# How does the Poynting vector factor into a normal circuit?

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• rumborak
In summary, the Poynting vector plays a crucial role in energy transmission in circuits, as it represents the flow of energy through the electromagnetic field. However, circuit analysis is based on simplifying assumptions and does not make any claims about the location of energy within a circuit element or how it crosses a boundary. The wires in a circuit create the magnetic field, which then guides the electric and magnetic fields to transmit the energy to the resistor. The energy transmission is guided along the wires, just outside of them, and then directed inward towards the resistor. The concept of electrons being pushed along the wire may not be entirely wrong, but it is not the full picture and does not fully explain the energy transmission in a circuit.
rumborak
So, the Wiki page on the Poynting vector has this image:

I remember hearing/reading somewhere that the energy transmission in a circuit like this is actually not traveling through the wire, but that it actually happens through the electromagnetic field, I.e. essentially the Poynting vectors.

At the same time, I feel this is too wild a claim given how something like Ohm resistance is due to "pushed" electrons bouncing against atoms.

So, is this a mixed influence, or is the usual analysis of circuits just a convenient, but eventually inaccurate way of looking at things?

rumborak said:
So, is this a mixed influence, or is the usual analysis of circuits just a convenient, but eventually inaccurate way of looking at things?
I wouldn't say that circuit analysis is inaccurate. It is based on certain simplifying assumptions, but as long as those assumptions are met it is accurate.

The problem is that people read too much into it. Circuit theory describes how much energy is transferred, but it never makes any claim about where energy is located within a circuit element nor where energy crosses a lumped element's boundary. There is no conflict here because circuit theory makes no claim on the question.

hutchphd, anorlunda and LLT71
So, what is the role of the wires, if not the means of energy transmission? Is it simply creating the field which then transmits the energy straight from the battery?

rumborak said:
Is it simply creating the field which then transmits the energy straight from the battery?
Yes. In particular, it contains the currents which are the source of the magnetic field.

hutchphd
How does this work transfer happen in a DC circuit like this? Since the magnetic field is constant, nothing gets induced in the resistor, so what part of the field actually transmits the energy to the resistor? That is, how does the "electrons bouncing off atoms" play into this which eventually transmits the energy to the atoms in the resistor?
If the electrons got accelerated by the field, thus providing the energy that makes them bounce against the atoms, I could see that. But a static magnetic field won't accelerate electrons.

Ooh, that is an excellent paper, thanks a lot!

In a nutshell.

The potential difference between the conductive wires guides the electric field.
The currents that flows along the conductive wires guide the magnetic field.

The cross product of the electric and magnetic fields gives the Poynting vector = energy flow.
The product of the voltage and the current gives the rate of energy flow = power towards the load.

The resistance of the wires, multiplied by the square of the current that is needed to guide the magnetic field, results in a loss of energy along the wires before that energy can reach the load.

rumborak said:
So, is this a mixed influence, or is the usual analysis of circuits just a convenient, but eventually inaccurate way of looking at things?

As @Dale said, simplified is a better word than inaccurate. I say thank god that we can do the simplification or else we could never design circuits.

Consider the following four pictures. They depict four different ways to lay out the wires implementing the simple circuit you showed in the OP. According the simplifications of circuit analysis, the could all be equivalent, and when checked with a voltmeter and ammeter, give identical results. Yet when analyzed with Poynting vectors, they are wildly different. It might take a lifetime of analysis to prove that the four are equivalent using Maxwell's Equations.

So, if I understand that paper @Dale linked to correctly, while it is true that the Poynting vector permeates all space around the setup, due to the surface charges on the wires, the electric and magnetic fields are guided in a way that a large fraction of the energy travels parallel to the wire, just outside of it. At the resistor these vectors then point inward, which means the energy flows inside the resistor.

@anorlunda , I think that guiding action is probably also the reason why all those circuits can be analyzed in the simplified manner, since virtually all energy flows right outside the wire.

On a side note, reference 8 of that paper quoted Feynman, who was stymied by this too. Amazingly, it seems only in recent years have people figured out how this works even for the most basic circuit.
EDIT: This lecture:. http://www.feynmanlectures.caltech.edu/II_27.html. The relevant section starts at 27-5.

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rumborak said:
o, if I understand that paper @Dale linked to correctly, while it is true that the Poynting vector permeates all space around the setup, due to the surface charges on the wires, the electric and magnetic fields are guided in a way that a large fraction of the energy travels parallel to the wire, just outside of it. At the resistor these vectors then point inward, which means the energy flows inside the resistor
Yes, you are correct.

So, to *some* degree this makes sense, but what is still unclear to me is how this eventually relates to the atomic level. Is the idea here that instead of the usual view of electrons being pushed along the wire, it is actually the external electric field enters the resistor and pushes those electrons? (since it is a resistor and not a wire, the electric field actually enterering the resistor)

Is the "electrons pushed along the wire" idea entirely wrong, or just negligible to the energy transmission?

