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I often see people saying "Ohm's law doesn't apply here", usually in relation to AC circuits and diodes. They describe certain situations as being "non-Ohmic". My understanding of Ohm's law is that it always applies all the time, only it's sometimes not useful to think about it that way.

I'm the kind of person that needs to understand from the most "technically-correct" point and work upwards from there. For instance, AC isn't some special form of electricity, it's just DC with a frequently-inverted voltage. For practical engineering in the real world, it's useful to treat the two as unique, but technically speaking, it's all just electricity.

So with that in mind, does Ohm's law always apply in all situations so long as you sample an infinitely narrow point in time?

Note: When I talk about Ohm's law, I'm talking about the relationship between voltage, resistance and current: given any two you can calculate the third. I've seen some people try to describe Ohm's law as defining that a linear increase in voltage would result in a linear increase in current (which might not be true because increasing current may generate heat and also increase resistance, for instance).

Clonkex
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    Your "infinitely-narrow point in time" may be too restrictive. Current is related to time: *coulombs-per-second*. I suppose you could measure current by measuring the time for one charge carrier to pass a point, but you'd have to do that many times to get an accurate estimation of current. – glen_geek Apr 12 '23 at 23:19
  • @glen_geek That's a good point. I forgot current describes rate of flow and thus must be measured over some amount of time. – Clonkex Apr 12 '23 at 23:42
  • Well, but that is taken care by infinitesimal calculus, isn't it? When you have a nonlinear resistor, like a diode or a transistor, you can always linearize at the operating point and use the incremental or differential resistance. – Sredni Vashtar Apr 12 '23 at 23:50
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    "My understanding of Ohm's law is that it always applies all the time" No it doesn't. Unfortunately, professors don't know that or don't think about it, so they incorrectly teach students that Ohm's law applies to all components. It doesn't. It only applies to resistors and a few other resistive components. – Davide Andrea Apr 13 '23 at 00:38
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    @DavideAndrea Professors? Ha, I'm entirely self-taught. Maybe I'd actually know better if I had gone to University haha :) – Clonkex Apr 13 '23 at 00:41
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    The grain of salt to take with resistors and Ohm's law is *noise*. – greybeard Apr 13 '23 at 02:13
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    Consider a circuit with reactive components such that voltage and current are out of phase. Ohm's law may be found to apply moment by moment but the results change continually. While the basis of the changes are well understood any given instantaneous result in isolation is meaningless (maybe "useless") when the system is considered over a time period of an integral number of AC cycles. The fact that Ohm's law may apply at any given point, or at all points in isolation may be true but of no practical value in understanding circuit operation. [That's not as clear as I'd like. Comment welcome]. – Russell McMahon Apr 13 '23 at 11:31
  • @RussellMcMahon That's very interesting. It may be of no practical value when designing circuits, but if it grants a better understanding the behaviour of electricity or a convenient way to think about the behaviour then it's not worthless. I learn best by understanding the behaviour at a lower level, even if it's not directly useful at a higher level (and typically frustrate people trying to help me learn because they don't see why it matters to me whether it's technically the same behaviour at the lower level in two different situations). – Clonkex Apr 14 '23 at 01:57
  • "When I talk about Ohm's law, I'm talking about the relationship between voltage, resistance and current: given any two you can calculate the third." – This is *not* the Ohm's law. R=U/I is the *definition* of R. Ohm's law states that R for a certain object is *constant* (i.e. does not depend on U or I). Now we can ask if it always applies (it doesn't) and for all objects (no); but the core of the Ohm's law is R being constant, not its relationship to U and I. R=U/I is trivially true by definition. – Kamil Maciorowski Apr 14 '23 at 17:59

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No, Ohm’s law only applies when considering constant-value resistive elements in a lumped-element circuit model.

Maxwell’s equations apply always in all situations. But that requires vector calculus, and it’s unwieldy for normal use. Ohm’s law is a simpler model that assumes there is a linear (affine) relationship between voltage and current for resistors.

For practical engineering work, first-order analysis assumes resistors have constant resistance. Sometimes we have to account for the way this constant resistance changes over temperature, so it’s not really linear. But it’s close enough.

