# Ohm’s Law Mellow

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At first I wanted this title to say “Ohm’s law is not a Law.”  But someone else used that phrase in a recent PF thread, and a storm of protest followed.   We are talking about the relationship between Voltage between two points in a circuit and the current between those same two points.

##R=V/I##, or ##V=IR##, or ##I=V/R##

I won’t explicitly talk about inductance or capacitance, or AC impedances, although all of those could substitute for ##R## in the following discussion.

### Semantics

Newton’s Second Law can be derived from fundamental symmetries in nature.  But Ohm’s law is not derived, it was empirically discovered by Georg Ohm.  He found materials that have a linear relationship of voltage to current within a specified range are called “ohmic.”  Many useful materials and useful electric devices are non-ohmic.

Beginners often believe that there is great significance in the use of words like “law” or “theory” In physics.  Not so.  Often it is just an accident of history or a phrase that slides easily off the tongue.  It could have been called Ohm’s rule or Ohm’s observation, but it wasn’t.

### Underlying Assumptions and Limitations

Really there is only one assumption behind Ohm’s law; linearity.   Not in the mathematical sense, but rather that a graph of voltage versus current shows an approximately straight line in a given range.

There are always limits to the range even though they may not be explicitly mentioned.  For example, at high voltages breakdown and arcing can occur.  At high currents, things tend to melt.    In the old days, we said that real-world resistance can not be zero. But now we know that superconductors are an exception to that rule.  ##R=0## is OK for superconductors.

Students often forget that limits exist.   A frequent (and annoying) student question is, “So if ##I=V/R##, what happens when ##R=0##.  Ha ha, LOL.”  They think that disproves the “law” and thus diminishes the credibility of science in general.  Their logic is false.

### Ohm’s law Has Several Forms

##I=V/R## may be the most familiar form of Ohm’s law, but it is far from the only one.

• In AC analysis, we use ##\bar V=\bar I\bar Z##, where ##\bar V##, ##\bar I## and ##\bar Z## are all complex vectors. ##\bar Z## is the complex impedance that can describe combinations of resistance, inductance and capacitance, without differential equations.  See PF Insights AC Power Analysis, Part 1 Basics.
• ##\overrightarrow E=\rho \overrightarrow J## is the continuous form of Ohm’s law. where ##\overrightarrow E## is the electric field vector with units of volts per meter (analogous to ##V## of Ohm’s law which has units of volts), ##\overrightarrow J## is the current density vector with units of amperes per unit area (analogous to ##I## of Ohm’s law which has units of amperes), and  ##\rho## is the resistivity with units of ohm·meters.   Use this form to model currents in 3-dimensional space.
• If an external ##B##-field is present and the conductor is not at rest but moving at velocity ##v##, then we use: ##(\overrightarrow E + v \times \overrightarrow B)=\rho \overrightarrow J##
• There are also variants of Ohm’s law for 2 D sheets, for magnetic circuits (Hopkinson’s Law), Frick’s Law for diffusion dominated cases, and for semiconductors.

Someone on PF once said that one form was the only “true” Ohm’s law.  I disagree.  All the forms are useful in different contexts.   We can honor Georg Ohm by using his name to cover all the variants, even if he didn’t personally invent all of them.

### Real Life Example, A Solar Panel

A solar panel is a real life device.  Below, we see a family of curves showing the Voltage V versus current I relationship of a solar panel.  Each curve represents a different value of solar intensity.  As you can see, the curves range from nearly an ideal voltage source (vertical), to a nearly ideal current source (horizontal), to everything in between.

We can draw straight line segments to define the average resistance over a particular range, like R1, R2 or R3.  As you can see, the panel ranges from nearly an ideal voltage source, to a nearly ideal current source, to everything in between.  R4 shows a resistance defined by the tangent to the curve at one particular point.  I call R4 the resistance linearized about a point on the curve.

We could use these resistances, plus a Thevanin’s Equivalent voltage ##V_{thev}##  to model the solar panel in a circuit to be solved using Ohm’s law. (##V_{thev}##  is  the place where the straight line intercepts the ##V## axis or where ##I=0##).

