Ohm’s law holding true on temperature-dependent resistances

Question

As far as I know, Ohm's law (in macroscopic form) states that the in some devices/conductors/materials, the (instantaneous) current through the device is directly proportional to the (instantaneous) voltage across the device. It can be proven this occurs only if the (static/DC) resistance of the device is constant; if it is variable, then the voltage and current aren't directly proportional, so Ohm's law is not satisfied.

In the above, I haven’t discussed about temperature. In metallic conductors, as we know their resistivity is dependent on the temperature of the conductor, which depends on the ambient temperature and the current flowing through the conductor (Joule’s law). Since the resistance depends on the resistivity, it follows the resistance of a metallic conductor depends on the current through it. Doesn’t that make metallic conductors not obey Ohm’s law (since the voltage is not directly proportional to current)? Or am I wrong (if so, how)?

(This question was originally longer, but people suggested to shorten it, so I did it. The content of the original post is in this Quora answer.)

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I read the following questions and corresponding answers, but they don’t address or answer my question:

• Ohm's law holds at constant temperature - what about Joule heating?: The question does address mine. The question and answer by Bill N say that temperature must be held constant in order for Ohm’s law to be satisfied. Is this true? I mean, if we change the current in a metallic conductor, the truth is its temperature will change and so its resistance will also.
• Do metals at low temperature follow Ohm's law?: The asker is asking regarding very low temperatures.
• Ohm's law in metals: The question does address mine. The answer by KF Gauss says that if temperature depends on resistance, it doesn’t contradict Ohm’s law. I think it does, as I explained above. I commented their answer but didn’t get a reply.
• Can Ohm's law break in metals?: The answers don’t seem to have addressed my question.
• Non-ohmic conductors: The answers don’t seem to have addressed my question.
• Ohm's law hold till what temp?: The asker is asking regarding very low temperatures.
• Ohm's law and Joules heating: The answer by BioPhysicist does address my question, and seems to be in agreement with what I said. Can anyone confirm if we’re correct?

Answer 1

There seem to be at least two valid ways in which physicists and engineers use the term Ohm's law, neither of which is merely a definition of resistance.

(a) If $I$ is proportional to $V$ then the conductor obeys Ohm's law, otherwise it doesn't.

(b) At constant temperature (and, strictly, at constant pressure) metallic conductors and most single-substance conductors exhibit $I$ proportional to $V$.

Clearly Ohm's law as defined in (a) is obeyed only by a narrow class of conductors. But (b) attempts to be a self-contained law of nature with few exceptions.

Your thermistor doesn't obey Ohm's law according to (a), for the reasons that you state.

On the other hand, (b) has nothing to say on whether or not a thermistor, self-heating or externally heated, obeys Ohm's law. This is because (b) specifically doesn't deal with conductors that aren't at constant temperature!

My own preference is for (a), but as nasu implies (in his comment on the question) the matter is of no great moment: you can usually work out from the context what someone means by Ohm's law.

Let the down-votes descend!

Answer 2

The recovering ex-engineer weighs in.

Note that for most common engineering metals, the temperature coefficient of resistance is small. This means that for temperatures not far from ambient, the shift in resistance caused by ohmic heating is small enough to ignore, and so engineers (even recovering ex-engineers) will indeed ignore it.

This means that engineers will use Ohm's law as a law and not just as a definition of resistance. A further reason for this practice is as follows:

In the field of dynamic systems modeling, an Ohmic (i.e., linear) dependence of a flow variable on a resistive element furnishes a satisfactory accounting for the behavior of electrical, mechanical, hydraulic and thermodynamic systems subject to small displacements- certainly good enough for first-order models to yield sufficiently accurate predictions (two-place accuracy) of system behavior for those models to be useful.

In general and in my experience, three-place accuracy requires accounting for second-order effects- and the dynamic systems model of a temperature-dependent resistance will replace a single-valued function with a value pulled from a lookup table that contains the (precalculated) second-order dynamics.

Even this is insufficient to prevent engineers and recovering ex-engineers from considering ohmic behavior as a law and not just a definition of resistance.