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In Newtonian gravity, electrostatics, and magnetostatics, we have some vector field $\textbf{F}$ with an energy density proportional to $|\textbf{F}|^2$. What is the simplest, most elementary argument to the effect that it would be unphysical to have an energy density proportional to $|\textbf{F}|$, without the square? I've come up with several arguments that either don't seem convincing or don't seem elementary enough. I'm trying to work out a simple argument that would work well for my students, who are college freshmen.

A non-elementary argument that I find pretty convincing is that for this energy density, the energy has a non-analytic form, and this makes the theory not have predictive value in certain cases. I think this could probably be developed into a more rigorous argument, but I don't think it's something that would fly with my audience.

For an energy density of this form, with a positive constant of proportionality, the force law would not be $1/r^2$, and the triangle inequality tells us that the force would always be attractive, never repulsive. This is contrary to experiment, but not obviously impossible a priori.

Like charges can be superposed without an input of energy. (The energy of two superposed charges is the same as the sum of their separate energies.) Opposite charges do interact, however. Again, strange, but not obviously impossible.

In electrodynamics, it's hard to see how we could preserve conservation of energy when, e.g., two traveling waves superposed. As with the argument about nonanalyticity, I think this could probably be developed into a more rigorous argument, but that would not work for my audience, who don't yet know about electromagnetic waves or induction. I want to limit this to statics.

The force law would not be additive, which would be incompatible with Newtonian mechanics. This is because $|\textbf{F}+\textbf{G}+\textbf{H}|$ doesn't break down into a nice sum of two-body interactions, as $(\textbf{F}+\textbf{G}+\textbf{H})^2$ does. This seems somewhat abstract, although maybe there is a more simple, concrete realization of it?

  • Perhaps similar but easier to explain is the dependence of the kinetic energy of a point particle on $v^2$ instead of $|v|$. My guess is that the explanation should be the same. The obvious link is Why does kinetic energy increase quadratically, not linearly, with speed?. – AccidentalFourierTransform Aug 27 '19 at 01:19
  • @TobErnack: The system of units would just have to be different. –  Aug 27 '19 at 02:26
  • @AccidentalFourierTransform: I assume you have in mind the answer to that question that gives an argument involving Galilean boosts. I think the equivalent of that for E&M would be something involving the relativistic transformation of the electric and magnetic fields. I'm sure that could be made to work. For instance, I could say that in order to make a stress-energy tensor that has the right boost behavior, I need to make it bilinear in the electromagnetic field tensor. But it seems like it would be hard to reduce that to an elementary argument for early on in freshman E&M. –  Aug 27 '19 at 02:31

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