Heaviside's Force Law

I give Heaviside more credit than others. He may have published Lorentz's force law first and he had so much to do with the telegraphers equations to say nothing of his influence on the way we right Maxwell's equations today with vector calculus.

The law agrees with Maxwell's equations and I thought I'd work through some educated speculation and some facts.

First, if a positive charge is located a positive distance along the x-axis from a charge carrying wire on the z-axis with the current flowing out of the page. The current flowing in the wire will generate a 'magnetic field' circulating in a counter-clockwise direction around the z-axis.

The electrons flow counter to the current. There are more than ten to the eighteenth power electrons per Coulomb so a more manageable 1 mA current will still have a massive more than ten to the fifteenth power electrons flowing per second. Try for a second to grasp how large that number is. That's a lot of electrons traversing what might be a small wire.

The telegrapher's equations tell us that there is a certain small G, C and especially L that sap some of the ten to the power of fifteen electrons per second from our current. I want to focus on the L which must escape the wire (nothing is really holding it in but a hot conductor). The escaped electron will generally be traveling in the same direction as the electron flow. This negative charge carrier will be a hot carrier injected into the flow causing flux. The electron will eddy out behind one or more positive nuclei causing curl. The cross product of this vector curl is said to be the fictitious magnetic field.

Heaviside's force law will take the cross product of the positive ion traveling out of the page along the z-axis with the 'magnetic field' which is pointed up at that point. The positive ion will move towards the conductor which agrees with three of Maxwell's equations. The 'magnetic field' isn't changing so one of Maxwell's equations is trivial.

Back to the telegrapher's view of electric matter. The flow around the conductor is in flux due to the hot carriers flying off the wire. These hot carriers are almost all electrons and they curl. The hot carriers congregating at the boundary of the new medium may cause the positive carriers to move in to balance the charge.