Ohms Law and Its Practical Limits: Investigating a 10um Steel Wire

[strokebow]

Does Ohms Law have practical limits?

For example:A piece of steel wire of length of length 10um and of CSA 20E-10.

Can we calculate the resistance of such a tiny piece of wire (resistor) in the usual way?

thanks

 

[vk6kro]

That is the formula for working out the resistance of a conductor given its dimensions and the resistivity of the conductor. So, it isn't Ohm's Law.

However, the formula itself applies for all conductors provided they are uniform.

There will certainly be practical problems with making perfect contact with some very short or very wide conductors.

 

[strokebow]

There will certainly be practical problems with making perfect contact with some very short or very wide conductors.

Hi, thanks for your reply.

What sort of problems would your envisage for making contact with such small conductors? and how would you attempt to overcome them?

cheers

 

[Floid]

I would imagine parasitic capacitance could be a problem.

I don't remember enough about inductance characteristics to be sure, but since the radius of your wire is greater than its length you might check what that means inductance wise as well.

Anyway, with a wire that short its going to be capacitance and inductance that cause you problems and not resistance.

 

[vk6kro]

Hi, thanks for your reply.

What sort of problems would your envisage for making contact with such small conductors? and how would you attempt to overcome them?

cheers

It would vary with the situation, but one example would be a conductor 1 meter square and 1 mm thick. Just making contact with the entire surface area equally would be difficult, but just doing it would compress the conductor enough to reduce the thickness of the conductor.

 

[strokebow]

Thanks for the responses.

It would vary with the situation, but one example would be a conductor 1 meter square and 1 mm thick. Just making contact with the entire surface area equally would be difficult, but just doing it would compress the conductor enough to reduce the thickness of the conductor.

I take your point. So, ideally how would we want to take resistance measurements? and in practise, what would realistically be the best way to do this?

I would imagine parasitic capacitance could be a problem.

I don't remember enough about inductance characteristics to be sure, but since the radius of your wire is greater than its length you might check what that means inductance wise as well.

Anyway, with a wire that short its going to be capacitance and inductance that cause you problems and not resistance.

Are capacitances and inductances relevant? Since the supply will only ever be direct current??

cheers

 

[sophiecentaur]

Ohm's Law includes the words "providing the temperature remains constant". This is very relevant for small / thin wires because, although they might have the same resistance as a big fat, long wire, they will need to dissipate the same amount of power (I²R) as the large wire. Being so small, they have a small surface area and will reach a higher temperature than the large wire and could melt before equilibrium is reached. There is a vicious circle at work here because, as the small wire gets hotter, its resistance will go up and it will then dissipate even more power... poooof!

If you could, somehow, cool that small wire, then it would 'follow ohm's law' over a much wider range of currents than it would in air.

 

[vk6kro]

I take your point. So, ideally how would we want to take resistance measurements? and in practise, what would realistically be the best way to do this?

You just have to accept that some measurements will be difficult and live with it.

One approach is to not measure these extreme cases at all. The properties of metals are already well known, so you can extrapolate the resistance of common lengths of wire to oddly shaped ones and simply calculate the resistance.

 

[jsgruszynski]

Ohm's law is a "Lumped Approximation Model" which presumes a certain level of physical aggregation and frequency of operation.

Ohm's law is limited in three ways:

1. The frequency of operation must be lower than wavelength of the signal - as you approach this limit so-called "distributed effects" start to dominate and Ohm's breaks down. Ohm's law is nothing but an approximation of Maxwell's equations and at this cusp, part of the wave natural of Maxwell's is taking over and you have to start deal with things as return loss and s-parameters instead of resistance, capacitance and inductance

2. The physical dimensions on the low end are dependent on the uniformity of statistical mechanics approximations. In condensed matter physics you are always using mixed classical-quantum approximations which as the "law of large numbers" breaks down results in quantum effects dominating.

3. As dimensions drop at the same time quantum effects like tunneling start to dominate as well separate from the statistical aspects. As these effects turn on they can appear as Ohm's Law (statistical aggregate) currents though the resistance cease to be linear (constant resistance vs. voltage). Like distributed, the wave nature of quantum mechanics starts to dominate.

 

[sophiecentaur]

Ohm's law is limited in three ways:

1. The frequency of operation must be lower than wavelength of the signal - as you approach this limit so-called "distributed effects" start to dominate and Ohm's breaks down.

I can't think what you mean here. How can a frequency be compared with a wavelength and what would the 'frequency of operation' be, if it was different from the 'signal'?

I stopped reading your post once I had read this. Can you clear it up?

 

[AlephZero]

I can't think what you mean here. How can a frequency be compared with a wavelength and what would the 'frequency of operation' be, if it was different from the 'signal'?

I guess he/she is referring to things like the http://en.wikipedia.org/wiki/Skin_effect , but soemthing got "lost in translation".

 

[sophiecentaur]

That makes sense. I must read the rest now . . . .

 

[jsgruszynski]

I can't think what you mean here. How can a frequency be compared with a wavelength and what would the 'frequency of operation' be, if it was different from the 'signal'?

I stopped reading your post once I had read this. Can you clear it up?

Sorry. Brain going faster than typing. Causes a written traffic accident. :-)

The operating frequency's equivalent wavelength must be "sufficiently" larger than the physical dimensions. Once the wavelength is shorter than physical dimensions, the lumped model is no longer valid: the concepts of resistance, capacitance and inductance as independent, separable component qualities are no longer valid or useful.

