Does Ohm's law always apply at any instantaneous point in time?

Question

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).

Answer 1

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.

Answer 2

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).

Answer 3

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.

Answer 4

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

Answer 5

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

Answer 6

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.

Answer 7

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.

Answer 8

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.