Sorry for the million questions, but I am mesmerized by that despite my Master's in EE, I apparently have no clue how even the most basic circuit operates.

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rumborak said:
how this eventually relates to the atomic level
I don't know enough quantum electrodynamics to answer anything at the atomic level. I can only help at the classical level. However, if you are willing to simply accept Ohm's law, ##J=\sigma E##, as a valid classical law in a resistor then I can help.

Personally, I find Ohm's law to be conceptually sufficient and don't see any need or value to talk about atoms or electrons. So I deliberately avoid mixing classical and quantum concepts.

rumborak said:
it is actually the external electric field enters the resistor and pushes those electrons?
The key is to recognize that the surface of the conductors and resistors have surface charges. A surface charge leads to a discontinuity in the E field. Outside the conductor the E field is roughly perpendicular to the surface of the conductor, but inside the conductor the E field is in the direction of the current flow.

Here is a reference showing this in a rather qualitative manner.
https://www.tu-braunschweig.de/Medien-DB/ifdn-physik/ajp000782.pdf

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rumborak
Although I think you can apply the poynting vector and get a result I don't think it is a valid intuition. If the energy in a DC circuit truly came from the battery to the fields to the load, then it would be subject to the inverse square law and circuits would not work well far from the battery.
Also, the em force is really only between two charges. If the power comes from outside the wire, then where is charge that delivered the force? It can't be outside the wire.

neobaud said:
If the energy in a DC circuit truly came from the battery to the fields to the load, then it would be subject to the inverse square law and circuits would not work well far from the battery.
This is not a valid argument at all. With the conductors attached the fields no longer fall off as the inverse square of the distance from the battery. That is the whole point of using the conductors.

neobaud said:
Also, the em force is really only between two charges. If the power comes from outside the wire, then where is charge that delivered the force? It can't be outside the wire.
The charge is located on the surface of the wire.

Dale said:
This is not a valid argument at all. With the conductors attached the fields no longer fall off as the inverse square of the distance from the battery. That is the whole point of using the conductors.

The charge is located on the surface of the wire.
The charge is also moving through the center of the wire. Otherwise why does the cross sectional area matter in calculating resistance?

Also, I think it is a perfectly valid argument. Are we to believe that the e field is somehow preserved just because it is in a conductor? No it is preserved because it is carried by the charge

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neobaud said:
The charge is also moving through the center of the wire. Otherwise why does the cross sectional area matter in calculating resistance?
The current moves through the center of the wire (in DC), but the charge is on the surface. Inside the wire, even where current is flowing, the charge density is 0 since the positive and negative charges cancel.

Dale said:
The current moves through the center of the wire (in DC), but the charge is on the surface. Inside the wire, even where current is flowing, the charge density is 0 since the positive and negative charges cancel.
There is non zero charge density all through the wire (due to imperfections in the structure.) This is what literally what causes resistance and produces heat

weirdoguy
neobaud said:
The charge is also moving through the center of the wire.
The surface of the wires guide the electric field from the source to the load.
The magnetic field is guided by it's reflection in the surface of the wires. The process of magnetic field reflection is the cause of the current in or on the wire.
The Poynting vector is the product of the electric and magnetic fields which gives the flow of energy = power.
The product of voltage between the wires, and the current in the wire, is also the power. The voltage is a proxy for the strength of the electric field. The current is a proxy for the strength of the magnetic field.

hutchphd
neobaud said:
There is non zero charge density all through the wire (due to imperfections in the structure.) This is what literally what causes resistance and produces heat
You seem to be talking about a microscopic scale. I am talking about the macroscopic scale. At the macroscopic scale the charge density is zero inside the wire, even with the imperfections since the imperfections are on the microscopic scale.

Dale said:
Here is a good reference on the topic.

http://depa.fquim.unam.mx/amyd/arch...ia_a_otros_elementos_de_un_circuito_20867.pdf

In particular, pay attention to the discussion of the resistance and the energy flowing into the wire.
It says I don't have permission to access that file. I wonder if it's a region-based restriction, or maybe it's because they noticed the increased traffic from our members and put up a wall, since they never meant to publish it on the open Internet in the first place?

rumborak said:
Sorry for the million questions, but I am mesmerized by that despite my Master's in EE, I apparently have no clue how even the most basic circuit operates.
Then it is good to ask these questions!
Consider the following:
I have a lamp plugged into the AC wall outlet and I have the 18 gauge cord across my workbench but I don't which end has the plug and which the lamp. If I "look" at the cord I see electrons wiggling back and forth but this motion is symmetric left to right. The motion of these charges does not tell you which way the power is flowing (I believe the motion is entirely agnostic).
How can this be? The fields are not symmetric and the Poynting vector near each conductor points towards the lamp.

