When was "Faraday's Law of Induction" first expressed in a quantified form?

Question

An often used definition of "Faraday's Law of Induction" goes something like this (found in Wikipedia)

The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path.

This law is often attributed to Faraday's 1832 Report to the Royal Society published under the name "Experimental Researches in Electricity"

However, in that document, I was unable to find any reference to an equality between a quantified electromotive force with a rate of change of a quantified magnetic flux. Rather, in section 42 of that document, he refers to

a law hereafter to be described (114)

and in section 114, describes a qualitative, but not quantitative law which he refers to as

The relation which holds between the magnetic pole, the moving wire or metal, and the direction of the current evolved

I will not repeat everything Faraday wrote about this relation. Suffice it to say, that the explication of this law turns out to be a far cry from a quantitative law equating the e.m.f. and the rate of change of magnetic flux.

Now, there is another published work by Faraday with the same name i.e. "Experimental Researches in Electricity", published in three volumes from 1839 to 1855. The first part of volume 1 of this larger work appears to be identical to that published in 1832. I did not read it all. However, a search of the words "flux" and "electromotive" shows they only appear 3 times each in the first volume, and none in conjunction with each other.

So, my question is, when was Faraday's Law of Induction, in the form of a quantitative equality, first expressed. And was it even first expressed by Faraday? Was it by Maxwell? Heaviside? Or did it come earlier than those two scientists? Is there something that Faraday wrote that entails a mathematical equality, even though it may have been expressed with words other than "flux" or "electromotive force"?

Answer 1

Faraday stated the law more quantitatively in the report titled On Lines of Magnetic Force read to the Royal Society in 1851. He does not specify the direction of the force (and he originally got it wrong in 1832), but the sign was already established by Lenz in Ueber die Bestimmung der Richtung der durch elektodynamische Vertheilung erregten galvanischen Ströme (1834). However, Faraday's formulation is still verbal, and the lack of mathematical gloss was generally an obstacle to the reception of his ideas. They were assimilated only after Maxwell gave them an explicitly mathematical formulation in 1861-2, although practical applications, like electric generators and telegraph, go back to 1830-s, see The birth of the electric machines. In particular, Maxwell reformulated the law of induction as a partial differential equation, which Heaviside originally dubbed Faraday's law, but which is now called the Maxwell–Faraday equation.

Faraday's report is printed in Experimental Researches in Electricity, vol.3, here is the relevant passage:

"3114. It is also evident, by the results of the rotation of the wire and magnet (3097. 3 106.), that when a wire is moving amongst equal lines (or in a field of equal magnetic force), and with an uniform motion, then the current of electricity produced is proportionate to the time; and also to the velocity of motion. 3115. They also prove, generally, that the quantity of electricity thrown into a current is directly as the amount of curves intersected."

The "quantity of electricity thrown into a current" is essentially the electromotive force, the "curves" are Faraday's lines of force, and "the amount of curves intersected" is proportional to the intensity of the magnetic field in modern terms. Whittaker explains the context in A History of the Theories of Aether and Electricity, pp. 190ff:

"Faraday conceived the idea of partitioning all space into compartments by tubes, each tube being such that this product has the same definite value. For simplicity, each of these tubes may be called a "unit line of force"; the strength of the field is then indicated by the separation or concentration of the unit lines of force, so that the number of them which intersect a unit area placed at right angles to their direction at any point measures the intensity of the magnetic field at that point."

He then paraphrases the boldface passage into a more recognizable form:"The induced electromotive force is, in fact, simply proportional to the number of the unit lines of magnetic force intersected by the wire per second". Symbolically, with Lenz's sign rule, $\mathcal{E}\sim-\frac{d\Phi_B}{dt}$, as written in modern textbooks.