# collinear current elements question

In Maxwell's treatise he says the following about force between collinear current elements:

It says $\alpha$ and $\alpha'$ are in the same straight line. The force between them must therefore be in this line.

Why should it be so? Why should the force between collinear elements necessarily be in their line?

If the force were not in the line of the current elements, than in which direction would you like it to point?

By the symmetry of the situation, any direction that is not "in line" is no different than any other direction. And since the force would have to point "somewhere" the only direction it can point is along the line of symmetry.

The normal motion of electrons within any material is erratic and random. Depending upon how tightly electrons are bound in the outermost shell of their atoms, they may be more or less free to wander about the material. In metals and other electrical conductors, the outermost electrons are so loosely bound that they are free to wander chaotically, can easily be influenced by outside forces, and may be considered to be free electrons.

As these free electrons move about, their motion registers as heat. Heat is the product of electrical resistance. Ohm's Law relates resistance, current (flow of electric charge), and voltage (electromotive force). Electric current is directly proportional to voltage, and inversely proportional to resistance:

I = V / R where I is current, V is voltage, and R is resistance.

So more current per unit time flows where resistance is least. The longer is the pathway between two elements, the greater the amount of vibration and movement that free electrons will undergo as they carry electric charge between the elements.

Electric current is free to follow any path, curvilinear or straight, but the greatest quantity of current per unit time will follow the straight path of least resistance.

In Maxwell's treatise, however, I think he was taking collinear to mean sharing the same circuit, rather than necessarily the shortest path within the circuit.