### Arbitrary Numbers of Arbitrary Electric Devices

When we go beyond a simple resistor made of some material with a linear V-I relationship, we find that very many real-world devices are non-linear.  For example, the constant power device (yellow) and the tunnel diode device (blue) curves seen below.   There are no physical constraints on the shape of a V-I relationship other than that both V and I must be finite.  Even a multi-valued curve that loops back on itself violates no physical law.

Students in elementary circuits courses often learn 0nly about the constant R (i.e. the resistor) element, plus L and C.  Other kinds of nonlinear circuit devices may not even be mentioned.

At first glance, you might say that Ohm’s law doesn’t apply to nonlinear devices, but that’s not true. Suppose we had a circuit containing a solar panel, a constant power and a tunnel diode.  A student using paper and pencil could not be asked to solve such a circuit, so courses that limit themselves to paper and pencil methods do not cover nonlinear elements.  But using computers, there is a relatively simple method:

1. Guess an initial V and I point along the curve for each device.
2. Find the linearized resistance (analogous to R4) for each device at the V and I guess point.
3. Solve the linearized circuit using Ohm’s law, calculating new values for each V and I.
4. Use the calculated V and I as the new guess and return to step 2.

An iteration like that is very easy to perform with a computer.  It isn’t guaranteed to succeed, but when it does succeed, after several passes through steps 2 and 3, it will calculate values for V and I that simultaneously satisfy the relationships of all linear and nonlinear elements in the circuit.    It is routine in power grid analysis to solve circuits with a million or more diverse nonlinear elements.  Thus, even when modeling devices that don’t seem to obey Ohm’s law, that we can make productive use of Ohm’s law nevertheless.  I’ll say it again using stronger words. Ohm’s law cannot be violated in real life; rather it can be adapted to nearly all real life situations in electric circuits.

Ohm’s law is not always a complete description of electrical devices, but Ohm’s law is almost always a useful tool.   Students are advised to learn to think that way.  Physicists and engineers seek usefulness and try to leave truth to philosophers.

### Electricity Study Levels

One can study and explain electricity at (at least) 5 levels.

In this article, my focus was circuit analysis.   One of the standard assumptions for circuit analysis is that Kirchoffs laws apply instantaneously to the entire circuit.  That makes ##V##  ##I## and ##R## co-equal partners.  None of the three can be said to be cause and the other effect.  All three apply simultaneously.

Students dissatisfied with Circuit Analysis sometimes yearn for physical explanations and invent false narratives about what happens first and what comes next.  If you are such a student, I urge you to study Circuit Analysis first, then follow-up with the other levels.  If that describes you, I advise that the next step is to abandon circuit analysis, Ohm’s law, Kirchoff’s laws, and to learn Maxwell’s equation as the next deeper step.  Fields, not electron motion are the key to the next deeper step.

Thanks to PF regular @Jim Hardy for his assistance.

41 replies
1. Handy Andy says:

Ohms law is based on an ideal world for students. All conductors have a complex impedance based on frequency, temperature, resistance, inductance and capacitance. These vary depending on how cables are run adjacent to each other, even climatic conditions can come into play etc. It isnt an ideal world, even a straight piece of wire has typically 10mH per metre. The complex impedances in a conductor become significant when switching very high currents quickly.

2. Mister T says:

The relation ##R=frac{V}{I}## is not Ohm's Law. Rather, it's the definition of ##R##.

Ohm's Law is the assertion that over a range of voltages, ##R## is constant.

Like all laws, it has limits of validity. There is no such thing as a law with universal limits of validity. Hooke's Law is an example of a law that can be compared to Ohm's Law when teaching this concept of limited validity.

3. Dilema says:

The article "Ohm's law Mellow" address a very impotent issue, Yet there is no evidence  for violation of  Ohm's law (within the limit of ohm's law). I think the confusion comes from the fact that  text book  do not elaborates on the preliminary assumptions under which Ohm's law valid.