Of course the fields and the charges in the wire depend upon each other in a complicated way,but that is not the important physics here. I will sow the wind and recommend highly the Veratasium video

/

Baluncore said:
The surface of the wires guide the electric field from the source to the load.
The magnetic field is guided by it's reflection in the surface of the wires. The process of magnetic field reflection is the cause of the current in or on the wire.
The Poynting vector is the product of the electric and magnetic fields which gives the flow of energy = power.
The product of voltage between the wires, and the current in the wire, is also the power. The voltage is a proxy for the strength of the electric field. The current is a proxy for the strength of the magnetic field.
In the case of a DC circuit the electrons transfer the E field. The electrons in the battery push the electrons near them which push the electrons near them and so on through the wire. This is why the inverse square law does not apply in a circuit.

hutchphd said:
Then it is good to ask these questions!
Consider the following:
I have a lamp plugged into the AC wall outlet and I have the 18 gauge cord across my workbench but I don't which end has the plug and which the lamp. If I "look" at the cord I see electrons wiggling back and forth but this motion is symmetric left to right. The motion of these charges does not tell you which way the power is flowing (I believe the motion is entirely agnostic).
How can this be? The fields are not symmetric and the Poynting vector near each conductor points towards the lamp.

Of course the fields and the charges in the wire depend upon each other in a complicated way,but that is not the important physics here. I will sow the wind and recommend highly the Veratasium video

/

Yes this video actually caused me to start googling. I have not seen a good answer to my two questions above. Also, I see the experiment in the video as being irrelevant. Even if you agree with the result it is because the moving charges near the battery led to the force that caused a current in the light bulb right? And this only works if the switch is near the light. If it is a light second away you don't get the same result.

Dale said:
You seem to be talking about a microscopic scale. I am talking about the macroscopic scale. At the macroscopic scale the charge density is zero inside the wire, even with the imperfections since the imperfections are on the microscopic scale.
Dale: Pretty sure there is no non-zero charge on the surface of the wire. If there were, then a charge on the outside would be attracted to (or repelled from) it. I can't get to the paper you linked but I am almost certain that this is not the case.

neobaud said:
Even if you agree with the result it is because the moving charges near the battery led to the force that caused a current in the light bulb right? And this only works if the switch is near the light. If it is a light second away you don't get the same result.

Yours is a chicken and egg argument. It is both. What occurs is a compendium of electromagnetic effects including charges and fields. The asymmetry manifests in the fields: Poynting Vector.

/

hutchphd said:
Yours is a chicken and egg argument. It is both. What occurs is a compendium of electromagnetic effects including charges and fields. The asymmetry manifests in the fields: Poynting Vector.

/
It seems like you are agreeing with me? "What occurs is a compendium of electromagnetic effects including charges and fields."

Could you give a little more detail of what you mean by: "The asymmetry manifests in the fields: Poynting Vector."
I don't understand this.

See the Veratasium video from 6:00 et seq.

neobaud said:
It seems like you are agreeing with me?
You will need to be a little more spoecific...!

neobaud said:
Dale: Pretty sure there is no non-zero charge on the surface of the wire. If there were, then a charge on the outside would be attracted to (or repelled from) it. I can't get to the paper you linked but I am almost certain that this is not the case.
The surface charge on conductors in ordinary circuits is both well known and completely essential for the functioning of the circuit. This topic is covered in depth by Jackson: https://doi.org/10.1119/1.18112 but it was well known prior to that.

Here is a good paper for developing some intuition about the surface charges: https://www.tu-braunschweig.de/inde...oken=90bbd237cd4ef87cf2941a99eb3c3d610de3d0d3

Dale

## 1. What is the Poynting vector and how does it relate to circuits?

The Poynting vector is a mathematical concept in electromagnetism that describes the directional flow of energy in an electromagnetic field. In a circuit, the Poynting vector represents the flow of energy from the power source to the load.

## 2. How is the Poynting vector calculated in a circuit?

The Poynting vector in a circuit is calculated by taking the cross product of the electric field and magnetic field vectors. It is represented by the symbol S and has units of watts per square meter (W/m2).

## 3. What is the significance of the Poynting vector in circuit analysis?

The Poynting vector is important in circuit analysis because it helps us understand how energy is transferred and distributed in a circuit. It can also be used to calculate the power flow in a circuit and determine the efficiency of the circuit.

## 4. How does the Poynting vector factor into the electromagnetic radiation emitted by a circuit?

The Poynting vector plays a crucial role in the emission of electromagnetic radiation from a circuit. It represents the direction and magnitude of the energy flow, and the rate of energy transfer is directly proportional to the magnitude of the Poynting vector.

## 5. Can the Poynting vector be manipulated in a circuit to improve its efficiency?

Yes, the Poynting vector can be manipulated in a circuit to improve its efficiency. By adjusting the electric and magnetic fields, the direction and magnitude of the Poynting vector can be optimized to increase the power transfer and minimize energy losses in the circuit.

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