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Say I had a component with a resistance of 100 Ω, and I attached a 12 V power supply across its terminals using:

  1. a thin wire
  2. a thick wire

Would the component in both cases really draw (12 V/100 Ω) A?

I would imagine the cross sectional area of width x for the two circuits would contain different amounts of electrons, so you would expect more current drawn using the thicker wire.

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    \$\begingroup\$ Is Ohm's law really accurate? <-- yes it is. \$\endgroup\$
    – Andy aka
    Jan 8, 2023 at 10:28
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    \$\begingroup\$ @20MikeMike If Ohms Law would not be accurate, then IMO you wouldn't have a functioning computer to be able to place that question here. \$\endgroup\$
    – datenheim
    Jan 8, 2023 at 17:31
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    \$\begingroup\$ Ohm's law is accurate when one accounts for the total resistance in a circuit branch. The question about thickness of wire relates the resistivity of its material to its resistance in the circuit branch as shown here: phet.colorado.edu/sims/html/resistance-in-a-wire/latest/…. \$\endgroup\$ Jan 8, 2023 at 20:51
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    \$\begingroup\$ In short, why do you believe that number of free electrons has anything to do with the current induced by a voltage, any more than number of available water molecules has anything to do with water currents induced by a pump? Can you explain? \$\endgroup\$ Jan 9, 2023 at 2:35
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    \$\begingroup\$ Ohm's law applies to ideal components, which don't have any distance-like properties such as thickness. The question is akin to having doubts in geometry because its laws appear to depend on the kind of pen you're using for drawing. \$\endgroup\$ Jan 9, 2023 at 9:23

9 Answers 9

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Ohm's Law says that the current through a resistance is the voltage across it divided by the resistance.

The key point is that it is the voltage across the resistance, not the voltage of your supply, that you use in the calculation.

So it doesn't matter what wires you use to connect it to the supply, because you would measure the voltage right at the resistor. With thinner wire this voltage would be less than with thicker wire because there is a voltage drop in the wires due to their resistance and the current through them.

schematic

simulate this circuit – Schematic created using CircuitLab

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    \$\begingroup\$ Thanks, thats a nice illustration. As i understand, when the wires are changed the resistances of the circuit also change, os the current through the R1 resistor will be different across the circuits. \$\endgroup\$
    – 20MikeMike
    Jan 8, 2023 at 9:30
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    \$\begingroup\$ @20MikeMike Note that the circuit shown above is almost exactly what we use for four-terminal sensing when trying to take particularly accurate measurements of resistance. (We just make the supply on the left supply a constant and very precisely defined current, so that we need not care what resistances happen to be present between that supply and the voltage probes on the right.) \$\endgroup\$
    – cjs
    Jan 9, 2023 at 12:32
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    \$\begingroup\$ Intuitively, I'd have thought a thicker wire has less resistance than a thinner wire. Why d you say the opposite? \$\endgroup\$
    – Designalog
    Jan 10, 2023 at 12:24
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    \$\begingroup\$ @ErnestoG I don't think I did. I said the voltage across the resistor will be less with thinner wire. \$\endgroup\$
    – GodJihyo
    Jan 10, 2023 at 12:30
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    \$\begingroup\$ Ah sorry, I thought you meant the voltage will be smaller in the resistor. \$\endgroup\$
    – Designalog
    Jan 10, 2023 at 12:32
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All models are wrong. Some models are useful.

First, you should calculate the resistance of the wires (Whether thick or thin) and include that in your model. Provided the thin wire isn't too thin, the model is likely now accurate enough for most purposes.

If the thin wire is indeed very thin or made from a material with a high TCR (thermal coefficient of resistivity), you might find that the current through it heats the wire enough (through Joule heating) to change the wire's resistance, and you must consider that effect also in your model to accurately predict the current through the 100 ohm resistor.

Similarly, the 100-ohm resistor also has a non-zero TCR which you may also want to consider in your model.

