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Also no real world resistor is a true resistor, they allways have parasitic elements in there, like series inductance and parallel capacitance. Those may be very small, but they are there.
Other thing is that every resistor has some level of noise, which induces a tiny noise voltage across that resistor, so effectively the resistance oscillates around the nominal value by tiny bit. See https://en.wikipedia.org/wiki/Johnson–Nyquist_noise
R=E/I describes what happens at one point. If the resistor is linear, then R=E/I describes what happens over wide range of voltages/currents. Not all resistors are perfectly linear, in which case they are described as "non-Ohmic" resistors.
As mentioned already, the formula itself is perfect, no matter what. It is based on the definition of voltage, current, and resistance. But no resistor, voltage source, or current source is perfect, and neither are the tools used to measure them. Voltage sources in the real world always have an internal resistance, which is like adding a resistor in series with a battery. Resistors are not perfect either. The value of resistance might change with surrounding temperature. If they're near other components, you'll get a lot of interference. So the formula is perfect, the world is not.
You might have to ask some more specific questions because there are different ways to interpret this question.
For example, if R is considered to be a lumped circuit element then V=R*I breaks down when the wavelength is comparable to the physical dimensions of the resistor.
Because you mentioned power, it might be that you are worried about something like the resistance changing as it starts to dissipate more power. Well, most resistors get warm when they start to dissipate power and that usually changes the resistance to some degree, so V=R*I again has to be modified to V=R(T)*I where T is temperature and now R is a function.
Also, in algebra we would consider:
y=a*x
to be perfect, but depending on the application we may have to investigate changes in a or x for example. The theory is perfect, but the application seldom is.
Because you asked this question you may be worried that it wont work for some given application or use, so maybe you can mention what it is that worries you most about this.
You might do some mind experimenting at a micro level. For example, if your own body was miniaturized down to atom level, there would be not such thing as things feeling warm or cold. The heat is felt by us big people is caused by molecular movement. Shrunken into a tiny person, I could not feel a "warm" molecule.
V=IR can be appplied to the world of pneumatics. Equivalence would be Pressure = gas Flow (current) * Constriction (resistance). If you have a gas filled container and you open a pet cock to let all the gas out into a vacuum, you cannot predict at what time the last atom of gas leaves the container. It could be bouncing around inside the container for years without bouncing toward the exit. I believe that our laws of physics for the macro world are an expression of the probability of massive events in the micro world. Our laws of physics are statistically accurate out here in the macro world..
Well, for DC the equation V=R*I always holds because it is a self fulfilling definition in that if you measure V and you measure I then:
R=V/I
and that's said to be the resistance.
Then, using algebra which is perfect in itself, we rearrange to get V=R*I and it has to work because R is defined that way too.
In the AC case however, although V=R*I holds for a large number of cases (audio for example), it may not hold when the wavelength (RF) becomes comparable to one of the physical dimensions of the resistor (such as length, width, or height). In that case it has to be handled as a sort of transmission line with distributed capacitance and inductance as well as resistance where the total resistance is broken up into pieces but even then we would allow the use of V=deltaR*I which is almost the same.
So decades of theory doesnt go down the tubes, it just has to be handled a little differently sometimes. In this forum we seem to rarely talk about the applications where this would be significant however.
As to a real world application, as mentioned as the resistor heats up the resistance often goes up by some degree so we have to change it a little:
V=R(T)*I
where R(T) is the resistance function of temperature. A brief example would be:
R(T)=Ra*(1+(T-293)/1000)
where
T is the temperature in Kelvin, and
Ra is the resistance at 293 Kelvin.
This function tells us that for temperatures over 20 deg C we get a slight increase in resistance, and for temperatures less than 20 degrees C we get a slight decrease in resistance.
As the resistor heats up the temperature rises, so the resistance rises.
Quotation (post#5): So the formula is perfect, the world is not.
I think this is a good answer - however, for my opinion it needs some kind of explanation/interpretation.
General rule (applicable not only to electronics):
You always should know WHAT you are doing and convince yourself WHY resp. IF you are allowed to do this.
This means, you always have to recognize that in the world of electronics no formula can be - under all circumstances - correct by 100% because it always contains simplifications and neglected parasitic influences.
Thus, for each application it is necessary to prove if the conditions allow the usage of the particular formula.
Simple example: The gain of an inverting opamp configuration (-R2/R1) is a sufficient good approximation ("perfect" gain formula) only as long as the operating frequency is far below the transit frequency (gain-bandwidth product) of the opamp.
Winterstone:
That's a pretty good answer really. I'd like to add to it here.
What else we have to do is be able to separate theory from the application of the theory. Most of the theory neglects certain things as Winterstone pointed out, and the theory itself is considered perfect. In short, the theory states that:
"Given the exact parameters of the theory it holds for every case".
But when we go to apply that theory we often have to apply it knowing that we may have to modify the final results somewhat depending on what things were not included in the theory. For a few examples: parasitic capacitance, internal op amp gain and roll off and slew rate, resistor temperature characteristics.
So it is good to know how to separate theory and practice, but also to know how to combine them.
the formula always works, even for nonlinear resistances, because the formula describes the relationship between voltage, current, and resistance (whatever that may be at the point E and I are measured) so, for a semiconductor junction (a very nonlinear resistance) it can be shown that if the voltage is 0.65 and the current is 10mA, the resistance is 65 ohms. when the current is increased to 100mA and the voltage is now 0.7V. the resistance has changed to 7 ohms. some semiconductor junctions have a negative voltage characteristic along a small part of their curve, which is occasionally put to use in microwave oscillators in the form of a tunnel diode.
I was not asking for a particular problem, but rather to simply ask lol.
And yes, I was maybe thinking in the impossible scenario of a perfect resistor with no parazites.
Like, if at some point there's a fourth variable we need to take account of.
Like, I don't remember what I read in a book last year so correct me if I'm wrong, Stephen Hawking was talking about the gravity here on earth and that our fomulas didn't make sense when used for the big big stuff (galaxies). I think it was gravity, not sure.. but clearly saying that quatum mecanics were required for something big.
I dunno if it makes sense with my V=RI here but you might know where I'm getting at
The ohms law equation will always be correct for any circuit consisting of purely DC elements.
One volt applied to one ohm of resistance will cause one amp of current to flow.
This is not because the equation was derived to match the observed behavior of a circuit, but because the values for voltage, current and resistance were chosen to make the equation work.
Any apparent discrepancy is due to measurement errors and/or parasitic affects.
It can be used for AC circuits with AC elements such as capacitors and inductors as well. Just change Resistance to Impedance. The relationship stays the same.
Oh, and at really low currents like pA or 1E-12 A you have to worry about the triboelectric effect. Motion of the conductors in the Earth's magnetic field introduce measurable currents. Been there, done that. In one thing, I was measuring, it was better to use a Coulomb meter and divide by the time. The actual resetting of the meter also injects charge. So you have to, zero the meter, wait a short time, unzero the meter, wait a short time, take a reading; wait 30 sec or so and take another; subtract them and divide by the time to get the current.
At low currents, you have to use guarding to keep leakages at bay.
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