Veritasium's Problem

Wow, I'm impressed by Veritasium's videos on electricity. The question he poses is so fundamentally interesting that we must review it here. Our goal in this post is to record what we believe Weber and Kirchoff's answer to this question would be.

Let's recall Veritasium's setup: We imagine a huge circuit, where the lightbulb and battery are spatially close, but where the circuit connecting the battery and lightbulb is extremely long. See the screenshot from his video below:

The question asked by Veritasium is this: "How much time is required for the lightbulb to turn on when the circuit is closed?"

The question is seemingly a contest between two ideas: is the electricity like a fluid that must necessarily travel through and along the wire, therefore traversing a large distance requiring a "large" amount of time. Or is electricity like the Maxwellian field view that the battery somehow transfers energy through the field to the lightbulb. And since the battery and lightbulb are very close (small distance) it's assumed that the lightbulb turns ON in a very "small" amount of time.

There have been many critical response videos. For example ElectroBoom makes some fair critiques. For example, the threshold for how much energy is required to turn the lightbulb ON needs to be specified, and in practice there is some leakage voltage coming from electrons in the environment. But this is not really the point.

The Maxwellian Field View of Energy Transfer

The following image shows Veritasium's Maxwellian field-theoretic viewpoint, however ElectroBoom makes a very significant point about the field, specifically about the Poynting vector field $S=\frac{1}{\mu}E \times B$. What Electroboom argues is that $S$ has a $1/r$ term -- but he does not explain where this $1/r$ term arises, so it's not clear that ElectroBoom's point is valid. The issue is that Veritasium is showing the flow of a Poynting field in a situation which is geometrically very different from the original problem.

In this image the yellow lines show the Poynting vector, which is assumed to represent energy flux. But the point is that the circuit depicted in the image is not at all like the original circuit.

Weber and Kirchoff's Response: the Surface Charge Distribution

The best reference for Weber and Kirchoff's work on electricity is AKT Assis' book on "The Electric Force of a Current" available at AKT Assis' webpage. Basically Weber and Kirchoff identify the time required for the lightbulb to turn ON to be equal to the time required for the battery to assemble the surface charge distribution representing the steady state current. This needs be elaborated further.

Briefly, what Weber and Kirchoff discovered is that the purpose of the battery is to work and sustain a surface charge distibution on the wire. Consequently, the time required for the lightbulb to turn ON is precisely the time required for the surface charge distribution to be assembled by the battery. The battery does work using either electrochemical energy or other sources of mechanical or non-electrical energy.

We do not precisely know the time required for this surface charge distribution to be assembled, and it's very interesting question. However this charge distribution is very quickly attained because the electric charge elements do not need travel very far. The surface charge does not require, for example, any negative charge elements to flow "from the battery". There is no "flow" from the battery to the lightbulb, contrary to the Maxwellian-Poynting conception. For Weber and Kirchoff, the battery is specifically used to assemble the surface charge distribution.

I suppose the interesting question is: "Why does the battery only do work on the surface charges when the circuit is closed?" In otherwords, "_What is the difference between open and closed circuits?"_

The Battery

At issue is the question, "What is the battery doing?" The idea of Weber-Kirchoff is that the battery does work, namely assembling and sustaining a surface charge distribution. This is treated in more detail in Assis' textbook cited above.

Now Veritasium are somehwat aware of surface charge distribution in wires. See the image below. Apparently they learned this from Shabay-Sherwood's textbook "Matter and Interaction", and this same text is influential on AKT Assis. However Assis demonstrates how Weber and Kirchoff effectively found the correct formulation of Ohm's Law, which leads to more precise equations for the surface charge distribution.

Confusion and Uncertainty in the Maxwellian Viewpoint.

We remark that Veritasium phrases his question in classical terms. And it's very interesting that we are comparing "conjugate variables" in the sense of quantum mechanics, i.e. Energy and Time. This is like asking for the velocity of a particle when it passes through a certain point. Heisenberg's Uncertainty Principle basically precludes any definite answer! So we must clarify that Veritasium is seeking a classical estimation of the time required. And we do not object to this at all! In otherwords, Veritasium's question is asking for the time required before a certain amount of energy is attained by the lightbulb. And Heisenberg Uncertainty does not allow us to precisely know both values simultaneously.

This quantum uncertainty is somewhat bypassed because the battery and the lightbulb are proximally "close", i.e. they are extremely close and not separated by a large distance. This somewhat allows us to bypass the controversial question of whether information is a priori bounded by the speed of light constant $c$.

Ultimately, when Maxwellians look at the power lines, they look at the empty space between the powerlines and imagine there is energy and a field there. But the field is not reified, i.e. it has no substance (no length or mass or area, etc.) The field has no temperature or pressure or anything definable. It is pure mathematical symbolism: $E$ and $B$. For indeed Maxwell must define two fields. And how do these fields interact with matter?

Conclusion

This post argues that Weber-Kirchoff interpret the time required in Veritasium's huge circuit problem as exactly the time required for the battery to assemble the surface charge distribution representing the stead state current. This is a mathematical problem to be studied further. Using the Weber-Kirchoff equations, this time should be amenable to mathematical analysis, and this to be subject of further study.

[To be continued ... -JHM]

To explain further the idea: the question is somewhat related to the following question: suppose we have a density of positive and negative electric charge distributions $\rho$ on a unit disk $D$. If we let the potential $\rho$ evolve according to Coulomb's force law to equilibrium, then either the charges neutralize or they are pushed through the boundary $\partial D$. Given that the disk has unit radius, we ask the question: "How much time before the charge distribution is neutralized (either in the interior or at the boundary)?" All this needs to be elaborated in more detail.