Finally, depending on the nature of the 12 V source, you might also have to consider the effect of its internal resistance before you get an accurate enough prediction of the current. (Where "accurate enough" depends entirely on how you are planning to use the model's prediction of the current)

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    \$\begingroup\$ It also depends on the accuracy of whatever instrumentation you plan to use, to verify the circuit current and voltages, They will also affect the measurement, and you will need to determine just what that is. \$\endgroup\$
    – PStechPaul
    Jan 8, 2023 at 8:00
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    \$\begingroup\$ Thanks go to George Box for your first line. \$\endgroup\$ Jan 8, 2023 at 13:42
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    \$\begingroup\$ "All models are wrong. Some models are useful." While technically true, there are few applications where Ohm's Law is inaccurate. Furthermore, since neither the example in question nor any of the hypothetical examples in this answer violate Ohm's Law, I think the opening line only adds confusion. \$\endgroup\$
    – Vaelus
    Jan 8, 2023 at 17:11
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    \$\begingroup\$ @Vaelus, the point of the quote is that you should never expect your model to be 100% accurate. You have to decide how accurate is accurate enough. \$\endgroup\$
    – The Photon
    Jan 8, 2023 at 23:07
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    \$\begingroup\$ Thermal noise in resistors is one aspect that does not follow Ohm's law, and is not that rare problem to have in real world circuits. \$\endgroup\$
    – jpa
    Jan 9, 2023 at 8:39
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Ohm's law is a model, that means it is aims at prediction of physical reality with a mathematical formula.

As all models, it is valid only under certain conditions.

First of all it is only applicable to circuit elements that are qualified as resistor. That is not really an explanation. It is more that a "resistor" is something that is designed to follow (or by nature follows) Ohm's law under the technical typical current and voltages.

Deviations from Ohm's law are very easy to find:

  • Self-heating. One often says resistance changes with temperature, but in reality one has a nonlinear current - voltage response when heating by current losses are non-negligible. Practically this means that if current is increased, we, at some point, measure a voltage which deviates from the expected linear response.

    When transient behavior comes into play actual U, I behavior may become very complicated to predict, as it depends on thermal properites, including wire shape and coupling with its currounding.

    Note: Even if one defines resistance at U/I at constant temperature, this is not really avoiding the effect. Self-heating may occur solevly on the inside of a conductor, even if the surface is held at same temperature. Still one gets non-linear U/I response. It is an inherent problem. It shows that outside of "typical" working conditions one easily gets deviations from Ohm's law. In fact one often purposefully operates wires and resistors only under working conditions where Ohm's law holds. So that something is following Ohm's law is not a property of the device but more a consequence of the operating conditions.

  • Self-induction and AC behavior: At high enough frequencies the conductor will create a magnetic field which will affect current flow. One finds that a wire is always an inductor, so, at least under AC operation, not following Ohm's law. Wether this effect is negligible depends on the shape of the conductor and the operation frequency. (One can extend Ohm's law and work with impedances instead of resistances, so this problem can be "fixed". But in a strict sense Ohm's law only holds under static conditions)

  • Semicondcutor devices (voltage dependent resistor, diode, transistors etc etc.) generally do not follow Ohm's law. This is of course on purpose, the devices are designed to do so. Normal wires do not show this behavior though.

  • At extremely low currents (think of single electrons) Ohm's law is useless.

    Ohm's law is only valid when a huge number of electrons behave like a continuous current flow. At low number of electrons we could still define a "current" (electrons per sec) but the physical quantity "resistance" makes no real sense.

    This is because Ohm's law is more or less the friction the electrons experience when they move through a conductor. They collide with each other an other things in the conductor and this causes the resistance. So resistance is only an average effect and Ohm's law cannot predict the path of a single or few electrons.

  • Ohm's law violates Theory of Relativity.

    This is relevant for instance in plasmas where the electrons may move very fast, then Ohm's law is no longer valid. (Well, this is not relevant for the wire example but just to demonstrate the point)

    That Ohm's law is incompatible with relativity is not really surpising. It is based on the Diffusions equation (Ficks law) which is non-relativistic [1].

So to answer the original question:

Yes depending on how thin the wire is, one likely sees a current different from 12V/100R.

  • If the wire is small you have self-heating. (This is a very realistic case)
  • If the wire is really really small (think in the order of single atoms) one probably sees very different behavior.
  • IF the wire is really hot (and we have an electron ion-plasma) one probably sees also very different behavior.

The last two are more a theoretical case, but the point is that outside of the typical operation conditions many things can (and do) happen. And this depends on the material and the environmental conditions (temperature, magnetic field etc. etc)


[1]: This can easily be seen when an impulse source is considered. Since the diffusion equation yields the Gaussian function as response this is volating relativity. The Gaussian, at infinitely short times after the impulse, has a response arbitrarily far away from the source. This contradicts that information can only propagate with the speed of light. Ohm's law (and other diffusion equation based laws) neglect this.

