How did Henry Cavendish deduce the inverse square law in electrostatics from his experiment in 1772?

Question

An elegant experiment in 1772 by Henry Cavendish. Cavendish charged a spherical conducting shell that contained within it, and temporarily connected to it, a smaller sphere. The outer shell was then separated into two halves and carefully removed, the inner sphere having been first disconnected. This sphere was tested for charge, the absence of which would confirm the inverse-square law. Assuming that a deviation from the inverse-square law could be expressed as a difference in the exponent, 2 + δ, say, instead of 2, Cavendish concluded that δ must be less than 0.03. This experiment of Cavendish remained largely unknown until Maxwell discovered and published Cavendish’s notes a century later (1876). At that time also, Maxwell repeated the experiment with improved apparatus, pushing the limit down to δ < 10^(-6). The present limit on δ is a fantastically small number – about one part in 10^(16);

From this experiment he deduced the inverse square law in electrostatics even about 14 years before Coulomb. But I want to know how exactly did he deduce it? The experiment isn't proving the law directly. So how did he do it exactly?

Answer 1

Alexandre Eremenko's answer is great, but I figure the page could benefit from an explanation of the method in general.

Cavendish's Experiment

The question Cavendish was facing was this: Given that we know that charged bodies exhibit attractive and repulsive forces, what is the likelihood that, like Newton's law of gravitation, the electric force obeys an inverse square law with distance? (Not probabilistic likelihood, I mean just inquisitively). Newton had shown that if the force law is an inverse square law, then a hollow uniformly massive sphere will not produce a net gravitational force at any point within the sphere.

Cavendish took the converse of this to investigate the behavior of the electric force. That is, he assumed if a hollow uniformly charged sphere does not produce a net electric force at any point within the sphere, then the electric force obeys an inverse square law. Note that, as mentioned by Alexandre, this is logically invalid, because a statement does not imply its converse. Cavendish may not have been aware of this, but nonetheless, his experiment was designed to prove that if the electric force obeys a power law with distance, then the power must be $-2$.

The experiment assumes that (as was known at the time) conductors allow electric "fluid" to move freely. Then a hollow conducting sphere, when charged, will have a uniform charge distribution. If we have a conducting sphere on the inside of this shell, but not in contact, then we expect that if the net electric force at any given point is not zero, then the electric "fluid" (charge) will be distributed about the inner sphere unevenly. If you then connect a small wire from the charged shell to the inner sphere, we would expect a net charge flow because of the net force on points in the sphere.

Net charge flow means that, when we remove the charged shell connection and remove the charged shell, the inner sphere will be charged if there was net force, so if the inner sphere is not charged, then there was no net force. The absence of net force is what Cavendish took as evidence of the inverse square law, so after finding that the inner sphere was not charged, he accepted that the electric force was nearly an inverse square law.

By estimating the sensitivity of his electroscope, Cavendish was able to approximate the uncertainty in his null-charge measurement.

Maxwell's Proof

Maxwell, in his Treatise on Electricity and Magnetism (vol. 1), provides a fairly rigorous (but still flawed) proof of the fact that if the power is an inverse power, then it must be $2$, which he credits partly to Laplace. One problem with the proof is that it depends not on simply testing the null-charge result for one sphere radius, but for every radius, to truly be confirmed. Another issue, is that while the electric force is clearly a monotonically decreasing function, it was not necessarily an inverse power law. This comes back to the issue of "a statement does not imply its converse." However, the evidence for an inverse-square electric force was building, and the assumption seemed increasingly valid.

References: Maxwell's Treatise on Electricity and Magnetism, art. 74, (p. 80).

Answer 2

Newton proved that if the attraction obeys the inverse square law, then the force inside a uniformly charged sphere is zero. It follows from the description that you give that Cavendish used the converse statement. In fact this converse statement is true though I doubt that Cavendish had a proof of it in full generality.

It is very common for physicists (and other non-mathematicians) to confuse a direct theorem with the converse one, necessary conditions with sufficient etc. (Even Newton did this). Of course, one can make various a priori assumptions with which this statement becomes easy: for example that the law of attraction is a power. Then the power must be -2. I suppose this is what Cavendish had in mind.

Ref. for a discussion of the converse of Newton's theorem: MR2125274 S. Stein, Remarks on the gravity equation. Amer. Math. Monthly 112 (2005), no. 4, 322–333. (He discusses another similar theorem of Newton, but this one can be treated in the same way).

Remark. Coulomb did more than that. He was able to measure the force, to determine that it is proportional to the charge, not only inverse to the square distance.

Remark 2. Physics book from which I studied electricity as a child credits this argument to B. Franklin. But gives no reference.

Remark 3. Here is an exposition of Newton's proof of his theorem. (The statement and an application are here.

Answer 3

The previous posters wrote a nice excerpt about the history of finding the electrostatic inverse law of charges.

For a more scientifically explanation I recommend the following paper Falconer: "No actual measurement … was required: Maxwell and Cavendish’s null method for the inverse square law of electrostatics".