4. Dilema says:

anorlunda submitted a new PF Insights post

Ohm's Law Mellow

Continue reading the Original PF Insights Post.

I disagree to the general Idea that Ohm's Law is not a law, yet I strongly support the importance of this issue which I think boils down to the very basics philosophy of what is a physical law, what are the therms under which it consider violated.

Following the article spirit, I would recommend  title like: Ohm's hypothesis or even better – Ohm's convention.

I understood the solar device is given in order to provide real example for violation of ohm's law. But it is not. the rectification profiles comes from a diode like element in the equivalent circulate. Up to date there is no system that violates "Ohm's law" . Note for example that Memristor, Thermistor, Warburg element or any constant phase (CPE) or non-CPE do not violates Ohm's law.

5. anorlunda says:

There are no rigid rules in science about the use of words like law, rule, theory, etc, so we are free to disagree

I view the definition of R as the ratio of V and I.  As a definition, it can't be violated by definition (pun intended :wink:)

Newton's Laws and the conservation laws are all derivable from the fundamental symmetries of nature.  I resist the idea of making Ohms Law comparable with those. But if you want to call it a law, go ahead and knock yourself out. But don't complain if I prefer different words.

6. vanhees71 says:

Ohm's Law is derived from many-body theory. It's defining a typical transport coefficient in the sense of linear-response theory. It's the "answer" of the medium to applying an electromagnetic field, and defines the electric conductivity in terms of the induced current, ##vec{j}=sigma vec{E}##, where in general ##sigma## is a tensor and depends on the frequency of the applied field. So Ohm's Law is a derived law and has its limit of validity (particularly the strength of the electromagnetic field must not be too large in order to stay in the regime of linear-response theory).

7. sophiecentaur says:

I view the definition of R as the ratio of V and I. As a definition, it can't be violated by definition (pun intended :wink:)

I wholeheartedly agree. It's a formula / definition and says nothing about whether or not Ohm's law happens to apply to what's connected to the terminals on the 'black box' we're examining. R could change, or not as V,I or T changes. If it doesn't happen to change then the component is not following Ohm's Law. But one calculation wouldn't tell you one way or the other.

This puts me in mind of the SUVAT equations with which we learned to calculate motion under constant acceleration. We don't refer to them them as 'Laws' of motion and we wouldn't dream of suggesting that a measurement of the change in velocity of an object in a given time would be the same under all conditions. But somehow, R=V/I is referred to as Ohm's Law. Teachers and lecturers can be very sloppy about these things. Aamof, I don't remember the constant acceleration thing being emphasised to me in SUVAT learning days, either.  They just drew a V/t triangle and did some calculations. It left me uneasy for quite a long while. But teenagers feel 'uneasy' about a lot of things so it was actually the last of my worries.

8. anorlunda says:

Ohm's Law is derived from many-body theory. It's defining a typical transport coefficient in the sense of linear-response theory. It's the "answer" of the medium to applying an electromagnetic field, and defines the electric conductivity in terms of the induced current, ##vec{j}=sigma vec{E}##, where in general ##sigma## is a tensor and depends on the frequency of the applied field. So Ohm's Law is a derived law and has its limit of validity (particularly the strength of the electromagnetic field must not be too large in order to stay in the regime of linear-response theory).

That's true for the specialized case of a linear and uniform mediums.  The article addresses the general case of circuits containing any components, linear/nonlinear, active/passive.  As the article says, you can always linearize about a point, define R=V/I, then use linear circuit methods to solve it.

9. sophiecentaur says:

Ohm's Law is derived from many-body theory. It's defining a typical transport coefficient in the sense of linear-response theory. It's the "answer" of the medium to applying an electromagnetic field, and defines the electric conductivity in terms of the induced current, ##vec{j}=sigma vec{E}##, where in general ##sigma## is a tensor and depends on the frequency of the applied field. So Ohm's Law is a derived law and has its limit of validity (particularly the strength of the electromagnetic field must not be too large in order to stay in the regime of linear-response theory).

