How Does Voltage Behave in Non-Conservative Time-Varying Fields?

[thegreenlaser]

In circuits, we have no problem saying that the voltage difference between two point is $\cos(\omega t)$, but what does that actually mean? What is "voltage" in this case? I ask because my understanding is that in time-varying electromagnetics, the electric field is no longer a conservative field which can be written as the gradient of a scalar. We have to write $$\vec{E} = -\vec{\nabla} V - \frac{\partial \vec{A}}{\partial t}$$ i.e. (if my understanding is correct) the notion that $V$ is some sort of potential energy (per unit charge) is no longer valid because the electric field is non-conservative and so it has no associated potential energy.

So I have two questions:

1) When we talk about "voltage" in time-varying circuits, do we mean the scalar field $V$ as it appears in the Lorentz gauge (or maybe Coulomb gauge), or do we mean some other quantity?
2) What is the physical interpretation of "voltage difference" in the time varying case (if there is one)? I'm assuming you can't just say "It's the potential difference per unit charge" or something similar because we're not working with conservative fields anymore.

Additional note: I know that in a lot of practical cases (e.g. 60 Hz household wiring) you can just invoke the quasi-static approximation and continue to interpret "voltage" as it's defined in electrostatics, but what I'm interested in is the regime where the quasi-static approximation is no longer valid (e.g. RF/Microwave).

 

[mfb]

I ask because my understanding is that in time-varying electromagnetics, the electric field is no longer a conservative field which can be written as the gradient of a scalar.

This remains a very good approximation even for most high-frequency applications. If it is not, you can still consider a local voltage (as difference between two points nearby), where the non-vanishing curl of the electric field lines does not matter.

 

[Curve Shifter]

In circuits, it's fairly easy to visualize electrons pushing on one another. Voltage is a potential for a charged particle to move between two points. Electro Motive Force (voltage). For a phenomenon that's physically analogous in circuits, maybe the concept of pressure?

That does sort of break down for EMR waves which can travel even through a vacuum, because there's no medium to transmit the 'pressure'. There are equations that relate induced voltage back to magnetic flux. Remember from Maxwell, it can be derived that transverse waves of $\vec{E}$ travel with companion perpendicular magnetic waves and vice-versa. So in an EMR wave (from the point of view of a stationary point along its path), E and B are always in flux. So the ε produced when one of these waves comes in contact with charged particles inside a conductor also varies with time. For something analogous? Maybe a "surfer" trying to catch one of the individual cos waves where ε would be the 'steepness' of the wave face? So your 'gradient' holds up even in high frequency EMR.

Or was it some other phenomenon you were posting about?

 

[thegreenlaser]

This remains a very good approximation even for most high-frequency applications. If it is not, you can still consider a local voltage (as difference between two points nearby), where the non-vanishing curl of the electric field lines does not matter.

Maybe I'm misunderstanding, but I thought the reason RF/Microwave circuits is a different area of study than "normal" circuits is because that kind of approximation does break down at high frequencies. Or is the main difference just that we have to look at a length scale which is smaller than most of the components in order to keep the approximation alive?

In circuits, it's fairly easy to visualize electrons pushing on one another. Voltage is a potential for a charged particle to move between two points. Electro Motive Force (voltage). For a phenomenon that's physically analogous in circuits, maybe the concept of pressure?

That does sort of break down for EMR waves which can travel even through a vacuum, because there's no medium to transmit the 'pressure'. There are equations that relate induced voltage back to magnetic flux. Remember from Maxwell, it can be derived that transverse waves of $\vec{E}$ travel with companion perpendicular magnetic waves and vice-versa. So in an EMR wave (from the point of view of a stationary point along its path), E and B are always in flux. So the ε produced when one of these waves comes in contact with charged particles inside a conductor also varies with time. For something analogous? Maybe a "surfer" trying to catch one of the individual cos waves where ε would be the 'steepness' of the wave face? So your 'gradient' holds up even in high frequency EMR.

Or was it some other phenomenon you were posting about?

I'm not really sure what you're explaining to me... I do understand what voltage is, but my understanding is that that interpretation breaks down once you leave the quasi-static regime. The introduction of the wikipedia page on electric potential seems to agree with me. It says that in the time varying case, you can't interpret the scalar potential in terms of potential energy. Apologies if I'm missing your point.

 

[lightarrow]

Maybe I'm misunderstanding, but I thought the reason RF/Microwave circuits is a different area of study than "normal" circuits is because that kind of approximation does break down at high frequencies. Or is the main difference just that we have to look at a length scale which is smaller than most of the components in order to keep the approximation alive?

