Power flow outside a wire - how close?

[Joseph M. Zias]

TL;DR Summary

power flow vs distance from wire

For at least a couple of decades a plethora of papers have presented power flow (via Poynting vectors) to be outside the wires. Wire resistance does cause some power to flow into the surface and cause heating, however. Given a DC circuit with low resistance wire I question how close to the wire is the power flow? Suppose a 10 volt source to 10 AWG wire with a 10 ohm resistor some distance away. What is the distribution of power vs perpendicular distance from the wire? So far, I have not seen this addressed.

 

[renormalize]

TL;DR Summary: power flow vs distance from wire

The Poynting flow of energy $\vec{S}=\mu_{0}^{-1}\vec{E}\times\vec{B}$ due to electric circuits generally extends to a rather large volume around the wires. This is can be seen from the figures in: Morris & Styer-Visualizing Poynting vector energy flow in electric circuits . To simplify the analysis to a 2D model over a finite region, the authors consider circuits consisting, not of conducting wires, but rather infinitely-long, conducting, hollow square cylinders with discontinuities on the boundary that represent voltage sources and resistances. Here's an example of the Poynting flow due to a battery connected to two resistors in series:

Evidently, the flow of energy spreads out from the source (the battery) to fill the space enclosed by the circuit and then converges into the sinks (the resistors). Of course, the details of 3D Poynting flow in the vicinity of a circuit of wires is more involved, but we can still anticipate that the lines of flow spread well away from the wires to fill both the interior and exterior of the circuit.

 

[hutchphd]

I note this is an undergrad research paper from 2000. Way to go Oberlin College. Great idea.

 

[Joseph M. Zias]

So, the analysis given suggest the power lines at the top of the poles are transmitting energy across the street? I will look into the reference given but at present I don't see any scale of watts/cm^3 at a given distance.

 

[renormalize]

So, the analysis given suggest the power lines at the top of the poles are transmitting energy across the street?

Not true, at least to the extent that we can ignore the energy that's lost as heat in the power lines. Per the plot, no energy streamlines begin or end on the "power lines" themselves (i.e., the outer box, which is modeled as a perfect conductor), so no energy flows between any of the (perfect) conductors.

I don't see any scale of watts/cm^3 at a given distance.

(Note that your units here are misstated because they denote power per unit-volume. The Poynting vector tangent to the streamlines has units of power per unit-area or $W/m^{2}$ in SI units.) There are no units shown on the plot, but the text of Morris & Styer states:

so the spatial scale of the plot is established. But lacking something like a color scale that distinguishes the power flow in individual streamlines, the most we can say is that an overall power $I^{2}R=0.5\text {W}$ is delivered across space via 15 streamlines from the battery to each resistor. From that, depending on where you draw your cross-section and the distance between streamlines, from the plot you can guesstimate the average power flowing per unit-area.