That's ramping it up a bit for a number of the audience, I think.  But also, if σ changes with some other variable, the relationship breaks down so any 'Law' has hit the rails. A Law that's worth its salt will involve all the relevant variables – Ohm's law, when stated fully, fits that requirement.

10. vanhees71 says:

That's true for the specialized case of a linear and uniform mediums.  The article addresses the general case of circuits containing any components, linear/nonlinear, active/passive.  As the article says, you can always linearize about a point, define R=V/I, then use linear circuit methods to solve it.

It's true for any medium in the linear-response regime. More completely written out the relation reads

$$tilde{vec{j}}(omega,vec{k})=hat{sigma}(omega,vec{k}) tilde{vec{E}}(omega,vec{k}),$$

where we have Fourier-transformed fields in the frequency-wave-number domain, and ##hat{sigma}## is a complex-valued symmetric 2nd-rank tensor obeying the analytic structure in the complex ##omega## plane such that it is a retarded propagator.

If you have "active" elements and non-linearities, you have to extend the approximation beyond the linear-response level, as far as I know.

11. vanhees71 says:

That's ramping it up a bit for a number of the audience, I think.  But also, if σ changes with some other variable, the relationship breaks down so any 'Law' has hit the rails. A Law that's worth its salt will involve all the relevant variables – Ohm's law, when stated fully, fits that requirement.

I don't know, what you mean. Electric conductivity is a typical transport coefficient, describing the response of the medium to a small perturbation around equilibrium (in this case by a weak electromagnetic field). It's restricted to weak fields in order to stay in the linear-response regime. Of course, it has a range of validity, as has any physical law (except the ones we call "fundamental", because we don't know the validity ranges yet ;-)).

12. sophiecentaur says:

I don't know, what you mean. Electric conductivity is a typical transport coefficient, describing the response of the medium to a small perturbation around equilibrium (in this case by a weak electromagnetic field). It's restricted to weak fields in order to stay in the linear-response regime. Of course, it has a range of validity, as has any physical law (except the ones we call "fundamental", because we don't know the validity ranges yet ;-)).

I just meant that your wording and representation takes it to a higher level of understanding and familiarity. Of course the equation is correct – but it doesn't pretend to be a Law. By the time one gets to the level that you are using to describe what happens, I doubt that one would bring in the term Law.

But I guess this will never lie down as it falls within the overlap between higher level Physics and down to Earth practicalities; the two have different agendas.

13. sophiecentaur says:
David Lewis

Yes, if it's an applied voltage. A voltage drop (symbol V) occurs when current passes through the resistor.
Conversely, with a voltage source, EMF (symbol E) produces the current that passes through the resistor.

Isn't that just a chicken and egg argument for describing a 'relationship' between two variables?

14. cabraham says:
LvW

…and we can think about the meaning of the form: V=I*R.
We are using this form to find the "voltage drop" caused by a current I that goes through the resistor R.
However, is it – physically spoken – correct to say that the current I is producing a voltage V across the resistor R ?
(Because an electrical field within the resistive body is a precondition for a current I, is it not?)

Yes it is physically correct to say that current I produces voltage V in a resistance. It is also correct that a voltage V places across resistance results in current I.
Either one can give rise to the other.
No, an electric field across the resistance is not necessary for current to commence. A switch is closed, a battery has an E field due to redox chemical reaction. Charges move through the cables towards the resistor. Current is already commenced by battery redox. When the charges reach the resistance, they continue into the body but incur collusions between electrons & lattice ions. This results in e lectrons droppii g from conduction band down to valence band. Polarization occurs with photon emission. When current is in a resistance it gets warm from this energy conversion. The E field across the resistor happens when charges emitted from the battery arrive. Positive battery terminal attracts electrons from cable. An electron vacating its parent atom leaves a positive ion behind or hole if you prefer. The atom next in line emits an electron towards this hole. Reverse happens at negative battery terminal. The charges & the associated E field arrive at the resistor. Current already is established, as the charges are in motion before the resistor receives them. Charges proceed through the resistor colliding with lattice ions resulting in polarization & photon emission. Polarized charges have an E field, & the line integral of said E field over the distance is the voltage drop.
At equilibrium the equation J = sigma*E, or E = rho*J, which is Ohm's law in 3 dimensions. I will elaborate if needed.