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    \$\begingroup\$ This should be the accepted answer. Also, taken to the extreme, counter-example #1 (destruction by overheating) is another failure of Ohm's Law. Much like how Hooke's Law fails at the elastic limit, and the capacitance equation fails at dielectric breakdown. \$\endgroup\$
    – DrSheldon
    Jan 9, 2023 at 1:59
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    \$\begingroup\$ The counter examples described may sound extreme, but for some real extreme versions, let's wait till we see this question in a future "What If?" book :) \$\endgroup\$ Jan 10, 2023 at 1:13
  • \$\begingroup\$ @DrSheldon, This answer speaks to the title of 20MikeMike's posting, "Is Ohm's law really accurate?" But, that's not actually what 20MikeMike wanted to know. The body of the post describes a particular circuit and asks how it would behave. GodJihyo's accepted answer focuses entirely on that behavior. \$\endgroup\$ Jan 10, 2023 at 16:10
  • \$\begingroup\$ @SolomonSlow I think the most relevant point to the body of the question (with the wire thickness) is the self-heating. (That is why I put it first). Power dissipation is 1.44 W, which certainly causes a measurable effect for a short thin wire. For me this is the answer to questions body. \$\endgroup\$
    – Andreas H.
    Jan 10, 2023 at 16:54
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    \$\begingroup\$ I was not trying to disparage your answer. It contains a lot of valuable information. I was only pointing out to DrSheldon that OP accepted the other answer because the other answer came closer to what OP really wanted to know. OP wanted to know how to apply Ohm's law. OP's comments suggest that they did not understand that the V in Ohm's law is the voltage at the resistor, which will be different from the voltage at the terminals of the power supply. \$\endgroup\$ Jan 10, 2023 at 17:44
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Ohm's law is an empirical law. It is not always obeyed unless you define 'resistance' in such a way to make a circular argument. It is very closely followed for metal conductors at constant temperature and for useful resistor materials and constructions.

In the case you mention, if you adjust the voltage such that the voltage across the 100 Ω resistor is exactly 12 V (compensating for lead wire voltage drop), then the current should be exactly 120 mA (to the extent your 100 Ω resistor is accurate and stable and free of such nonidealities as a voltage coefficient).

Incidentally, the voltage coefficient is typically not a factor at lower resistances and voltages, but it can be a very significant effect for high voltage resistors. That means if you calibrate your meter (which uses a high voltage precision resistor in the front end) with 1 kV in, it may not be accurate with 5 kV in. The voltage coefficient is independent of the temperature coefficient and related changes due to self-heating. Speaking of which...


If you replace the ideal 100 Ω resistor with a coiled thin wire of pure platinum wound on a ceramic former and encased in ceramic like this:

Enter image description here

And allow the current to stabilize (maintaining precisely 12 V across the 'resistor'), you'll read something like 65 mA (almost half). That's because the temperature of the wire will increase by about 230 °C, and with it the resistance of the platinum wire.

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    \$\begingroup\$ +1 only this and the answer of @Andreas H. I consider correct answers here so far. Ohm's law is in many (by far not the most) cases somewhat accurate, but if you just measure accurately enough you could always find that reality deviates from it. \$\endgroup\$
    – Curd
    Jan 10, 2023 at 18:02
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    \$\begingroup\$ Ohm's law is a totally different category of physical law than e.g. Maxwell's equations. If you can find a single case where Maxwell's equations are not obeyed (and you can explain why) probably you will be the winner of one of the next physics Nobel Prices. If you find a case where Ohm's law is violated (no matter if you can explain why or not) nobody cares. Ohm's law's accuracy and validity is about comparable that of Hooke's law: it is just good enough in many carefully selected/protected situations, but it's easy to find cases where it is not valid at all. \$\endgroup\$
    – Curd
    Jan 10, 2023 at 18:02
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Ohm's law defines resistance (which like position or inertia or temperature or voltage isn't an inherent physical entity but often enough a stable enough abstraction of some complex ensemble that working with it happens to be very convenient). As such, particularly in its differential form, it is more accurate than reality. Where dependable accuracy of reality is required, special precision resistors can provide them across significant ranges of voltage, temperature, operating frequency and other variables.

