How To Translate Faraday's Law of Induction into Math

I know what you're thinking. What does he get up to that stops him posting promptly and prolifically? I wish it had something to do with Instagram models and / or  staying up late making music via Garageband, but it is much more mundane than that. Here's a short passage about the translation of Faraday's Law of Induction into mathematical notation that I've been working on for far longer than you might think. If I've done my job well, it should seem obvious. (Some of the original \LaTeX has been butchered to accommodate Blogger.

(starts)

Faraday's Law, more or less as stated by Faraday, is: 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. How does this get translated into mathematical notation? We need to know that the `electromotive force' is, in the case of magnetic induction, the work done on an elementary electric charge (such as an electron) travelling once around the loop. Work done moving along a path is always a line integral of the product of a force and a displacement (since `work = force times distance').

As a first step, we re-name those things as variables or constants:

let $\mathcal{E}$ be the electromotive force

let $B$ be the magnetic flux

let $\partial A$ be the path, enclosing an surface A

let $ds$ be a small displacement along $\partial A$

let $E$ be an electric flux field

We can write down the equations quite easily if we are familiar with the vector calculus. Work done is given by the mantra `work = force times distance'. For a small displacement $ds = (dx, dy, dz)$ and a force $E = (E_x, E_y, E_z)$ the product is $E_x dx + E_y dy + E_z dz$ which is $E \cdot ds$ in vector notation. The work done along a line is the sum of such displacements along it, which is conventionally shown by the integral $\oint_{\partial A} E \cdot ds$, giving us $\mathcal{E} = \oint_{\partial A} E \cdot ds$.

Translating the other side of Faraday's Law, Faraday thought of electromagnetic fields as `lines of force' - the more lines, the more force - and the flux of a field through an area was the number of lines of force through it. This was Faraday's way of thinking about line and surface integrals without having to actually use either. 

The number of lines of force within a path is the integral of the (strength of the) vector field over any smooth surface enclosed by that path. (The `any' has to be proved, but it is becomes intuitively obvious after visualising a few examples.) So if we take a surface $A$, divide it into non-overlapping patches $dA(n)$, calculate $\frac{\partial B}{\partial t}(n)$ for the centre of the $n$-th patch, and add the total, we get an estimate of the electromagnetic field strength. Make the patches smaller, and we get a better estimate, which in the limit is the integral 

$\mathcal{E} = -\iint_{A} \frac{\partial B}{\partial t} \cdot dA$

That can also be turned into a conventional double integral by substituting coordinates. Hence Faraday's Law of Induction is translated into mathematical notation as

$ \oint_{\partial A} E \cdot ds = -\iint_{A} \frac{\partial B}{\partial t} \cdot dA$

The left-hand side is the work done, and the right hand side is the negative of the time rate of change of the magnetic flux enclosed by the path. This completes the translation of Faraday's Law into mathematical notation.

This is no more conceptually complicated than if we had translated, say, a passage of Freud from German to English. There is no word-for-word mapping of the two languages, and there are many concepts for which there is a German word, but not an English one, and one must attempt to explain the German concept in English. Using an integral to denote the result of a limit of finite sums is no more exceptional that using a derivative to denote the result of take rates-of-change over ever small intervals. 

We can use some maths to go further. By Green's theorem, assuming the fields are sufficiently smooth, we have

$\oint_{\partial A} E \cdot ds = \iint_A \nabla \times E \cdot dA$

So we can put

$\iint_A \nabla \times E \cdot dA = -\iint_A \frac{\partial B}{\partial t} \cdot dA$

which gives us immediately one of Maxwell's equations

$\nabla \times E = -\frac{\partial B}{\partial t}$

We can prove that, with the rest of Maxwell's equations, this another statement of Faraday's Law of Induction. 

This is no more conceptually complicated than if, having translated the passage of Freud, we then drew a conclusion from the translation and some background knowledge that was not in the original, but helps us understand what Freud was saying. It just looks impressive / mysterious / difficult because it uses undergraduate maths.

(ends)

My thesis is that translating from a natural language into math notation is the same as translating from one natural language to another. It's just that maths is the language in which it is easier to see the patterns and make the deductions.