Diodes, PN junctions, transistors, saturated magnetic, etc, do not have a linear relationship between voltage and current, so we cannot use the simple linear Ohm’s law relationship to describe those elements. Diodes and transistors often require exponential equations instead of linear.

All of these are lumped-element models, where we assume ideal components connected by ideal wires. Again, this is done because the math is much simpler than trying to apply Maxwell’s equations to a complicated design. We can use lumped-element model as long as the signal bandwidth is not too high, so the wires and components are much smaller than the smallest wavelength we care about.

MarkU
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    Note that this is the opinion of 99% of professional engineers. There are some outliers who will argue, but this is certainly the definition that most people find most useful. – TimWescott Apr 12 '23 at 23:22
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    @TimWescott what is the source of your statistic? Also, do you think that the 99% of engineers would disagree about what law they apply to compute the current in a 12 V, 12W incandescent bulb? – Sredni Vashtar Apr 12 '23 at 23:37
  • Say at least 99%. Possibly 99.99%. But there's always somebody who will argue. For backup data, just do a search on this stack exchange for Ohm's law. – TimWescott Apr 12 '23 at 23:39
  • My assumption was that Ohm's law was simply a description of physical laws. I see that it's more a simplification only useful in specific situations. I know electricity is complex but just when I think I'm starting to understand how much I don't know I discover there's a whole new level of complexity just waiting to confuse me haha. – Clonkex Apr 12 '23 at 23:40
  • @TimWescott I can't say whether you're right or wrong, but I would like to see actual evidence for that sort of claim. What you've provided is anecdotal at best. – Clonkex Apr 12 '23 at 23:40
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    I should have mentioned, too, but this is the sensible and useful definition for Ohm's law. It isn't what Ohm wrote down, because when he was doing his investigation, it was not clear what the current versus voltage relationship was in a wire. We have just found it useful in linear circuit analysis to define an ideal resistor as being a perfectly linear component, so we name that usage after Ohm. – TimWescott Apr 12 '23 at 23:42
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    @Clonkex: here. https://electronics.stackexchange.com/questions/339172/general-definiton-of-ohms-law – TimWescott Apr 12 '23 at 23:46
  • @TimWescott but the most voted answer by TrevorG contradicts what is written in this answer; Maybe 99.9 is a bit exaggerated? – Sredni Vashtar Apr 13 '23 at 00:02
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    @MarkU: Ohm's law has nothing to do with lumped element circuit model. There is also a version of Ohm's law that is compatible with field based models: \$\vec{j} = \sigma \vec{E}\$, where \$\vec{j}\$ (current density) and \$\vec{E}\$ (electric field) are vectors of vector fields and \$\sigma\$ (conductivity) is a scalar of a scalar field. Note also that Maxwell's equations alone are not be sufficient to derive Ohm's law. It requires an additional theory about scattering of charge carriers (electrons) in a conductor (metals). – Curd Apr 13 '23 at 08:21
  • The very fact that we habitually discuss the resistance of a diode at some point on its VI curve makes a nonsense of this argument, for all practical purposes. Diodes do not "obey" Ohms law, yet we use the concept of resistance all the time to explain their behavior. Ohms Law doesn't even hold, as an absolute, for resistors, if we get into temperature or extreme voltages. You can't have it both ways. – danmcb Apr 13 '23 at 09:27
  • @danmcb The diode junction itself is non-ohmic, but the parasitic elements like lead resistance, bulk semiconductor resistance, and so on *are* ohmic, and that's what people are referring to when talking about the diode's ESR. – Hearth Apr 13 '23 at 12:56
  • @Hearth by "Ohmic" you simply mean that the resistance is (more or less) constant - the VI graph is a straight line, right? – danmcb Apr 13 '23 at 13:45
  • @danmcb Yes, that it can be approximated as an ideal resistor. – Hearth Apr 13 '23 at 14:35
