# Why Ohm’s Law Is Not a Law

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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.