Claude

15. cabraham says:
sophiecentaur

Isn't that just a chicken and egg argument for describing a 'relationship' between two variables?

Yesz it is. I & V generally h ave a circular relation. Either can come first & produce the other.

16. sophiecentaur says:
cabraham

Yesz it is. I & V generally h ave a circular relation. Either can come first & produce the other.

We are so used to using Batteries, which are essentially Voltage Sources, that it is hard to avoid think of Voltage as the senior member of the VI pair.

17. cabraham says:
sophiecentaur

We are so used to using Batteries, which are essentially Voltage Sources, that it is hard to avoid think of Voltage as the senior member of the VI pair.

I would agree. One thing worth noting is that a battery can be produced for constant current as well. A short across the terminals results in the off state, or no load. But losses would be greater than an open voltage source. So primary cells have been built for constant voltage operation for over a century.
Nuclear fission batteries have been produced, searching for them using "nucell" will give details. These nuclear cells are not only current sources, but a.c. instead dc. An a.c. current source battery, that is different. Apparently nuclear cells function better as a.c. current sources. They use fissionable material, so I won't hold my breath waiting for them to be available to the general public.

Claude

18. anorlunda says:
sophiecentaur

We are so used to using Batteries, which are essentially Voltage Sources, that it is hard to avoid think of Voltage as the senior member of the VI pair.

Try thinking of a superconducting loop with a magnetically induced current. No voltage in the loop before or after the current is induced.

We have disagreed before on the topic of teaching electricity. I think that we should stick to the 3 valid levels, QED, Maxwells, and Circuit Analysis (CA) [including Kirchoff's laws]. One of the key assumptions of CA is

• The time scales of interest in CA are much larger than the end-to-end propagation delay of electromagnetic waves in the conductors. [A simple rule of thumb, for 60 hertz AC circuits should have lengths of <500 km.]

For Voltage to start before Current, explicitly violates this assumption. That's OK in Maxwell's equations, but we should not mention it within the context of using CA. That is not helping students of CA, it is feeding them contradictory and confusing information.

19. anorlunda says:
vanhees71

I don't know, what you mean. Electric conductivity is a typical transport coefficient, describing the response of the medium to a small perturbation around equilibrium (in this case by a weak electromagnetic field). It's restricted to weak fields in order to stay in the linear-response regime. Of course, it has a range of validity, as has any physical law (except the ones we call "fundamental", because we don't know the validity ranges yet ;-)).

I think you ignored the following paragraph from the article that inspired me to write the article in the first place.

Students often forget that limits exist. A frequent (and annoying) student question is, “So if I=V/R, what happens when R=0. Ha ha, LOL.” They think that disproves the “law” and thus diminishes the credibility of science in general. Their logic is false.

Students wouldn't ask that dumb question if they understood that Ohm's Law only applies to a limited region. I don't believe that their teachers understand that. I suspect that the teacher's teachers don't understand that. In basic electricity Ohm's Law is being taught as absolutely true as if it had a foundation like the principle of least action underlying Newton's Laws of Motion. Somehow, the message that limited ranges are obvious is not getting passed down the ladder. Perhaps is is related to the fact that conduction in bulk materials requires quantum effects to accurately describe and that is just too difficult for most students and most teachers. That is what this article tried to address.

Even grad students and profs could stand a reminder and a moment of reflection on the fact that there is not physical principle that says that there has to be any wide region where voltage and current are linearly proportional. It could have been nonlinear all the way. If that were true, then simple algebra could not have been used to analyze simple circuits, and the evolution of electricity, electronics and computers in the 20th century would have taken significantly longer. If computers had been delayed, so would all of science. Therefore, IMO we should all thank our lucky stars for the accident that Ohm's Law is useful at all.