# Historical and experimental roots of the equation $Q = C V$ for a capacitor

As typically taught in high school, the voltage ($$V$$) across a capacitor is proportional to the charge ($$Q$$), which is expressed by the formula $$Q=CV$$, where $$C$$ is the capacitance.

I am interested in finding the earliest papers where this equation was stated and how it was demonstrated experimentally. Specifically, I would like to know how the charge and voltage were measured in these experiments.

You might argue that this is just the definition of capacitance, but if $$Q$$ were not proportional to $$V$$, it would not make sense. Therefore, some experimental work must have been done to verify this relationship.

I started by looking into Faraday's books on electricity but did not find relevant information.

• Surely this is somewhere in the following book, but I haven't tried to specifically locate it there: A History of the Theories of Aether and Electricity. From the Age of Descartes to the Close of the Nineteenth Century by Edmund Taylor Whittaker (1910). Commented Aug 31 at 17:08
• Scattered thoughts: Experimentally, capacitors were in use from the mid-1700s (Leyden jars) and this was the focus of a lot of research on the phenomena e.g. by Franklin. I can't say what the state of the mathematical relationships was at this time, since a proper mathematical framework of electrostatics wouldn't come until the turn of the 19th century. But it also depends on the understanding of voltage and charge, and for all I know (without doing some research), there was already a sense of proportionality in the mid-late 18th century. Commented Aug 31 at 21:43
• William Thomson published "On Transient Electric Currents" in 1853 (babel.hathitrust.org/cgi/pt?id=coo.31924066246632&seq=421) which contains a mathematical treatment of electric capacity (capacitance). This is the earliest I've found yet from a quick search Commented Sep 1 at 0:28
• One problem here is that the charge Q is more precisely a charge distribution over a geometric configuration. One will find many explicit formulas for geometric configuration in the literature early on. I suspect that just saying Q has become a pedagogical shorthand for pre-university physics expositions and might well be fairly modern. Commented Sep 2 at 12:17
• @GeorgEssl the definition of capacitance as a constant of proportionality between charge and potential of a metallic body does not require any discussion of charge distributions. Calculating capacitance would, but you would calculate it based on the ratio of charge to voltage (at least for self capacitance). Anyway, since the question of "what is the relationship between charge and potential" is so natural, I would expect it to appear earlier after Poisson. But so far Ive only narrowed it down to between 1813 and 1853. Commented Sep 2 at 17:37