# Confusion in the deduction of inverse square law of current elements

In Maxwell's treatise, he discusses the inverse square law of current elements in the following way:

It is observed from the experiment that the circuit B is in equilibrium, i.e. net force on the circuit B due to circuit A and C are equal. From here how can we conclude that force between corresponding current elements $(A_{2}B_{1} \text{and} B_{2}C_{1})$ are equal?

The key phrase is "whatever be the forms and distances" of the circuits. For any given particular form, one could imagine that the overall circuit is balances without deducing that each circuit element is balanced. But if you can make the statement for any shape of the circuits, then intuitively it is clear that the statement of balance must be true for some underlying reason, namely, balance between each infinitessimal circuit element.

You can in fact prove this, assuming only fairly weak smoothness conditions on the local currents. I'm not convinced Maxwell had a rigorous mathematical proof in hand but the conclusion must have been as obvious (or more) to him as it is to us today.

• Why is the underlying reason " balance between each infinitesimal circuit element"? Can you explain more elaborately or give an outline of the mathematical proof? Oct 20, 2016 at 1:23

Ampère hypothesized that the force between two circuits is due to the integrated forces between the circuits' infinitesimal circuit elements:

First hypothesis. — Let C be a uniform current that acts on an uniform current element $ds'$. We decompose in our mind the current C into elements $ds_1$, $ds_2$, … The action of the current C on the element $ds'$ is the resultant of elementary actions exerted by the elements $ds_1$, $ds_2$, … on the element $ds'$.

—Pierre Duhem, “Appendice au Livre XIV: Sur la loi d’Ampère,” in Leçons sur l’électricité et le magnétisme, vol. 3, p. 309 (p. 51 of English translation).

One cannot uniquely decompose an integrated force into forces between current elements. (There are infinitely many antiderivatives of a function.)

Ampère's "loops experiment" you show from Maxwell's book only sought to establish the inverse square nature of circuits' mutual forces.