’s phenomenological derivation of Coulomb’s law

May we not infer from this experiment, that the attraction of electricity is subject to the same laws with that of gravitation, and is therefore according to the squares of the distances; since it is easily demonstrated, that were the earth in the form of a shell, a body in the inside of it would not be attracted to one side more than another. ~ Joseph Priestly, The History and Present State of Electricity, Vol. II. 1767.

Charles de Coulomb’s  (fl. 1735-1806) justification for the law that bears his name in electrostatics is neither the law’s first nor its most elegant. In 1785, Coulomb put forward the following hypothesis, which is universally understood by all grade-school students of physics as Coulomb’s law:

The electrical force exerted between two static charges drops off as the square of their separation.

However, Coulomb’s law is somewhat of a misnomer; Henry Cavendish (fl. 1731-1810) discovered the law two years prior to Coulomb’s announcement. Furthermore, Cavendish’s experimental justification is simpler and  more elegant than the justification that Coulomb concocted, which required an experimental setup consisting of a somewhat complicated torsion balance:

Cavendish took a different approach. In his experimental set up, Cavendish did not measure electrostatic force; he simply confirmed there lack of. That is, he confirmed the following phenomena, which is now very well known:

A charged conductor produces no electric field in its interior.

Although this observation is correct, its justification in popular literature is often not. In many well respected physics textbooks, even at GSU’s hyperphysics, the justification is usually a sketchy qualitative “proof” using vague arguments about equilibrium and how an interior electric field is somehow a violation of that hypothesis. These arguments, however, are total bogus. If Coulomb’s inverse-square law were to be replaced, for example, by some sort of inverse-cube law, then charged conductors would produce internal electric fields. Therefore, any justification of Cavendish observation must involve a critical analysis of the inverse-square relationship between electrostatic forces and the separation of static charges.

However, it is possible to work in the other direction. Cavendish shown in 1773 that the experimental failure of charged conductors to produce internal electric fields is a strong justification that electrostatic forces must obey an inverse-square law of attraction/repulsion.

### The Experiment

Cavendish constructed the following apparatus to determine the electric field within a charged conductor:

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 electronometer failed to show any significant charge on the inner sphere, confirming the hypothesis that electric conductors cannot produce interior electric fields.

### Justification of Coulomb’s law

[[ For the sake of historical interest, Joseph Priestly (fl. 1733-1804), the discoverer of oxygen, first shown in 1767 that the absence of internal electric fields in conductors gives rise to the inverse-square law. ]]

Consider a conducting sphere $\displaystyle S$  and a small positive test charge $Q$ in its interior. And consider points $A_1$ and $A_2$ on the sphere such that $A_{1}SA_{2}$ forms a line. And consider arbitrarily small surface elements $dA_1$ and $dA_2$ that are disjoint neighbourhoods of $A_1$ and $A_2$, respectively. Pictorially:

The fact that the charge $Q$  remains in static equilibrium regardless of its position within the sphere  suggests that the the net electrical force on $Q$ applied by some charged spherical patch is opposed by an equal and opposite electrical force produced from another charged patch. Both patches, by the law of action-reaction, must produce the same solid angle when projected onto from $Q$ since the surgace charge density of a conductor is uniform. More strongly, the force applied on an interior charge from a charged spherical patch must be proportional to the corresponding solid angle. [[ Why? ]]

Suppose that  $dA_1$ and $dA_2$ produce equal and opposite forces acting on $Q$, so these patches correspond to a solid angle $d\Omega$ when projected onto from $Q$. The geometrical definition of a solid angle gives rise to the equations

$\displaystyle d\Omega =\frac{d{{A}_{1}}}{{{\left( Q{{A}_{1}} \right)}^{2}}}\cos \phi$

$\displaystyle =\frac{d{{A}_{2}}}{{{\left( Q{{A}_{2}} \right)}^{2}}}\cos \phi$,

where $\displaystyle \phi =\angle S{{A}_{1}}Q=\angle S{{A}_{2}}Q$. However, since these patches may be arbitrarily small, we set $\displaystyle \cos \phi =1$, from which it follows that the net force applied by the patch $dA=dA_1$ on the test charge $Q$ is proportional to

$\displaystyle Qd\Omega =\frac{QdA}{{{\left( QA \right)}^{2}}}$,

which is exactly Coulomb’s law.

Bibliography

* A Cultural History of Physics, by K. Simonyi. 1978.

* The Electrical Researches of the Honourable Henry Cavendish, an anthology of Cavendish’s research edited by J.C. Maxwell. 1879.

* The History and Present State of Electricity, Vol. II, by J. Priestly.1767.

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