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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:
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.
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.
simulate this circuit – Schematic created using CircuitLab
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)
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.
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.
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:
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.
Accurate? Yes as long as you evaluate ALL the possible losses:
internal resistance of power supply
connection resistance
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.
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.'
I α V
V/I = R
where 'R' is a constant, known as the resistance of the conductor at that temperature.
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.
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?
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.