If you can confine the time-varying magnetic fields or however the rot(E) ≠ 0, in small regions of space, then ok; if you can't do it and all your dipoles/devices are immersed in such fields, the concept of "voltage" applied to one of these dipoles/devices looses significance. It's as if you had to consider the "mutual induction" of every component with every other component. Clearly, who designs high frequency circuits has ways to compute approximations (I'm not an expert at all) but the common approximation of "lumped elements" circuit breaks down.

For example, try to measure the potential difference between two points with a voltmetre, in a region of space where there are variable fields, even at 50 Hz, if the currents (and so the fields) are quite high: you will discover that the result of your measure depends on how you position and orient the tips, the cables and the instrument itself (= the line integral of the electric field E depends on the path).

 

[Jano L.]

1) When we talk about "voltage" in time-varying circuits, do we mean the scalar field V as it appears in the Lorentz gauge (or maybe Coulomb gauge), or do we mean some other quantity?

In ordinary low-frequency circuits, it is the difference of potentials in the Coulomb gauge (instantaneous function of charge distribution) due to charges which do not produce significant rotational electric field. For example, if a copper coil is connected to 20 V AC voltage source contacts, we say the voltage across the coil is 20 V. What it actually means is that the electric field in the surroundings of the AC source is given by potential, and the absolute value of the integral of this electric field from one end to another along any path is 20 V.

Of course, the alternating current in the coil will produce its own rotational field, which cannot be described by potential function. However, if the coil is assumed to be a perfect conductor, similar integral of this rotational field along path passing through the wire all along will have the same magnitude but opposite sign, because inside perfect conductor, the electrostatic field of the source and the rotational field of the coil will cancel each other.

2) What is the physical interpretation of "voltage difference" in the time varying case (if there is one)? I'm assuming you can't just say "It's the potential difference per unit charge" or something similar because we're not working with conservative fields anymore.

Integral of the quasi-electrostatic field due to charges whose field is almost electrostatic (source contacts, straight wires, capacitor plates, but not inductors.)

 

[thegreenlaser]

In ordinary low-frequency circuits, it is the difference of potentials in the Coulomb gauge (instantaneous function of charge distribution) due to charges which do not produce significant rotational electric field. For example, if a copper coil is connected to 20 V AC voltage source contacts, we say the voltage across the coil is 20 V. What it actually means is that the electric field in the surroundings of the AC source is given by potential, and the absolute value of the integral of this electric field from one end to another along any path is 20 V.

Of course, the alternating current in the coil will produce its own rotational field, which cannot be described by potential function. However, if the coil is assumed to be a perfect conductor, similar integral of this rotational field along path passing through the wire all along will have the same magnitude but opposite sign, because inside perfect conductor, the electrostatic field of the source and the rotational field of the coil will cancel each other.




Integral of the quasi-electrostatic field due to charges whose field is almost electrostatic (source contacts, straight wires, capacitor plates, but not inductors.)

I think I get what you're saying, (and it helps) but this is only true in the quasi-static regime, right? Wouldn't it break down at high frequencies?

 

[Jano L.]

Yes, at high frequencies the fields due to most parts may have significant deviations from $\nabla V$ (V in the Coulomb gauge). I read somewhere that despite this, in the wartime people were able to use the simple theory to advance in the development of understanding of wave-guides, using effective lumped elements instead of full Maxwell equations.

However, the analysis based on the Maxwell equations should lead to more accurate description, and sometimes it is probably necessary. For example, in the design and modelling of antennae, the fields are too complicated and to be accurate the Maxwell equations should be used.

 

[stevenb]

In circuits, we have no problem saying that the voltage difference between two point is $\cos(\omega t)$, but what does that actually mean? What is "voltage" in this case?

It seems like you have a grasp on this now, but I can make an additional comment.

Notice how you prefaced your question with the words "In circuits". As soon as you say this, you invoke a number of assumptions about circuit theory and treat the structure as a topological entity. In a nutshell, you are saying (i) the physical extent of the circuit is irrelevant, (ii) all components can be treated as lumped elements, (iii), there is no net charge build up in any lumped element or component, and (iv) a unique voltage can be defined across the component terminals and these terminal voltages are path independent outside the component.

The last assumption is a key part of your question. A simple coil clearly has nonconservative fields, but if we lump the effects into a terminal behavior based model and assume coupling to the outside of the device is negligible, the circuit model holds accuracy very well.

However, circuit assumptions do break down often. Electrical engineers run into this problem and when they do, they ussually end up scratching their head for a while before they remember to check the circuit assumptions. However, over a very wide range of situations, circuit theory is quite accurate for practical work and the concept of a time varying voltage based on potential theory is useful. Higher frequency, larger circuits, components with large coupling of magnetic fields or components that can develop a net charge will throw a wrench into the works of a model based on circuit theory.