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    \$\begingroup\$ No, resistance is not required to be constant (as in Ohm's law). \$\endgroup\$ Jan 8, 2023 at 21:17
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    \$\begingroup\$ @PeterMortensen Ohm's law says nothing about resistance being constant. In practice, it often is close enough. Like rotational inertia mostly is (it depends on rigidity which is never absolute and can be rather iffish for things like fly wheel levers or fluids). Differential resistance allows for voltage-dependent resistance but there are other dependencies that can come into play. \$\endgroup\$
    – user107063
    Jan 8, 2023 at 21:23
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    \$\begingroup\$ Ohm's law states that if one passes varying amounts of current through a conductor, or places a varying voltage across it, the voltage drop and current will vary in proportion to each other. This is an approximation. On the other hand, it is also used to define the resistance of a conductor under particular conditions as being the ratio of voltage to current; that definition is, by definition, exact, though the number thus computed won't always be particularly meaningful. \$\endgroup\$
    – supercat
    Jan 9, 2023 at 23:28
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Accurate? Yes as long as you evaluate ALL the possible losses:

  1. internal resistance of power supply

  2. connection resistance

  3. wire resistance

Also change in resistance for any part due to any temperature change.

You can work out if I missed anything.

Also you need to define “accurate” - one reason resistors are available with a range of tolerances.

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    \$\begingroup\$ Yes, but it is still only a model. There might be nonlinearities in a real physical circuit (and RUD if the voltage is too high). \$\endgroup\$ Jan 8, 2023 at 20:45
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Is Ohm's law really accurate?

Yes, it is.

Here's what Ohm's Law states.

'At constant temperature, the current flowing through a conductor is directly proportional to the potential difference applied across its ends.'

enter image description here

I α V

V/I = R

where 'R' is a constant, known as the resistance of the conductor at that temperature.

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    \$\begingroup\$ You should probably include the word "metallic" in front of "conductor" if you want to give the law as stated by Ohm. \$\endgroup\$
    – The Photon
    Jan 8, 2023 at 23:09
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    \$\begingroup\$ @The Photon - Hi, Not necessarily. Graphite is a non-metal but a good conductor of electricity. \$\endgroup\$
    – vu2nan
    Jan 9, 2023 at 5:48
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    \$\begingroup\$ Graphite (and many other materials) may follow Ohm's law, but Ohm didn't claim it would (at least not in the original paper that established the law) \$\endgroup\$
    – The Photon
    Jan 9, 2023 at 16:19
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Is Ohm's law really accurate?

As the other answers have said, yes, it is extremely accurate if used correctly (much as e.g. antenna theory is extremely accurate if used in the right context, or a calibrated mercury thermometer is extremely accurate if you don't try to use it to measure the temperature of four molecules of water).

There is another question we're all skirting here, though:

Is Ohm's law correct?

Does Ohm's law get at some physical reality about the world (there really is resistance (coupled with other physical effects, which are sometimes negligible for given purposes, and obey their own laws), or is it (as has been suggested) simply a circularly defined abstraction which is sufficiently good at predicting behaviour under some conditions (=really accurate)?

This isn't a scientific question: your circuits will work just as well whatever your metaphysics if you apply the right theories correctly, or equally badly if you apply them incorrectly. Whether you're going to be a scientific realist or something else is off-topic here (although probably on-topic on Philosophy.SE). Nonetheless it's the fundamental question behind any (successsful) theory: what grounds the success of this model?

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Suppose one observes that 2.0 centimeter cube of some liquid has a mass of 10.0 grams, and wishes to estimate how much a cubic meter of that liquid would weigh. One could compute a quantity, commonly called "density", by dividing the mass (10.0 grams) by the volume (8.0 cubic centimeters) of a sample, and then estimate the mass of a certain volume (1,000,000 cubic centimeters) of that liquid by multiplying that volume by the computed density (1.25 grams/cubic centimeter), yielding a mass (1,250,000 grams).

If all parts of the liquid in both cubes have identical characteristics, the mass of the large cube should be precisely 125,000 times as large as the mass of the small cube. If, however, some parts of the liquid in the large cube are compressed more than the liquid in the small cube, the mass of liquid necessary to fill the large cube might be greater than predicted.

Note that the fact that average density of any sample of liquid is equal to the total mass mass divided by the total volume will hold whether or not the liquid is uniform, but knowledge of the average density of a sample will be most useful when dealing with uniform materials.

Ohm's Law effectively defines a unit called an "ohm" which describes a ratio between the voltage across something and the current flowing through it. As with "density", such a quantity may be extremely useful in some situations, and far less useful in others. Ohm's law can usefully predict how things will behave if resistance stays relatively constant, and is useful in systems that behave as though resistance is relatively constant, but tends to be less useful in systems where the apparent resistance is highly variable.

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