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    OK, that's fine. But we have an issue of terminology here, because things which (according to that definition) are "non-ohmic" (e.g. a diode) are routinely described in terms of their "resistance" (at some point on their VI curve), and we use the unit of Ohms. Now I understand the distinction, but can you see how utterly confusing this is to someone learning the subject? Somewhere, according to this viewpoint, our terminology is broken. – danmcb Apr 14 '23 at 07:24
  • "Maxwell’s equations apply always in all situations." As a nitpick, it doesn't. Just like a resistor explodes at high voltages, Maxwell's equations break at high field strengths, when electromagnetism becomes non-linear, since photons begin to interact with each other. In a vacuum, this is called the [Schwinger limit](https://en.wikipedia.org/wiki/Schwinger_limit). It also happens at lower powers in media, a common example are green laser pointers which use a crystal to double the optical frequency. – user71659 Apr 15 '23 at 05:20
  • @TimWescott and others in this comment thread: yes, we have terminology issues when speaking about "Ohm's law" because what we call today "Ohm's law" not only is not necessarily what Mr.Ohm established in his works, but depends on context. Authors call Ohm's law what MarkU explained, others refer to the historical version (rarer), others speak about "AC Ohm's law", other about microscopic OL (in the context of Maxwell's equations). ... – LorenzoDonati4Ukraine-OnStrike Apr 15 '23 at 09:58
  • @TimWescott The point is, the term "Ohm's law" has become an umbrella term to identify a series of equations that relate "current/current density" with "voltage/potential/electric field" *in a **linear, time-invariant** fashion*. There are also authors that extend the concept of resistance to non-linear situations, just keeping the **time-invariance**, however they no longer use the term "Ohm's law" for that. In the end we would be better off to be very careful to specify the exact context when speaking of Ohm's low, lest we enter endless debates about who's right, which are moot in the end. – LorenzoDonati4Ukraine-OnStrike Apr 15 '23 at 09:58
  • Oh, I'm trying to stay out of the "who's right" debate -- just pointing out that if, in a work situation, you say "Ohm's law", you're safest if you mean ideal resistors. To your point, you're safer still if you say "ideal resistor". – TimWescott Apr 15 '23 at 14:36
  • And making any claims that we need to stick to what Dr. Ohm described is shaky. Maxwell never grouped all four of "Maxwell's Equations" and said "here it is". Newton's and Liebnitz's notations for calculus have evolved considerably since the 17th century, yet we still attribute \$x'(t)\$ to one and \$\frac{d}{dt}x(t)\$ to the other. Fourier's work on differential equations does have the classical Fourier transforms buried in it, somewhere, but not the way we state them now. So asking us to stick to Ohm's original formulation when we don't with anyone else is questionable. – TimWescott Apr 15 '23 at 14:41
  • @TimWescott Sorry if I'm misinterpreting your last two comments. They are not explicitly addressed to me, but they seem to be referring to my comments. If this is the case, maybe I didn't make myself clear. In my first comment I addressed you explicitly because you were the first to initiate this comment thread, but I also mentioned "others". My point was simply that Ohm's name is (in a sense) abused. Without at least a bit of context, talking about "Ohm's law" can be misleading and two people could end-up arguing against each other without realizing they are both right in their own context. – LorenzoDonati4Ukraine-OnStrike Apr 15 '23 at 22:29
  • @TimWescott BTW, I do know what we usually call the "Maxwell's equations" are actually due to Heaviside, who used vector calculus to simplify a lot the mess of scalar equations that appeared in the work of Maxwell. That's another case of historically inaccurate use of a name that however causes usually less headaches than that of Mr.Ohm. – LorenzoDonati4Ukraine-OnStrike Apr 15 '23 at 22:33
  • "Without at least a bit of context, talking about "Ohm's law" can be misleading and two people could end-up arguing against each other without realizing they are both right in their own context." That was actually the point I was trying to make. – TimWescott Apr 15 '23 at 23:24
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For instance, AC isn't some special form of electricity, it's just DC with a frequently-inverted voltage.

You would be better off thinking of DC as a special case of AC, where the rate of change of the signals goes to zero.

When I talk about Ohm's law, I'm talking about the relationship between voltage, resistance and current: given any two you can calculate the third.

If you choose to define the resistance as the ratio between voltage and current, then of course you can say that Ohm's law applies anywhere at any time.

But that isn't the "most tehcnically-correct" way to define the resistance, and it isn't the most useful for understanding how circuits work or predicting how new circuits you haven't built yet are going to work.

The most technically correct definition of Ohm's law is the one originally written by Ohm. It applies only to metallic conductors (wires, essentially), and it was an observation that (if the temperature of the wire is kept constant) the voltage increases proportionally with the current, or vice versa. Any statement about a device other than a metallic conductor isn't "technically" Ohm's law.

The more useful definition of Ohm's law is that it describes many types of devices (including, but not limited to, metallic conductors kept at constant temperature) that behave similarly: the voltage varies proportionally to the current (or vice versa). If you have such a device, then you can use Ohm's Law to predict the circuit's behavior, using methods such as nodal and mesh analysis.

On the other hand there are many devices that whose I-V characteristics aren't linear, and therefore can't be considered as following the more useful form of Ohm's law. In those cases we need to use more involved methods of predicting their behavior. But those methods might still involve modeling the device as a combination of an ideal (linear, "Ohmic") resistor and some other device such as a voltage source or capacitor, at each step of the analysis (which can be done so as to be equivalent to using Newton's method to solve the nonlinear equation describing a nonlinear circuit).

The Photon
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  • _"You would be better off thinking of DC as a special case of AC, where the rate of change of the signals goes to zero."_ Why? I don't think of AC as a "subset" or "special case" of DC; instead I think of DC as just steady-state electricity. AC is just electricity with constantly varying voltage. There's effects that only appear when changes occur and therefore some effects are most commonly observed with AC, but that doesn't mean DC or AC is more special than the other. (I'm uneducated and self-taught. Feel free to correct me, but also expect me to defend my thinking.) – Clonkex Apr 13 '23 at 00:49
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    @Clonkex, I can't read your mind, I can only respond to what you write in your question post, which is where you wrote that AC is "just DC with a frequently-inverted voltage." If I misunderstood, feel free to ignore that part of my answer and focus on the other part. – The Photon Apr 13 '23 at 01:23
  • @Clonkex Maybe http://www.cwladis.com/math100/Lecture4Sets.htm can visualize the Problem. Please have a look at the Figure under Nr. 1. See the "A" as electricity in general, and "B" as steady-state electricity. But please do not make the mistake and assume, that there is no effect present in "B" which is not altered by some other effect only present in "A". Or in Math terms: There is no bijection between A and B, rather a surjection from A to B. See https://en.wikipedia.org/wiki/Surjective_function – ElectronicsStudent Apr 13 '23 at 02:05
  • @ThePhoton My comment asked "why would I be better off thinking of DC as a special case of AC?" and then explained my current mindset. I'm a little confused by your response. However I'm starting to realise there are many different ways to think about electricity, and all of them can be correct depending on your viewpoint. Unfortunately it seems like my main question about Ohm's law was built on a flawed assumption of what the law represents, so I'm finding it hard to wrap my head around exactly why it doesn't apply. – Clonkex Apr 13 '23 at 02:16
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    @Clonkex, As for why it's better to look at it one way than the other: The methods used to solve AC circuits can be used to solve linear DC circuits but not vice versa. Similarly, the methods used to solve nonlinear circuits can be used to solve linear circuits but not vice versa. So if you understand AC circuits then you already understand DC circuits. But not vice versa. – The Photon Apr 13 '23 at 02:54
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    Even a "linear, ohmic resistor" is not, if you heat it, or apply too many volts. So the "for a resistor" definition of Ohm's Law is idealised. There are also many non-linear devices which are analysed with a V-I or V-R curve. Thus, the more realistic and useful view is that R (instantaneous) is simply the ratio of V and I at that time, is more useful and less confusing. Basically, R is a man-made concept, which is *sometime* "constant" within limits. Yes, Ohms Law applies if can measure V and I. R is only constant within limits for "Ohmic resistors". – danmcb Apr 13 '23 at 08:03
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    "If you choose to define the resistance as the ratio between voltage and current ..." that is exactly what we do. Many non-linear devices are characterised by a VI curve, and as soon as we starting discussing that it won't be long before someone talks about slope, which is resistance. Even L and C have reactance, also in Ohms, even though they don't obey "classic" Ohms Law - they do once we account for phase. It's time to stop confusing people like the OP by trying to use a too-narrow definition. – danmcb Apr 13 '23 at 09:17
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    @danmcb, First, I explicitly mentioned the issue of thermal dependence of resistors in my answer. Second, it's easier to teach someone to understand a simple model of a system and then expand that knowledge to include more subtle aspects of the system behavior than to teach them to understand everything about the system and then point out how it can be simplified in specific circumstances. – The Photon Apr 13 '23 at 15:13
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    @danmcb: Effective series resistance is the radio of dV/dI, which cannot be ascertained merely by looking at the instantaneous voltage and current. – supercat Apr 13 '23 at 15:38
  • Re the relationship between DC and AC: Think about, say, Newtonian mechanics for comparison. “A system in stationary equilibrium” and “a system in oscillating motion ” are both special cases of “a system in motion”. But the stationary case is *much more special*, and the mathematics of it is much more degenerate, since there’s no velocity/acceleration component. The maths for an oscillating system is shows much more of the general case. Similarly, DC really means “*constant* voltage”, and is a very degenerate special case because of that. – PLL Apr 15 '23 at 09:29
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It seems to me that you are looking at this backwards.

Given a certain voltage and current, the ratio of the two at any point in time (and I am not going to get bogged down in your "infinitely small" statement - you have to sample across a finite period in order to even quantify current) the ratio of the two is known as resistance, unit being ohms.

It so happens that some types of devices have (more or less) constant resistance (if we don't worry too much about the effects of temperature and so on).

Others don't.

In other words - resistance is a useful ratio, devised by engineers, to help us classify types of material or devices, and make numerical predictions about how they will behave.

So in any situation where you can measure a current and voltage across a defined period of time, you can define a resistance (at that time). Thus, Ohms Law applies - because we say it does.

danmcb
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Ohm's law does not apply to other materials and devices, including insulators, capacitors, inductors, switches, transistors, vacuum, voltage sources, current sources, dielectrics, semiconductors, and many others. All of these devices and materials violate Ohm's law

Source: https://physics.stackexchange.com/questions/562633/why-ohm-s-law-doesn-t-work-in-these-scenarios-inside-ideal-battery-and-in-vacuu

So that's usually the case where ohms law doesn't apply, in those materials, there is no simple relationship. Ohmic losses are usually from random interactions in the material, a crude way of thinking about is that the electrons are bumping into atoms and imparting energy which is measured as heat.

Materials with ordered electronic states have nonlinear profiles and ohms law doesn't work to approximate their V-I curve.

For instance, AC isn't some special form of electricity, it's just DC with a frequently-inverted voltage.

Actually DC is AC with no frequency according to Fourier. DC would be a sine wave with infinite wavelength

Voltage Spike
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    _"Actually DC is AC with no frequency according to Fourier. DC would be a sine wave with infinite wavelength"_ That sounds like a way to describe DC mathematically. I like this, but with that definition, AC and DC are the same. You just see more or less of the AC-like effects depending on the rate, frequency and amplitude of change. I guess I just struggle with people trying to use clearly-defined boxes to say "this is AC and this is DC and they are different things" when it's actually an infinitely sliding scale. – Clonkex Apr 13 '23 at 01:14
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    @Clonkex Hmmmmh yes and no. Frequency can not be infinite, as this would mean wavelength is zero. So the infinite slider, from a frequency perspective, becomes finite. The defintion of DC is: No changes over time. So frequency = 0. Therefore, there are "two boxes". The one box beeing every electrical effect with f not 0 (AC) and every effect with f = 0 (DC). The difference beeing: Are electro/magneto- DYNAMIC effects at play or not. Sure, you can say: AC is DC with these effects inseatd of saying DC is AC without these effects. But: There are many effects at play which favor option 2. – ElectronicsStudent Apr 13 '23 at 02:13
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    @ElectronicsStudent I guess I was imagining representing the wave as length and amplitude. From that perspective, there's nothing stopping length being infinite and thus frequency being 0. It probably stops representing a wave, but that's accurate because there is no longer a wave. I guess you could also lower the amplitude to 0 for the same effect, except then the frequency no longer has to be 0 and it could probably still be considered a wave. So imagine that inductance is the slope of the wave at some point. If the wave is flat, the slope is flat and so there is no inductance. – Clonkex Apr 13 '23 at 02:28
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Hmmmh, you have some misconception here.

AC is not just DC with a frequently-inverted voltage .

The deeper you go (Waves, Signal-Theory and so on) the more they differ!

In fact, DC is a special form of AC and not the other way around. Many conceptions do not apply for DC, which have to be taken into account with AC.

To make it all worse, DC-Signals like DC Square-Waves are a fascinating topics of AC-Analysis.

So with that in mind, does Ohm's law always apply in all situations so long as you sample an infinitely narrow point in time?

No. Not at all. Maxwell Equations apply. DC Ohms Law is a special case (strongly defined) of these equations

ElectronicsStudent
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    I find it hard to believe that AC and DC could be considered so different. That's like saying a stationary wheel is a special case of a turning one. Yes, the turning wheel exhibits effects not visible on the stationary wheel (vibration, resonance, maybe the tyre expands if it's turning fast enough) but that's because it's turning. It's still a wheel, but depending on the rate of rotation you see more or less of the effects. Electricity is still electricity, but depending on the rate of change you see more or less of the effects commonly associated with AC. (I'm uneducated, please correct me.) – Clonkex Apr 13 '23 at 00:55
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    @Clonkex If you consider electricity beeing electrons flowing through conductors - then yes! i can see your point. As soon as you start seeing electricity as fields with gradients in respect to time and position in space, one dimension (the dimension of time) is dropped for pure DC considerations. So DC is a much more well constraint case of AC, as less effects can take place. – ElectronicsStudent Apr 13 '23 at 01:30
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    @Clonkex Sure, this is a purely theoretical approach. Think about electricity beeing a form of "light" focused through lenses (cables) made of conductive materials. With light beeing a wave and DC having infinite wavelength, DC makes no sense at all. It is some sort of: Actually its AC, but we make our lives easy (Math and so on) and introduce DC for this. – ElectronicsStudent Apr 13 '23 at 01:30
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    @Clonkex Also, it speaks alot about your mind to state on the internet, that you are "uneducated" (In your words, i dont think they apply here!). I like it. – ElectronicsStudent Apr 13 '23 at 01:33
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    Ah, I get your point. If they are the same (DC is just AC with infinite wavelength), then everything is AC and DC is just a special form of AC. I hated maths in school and never went beyond algebra, trig and some extremely basic calculus, so I have no idea what "gradients in respect to time and position" would be haha. Maybe one day I'll do a maths course to catch up since I actually love maths these days and rely on it as a computer programmer. – Clonkex Apr 13 '23 at 02:07
  • @Clonkex See https://www.wolfram.com/mathematica/new-in-8/new-and-improved-scientific-and-information-visualization/show-the-gradient-field-on-a-surface.html The arrows on the surface are "gradients". The bigger the change of "field-intensity" between two points, the bigger the gradient is. It is the same in 2D https://vitalflux.com/gradient-descent-explained-simply-with-examples/ – ElectronicsStudent Apr 13 '23 at 02:18
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Others have already spoken about Ohm's law with respect to different devices, so I'll address an aspect of Maxwell's equations and AC vs DC.

Something that I did not appreciate for a very long time is that the seemingly basic concept of voltage only really makes sense when your magnetic field is not changing (i.e., when you are dealing with magnetostatics). This is because Faraday's law of induction implies that Kirchoff's loop rule does not need to hold when there is a changing magnetic field. If you have alternating currents, then Ampère's circuital law implies you have changing magnetic fields.

Looking at instantaneous points in time doesn't save you here: when there are alternating currents, there is simply no such thing as "the" voltage difference between two points in a circuit anymore -- there can be current loops! At best, you can hope that your currents are alternating slowly enough that magnetostatics gives good enough of an approximation to what's going on -- that is, you hope your current loops have negligible net current.

For an ideal resistor, it still makes sense to talk about "the" voltage across it despite these complications. However, for physical resistors, for example those with a spiral of conductive material around a ceramic insulator, at high enough frequencies you are going to have to face the fact that they have non-negligible inductance since they internally have a time-varying magnetic field. Ohm's law does not describe the behavior of inductors.

I'll mention that there are some tricks where you take advantage of a mathematical property called linearity and, rather than resistance, you consider a frequency-dependent complex-valued quantity called impedance. You can analyze what a resistor does to each frequency component of a signal independently (how much does it scale it and how much does it time shift it), and then to see what a resistor will do to a particular signal you bring in Fourier transforms. There are linear relationships in the analysis, but I think it's safe to say we're outside the realm of Ohm's law.

Kyle Miller
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    In a circuit containing only linear elements driven by a periodic sinusoidal waveform, magnetic and electric field effects exhibit themselves in the form of inductance and capacitance. If the voltage is viewed as a complex number, with the instantaneous portion being the real component and the 90-degrees-out-of-phase portion being the imaginary component, and if passive linear elements are viewed as having complex impedance, ohms law will work when applied to any individual frequency. For waveforms containing the sum of multiple frequency components, ohm's law may be applied... – supercat Apr 13 '23 at 15:34
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    ...to individual components and the results summed together. I don't know that Ohm learned in his lifetime that his Law could be applied this way, but it does work for any device that can be modeled as a combination of ideal resistors, capacitors, and inductors. – supercat Apr 13 '23 at 15:35
  • @supercat Aw this is where the gaps in my knowledge of maths start to show through. I can google the definition of complex numbers but I'd have to work with them for a while to really grok/understand them. I have deep understanding of some things but giant gaps for others. I guess I should find a good maths course to fill in some of the gaps and then come back. – Clonkex Apr 13 '23 at 22:25
  • @Clonkex: At any particular frequency, perfect inductors and capacitors both behave with respect to Ohm's Law as though their resistance is an imaginary number (inductors and capacitors have opposite signs). If one has an inductor in series with a sinusoidal voltage source with an entirely-real component, the current will be an imaginary value, indicating that its phase lags the voltage by 90 degrees. A capacitor would have the opposite sign, and yield a current with the opposite sign, indicating that its phase leads the voltage by 90 degrees. – supercat Apr 13 '23 at 22:28
  • @Clonkex: What's amazing is that if one apply's Ohm's Law to any network of series and parallel resistors, capacitors, and inductors, using the rules for multiplying and dividing complex numbers, everything will "work" to model the combined effects of resistors, capacitors, and inductors at any particular frequency. – supercat Apr 13 '23 at 22:31
  • @Clonkex The high-level idea is what I mention in the last paragraph. A given frequency component gets amplified/attenuated by some factor and gets phase shifted; complex number arithmetic happens to be perfectly suited to carry out this calculation. It's a bit of a stretch to say Ohm's law is applying here exactly, because we're working with amplitudes and phases, not instantaneous voltages. Still, the formulas look very similar, even the series/parallel resistor network analysis ones. – Kyle Miller Apr 14 '23 at 02:18
  • If you're interested, some keywords are signals and systems, Laplace transforms, transfer functions, impedance, passive filters. There's a differential equation that describes the behavior of a circuit, and these are tools engineers and physicists use to efficiently solve them. – Kyle Miller Apr 14 '23 at 02:21
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Generally speaking simple linear models of resistance like Ohm's law are making the implicit assumption that the power dissipated by the circuit element (and thus its voltage) depends only on the instantaneous current, and not its history (i.e. time derivatives of current). However, all materials ultimately are composed of microscopic particles, and their resistance is caused by scattering due to these particles.

Therefore, such a model will only hold if the time resolution of our measurements is low compared to the relaxation times of whatever is scattering our current and causing the dissipations. If we measured instead across a time scale this small, we would observe small fluctuations around the Ohm's law voltage, due to the fact that the energy loss and gain of individual scatterers becomes resolvable in such short time scales.

So to summarise, even for an Ohmic material, Ohm's law itself is describing the average of a stochastic process caused by some underlying microscopic process, and at sufficiently high time resolution one can measure the fluctuations of this process directly, and see small random deviations around the Ohm's law prediction.

AwkwardWhale
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If you consider complex values of U and I and impedance instead of resistance, you'll start getting close. Considering real values, no. A coil or capacitor in an AC circuit puts the voltage and current out of phase with one another. Basically, the instantaneous current is proportional to the instantaneous voltage at a different point in time. The moments when the voltage is zero, the current can be close to maximum in alternating directions.

Divizna
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  • Just a nitpick: since this site language is English and in almost all English documentation about electronics voltage is indicated with V and not U, I suggest you to change that symbol, otherwise your answer could be misinterpreted (U may be seen as potential energy, for example), especially when talking to a newbie. And yes, I know there are countries where U is used to indicate voltage (e.g. Germany). – LorenzoDonati4Ukraine-OnStrike Apr 15 '23 at 22:41