The model that still works involves the concepts of power reflection and transmission of complex variables describing waves. This is what return loss and s-parameters are about. Strictly this "distributed model" is still only an approximation of Maxwells' equations. There are cases when it becomes Epic Fail also and you need to drop back and start from Maxwell's directly.

3 GHz is 1 cm, so any circuit element larger than 1 cm is 100% "distributed" and not "lumped" while at 300 MHz (1/10 3 GHz), 1 cm sized circuit elements are still "lumped" to a good approximation. In the in-between of 300 MHz-3Ghz, things get dicey because some aspects are lumpy enough while other aspects are distributed.

A case in point: all square wave or "pulse-y" waveforms have odd harmonics to make them "square". Basically all digital waveforms. So you need "sufficient" harmonics to get a square-ish edge on any digital waveform. The rule of thumb from this is digital waveform edges (rising or falling edge time) require bandwidths that 10x the clock rate.

So if your clock rate is 300 MHz, then you need 3 GHz for the edges. If you are at 3 GHz, you need 30 GHz for the edges. This is actually central to why microprocessor clocks hit a brick wall around 2000: how big is that average microprocessor die, diagonal corner to corner? ~1 cm. So you have edge issues already that are distributed rather than lumped.

And that's a problem: the entire concept of digital logic 0 or 1 is itself a lumped model on top of the analog lumped model that approximates Maxwell's equations. If the foundation turns to quick-sand, then the building itself will start having problems.

 

[sophiecentaur]

That is surely a transmission line issue, isn't it?

 

[jsgruszynski]

That is surely a transmission line issue, isn't it?

Yes. When you have to worry about transmission lines, Ohm's law has broken down.

 

[phinds]

Yeah, my best EE prof always referred to "Ohms Law" as "Ohms Approximation" for the reasons mentioned above and cautioned care when working with "extreme conditions", saying that it's a TERRIFIC approximation for "normal" circuits.

 

[sophiecentaur]

Ohm's Law only says what it says. It applies where it applies. Nothing more.

 

[vk6kro]

3 GHz is 1 cm, so any circuit element larger than 1 cm is 100% "distributed" and not "lumped" while at 300 MHz (1/10 3 GHz), 1 cm sized circuit elements are still "lumped" to a good approximation. In the in-between of 300 MHz-3Ghz, things get dicey because some aspects are lumpy enough while other aspects are distributed.

A case in point: all square wave or "pulse-y" waveforms have odd harmonics to make them "square". Basically all digital waveforms. So you need "sufficient" harmonics to get a square-ish edge on any digital waveform. The rule of thumb from this is digital waveform edges (rising or falling edge time) require bandwidths that 10x the clock rate.

So if your clock rate is 300 MHz, then you need 3 GHz for the edges. If you are at 3 GHz, you need 30 GHz for the edges. This is actually central to why microprocessor clocks hit a brick wall around 2000: how big is that average microprocessor die, diagonal corner to corner? ~1 cm. So you have edge issues already that are distributed rather than lumped.

And that's a problem: the entire concept of digital logic 0 or 1 is itself a lumped model on top of the analog lumped model that approximates Maxwell's equations. If the foundation turns to quick-sand, then the building itself will start having problems.

The wavelength of a 3 GHz radio wave in air is close to 10 cm or about 4 inches. (³⁰⁰/₃₀₀₀ * 100 cm)This is quite a lot bigger than the average computer CPU chip.

Transmission line effects cause apparent Ohm's Law violations all the time.
If you connect a quarter wavelength of 50 ohm transmission line to a 100 ohm resistor, at the other end of the line it will appear to be 25 ohms.
Put 100 volts across it (at the right frequency for that wavelength) and it will draw 4 amps. That is 400 watts and the resistor will get 400 watts too, (ignoring losses), but at 200 volts and 2 amps.
Ohms Law still applies locally, but transmission line effects make it look as if it doesn't apply.
After all, you put 100 volts in and the resistor drew 2 amps, that is only 200 watts, so where did the other 200 watts go?

 

[sophiecentaur]

I don't see that any of this relates to Ohm's Law at all. There's nothing inherently non linear about any of these effects and that's all Ohm's Law tells you. It deals specifically with resistance and doesn't concern any transforming effects of distributed components. In all these examples, doubling the Volts will double the Current. So where's any violation?

 

[vk6kro]

I don't see that any of this relates to Ohm's Law at all. There's nothing inherently non linear about any of these effects and that's all Ohm's Law tells you. It deals specifically with resistance and doesn't concern any transforming effects of distributed components. In all these examples, doubling the Volts will double the Current. So where's any violation?

That was the point I was making. You can get apparent anomalies, but Ohm's Law continues to apply.

If fact, I cringe when I hear talk of "non-ohmic" resistors. Just because the resistance of a resistor changes (due to temperature, for example) does not mean Ohm's Law no longer applies.
A small change in voltage still produces a small change in current and the ratio of these two still gives the resistance of the resistor.

 

[sophiecentaur]

It would have been better if they had called it 'Ohm's Behaviour', perhaps.

 

[Nevertamed]

ohms law fails when the resistance of the conductor increases as it gets hot by electron friction

 

[phinds]

ohms law fails when the resistance of the conductor increases as it gets hot by electron friction

That doesn't seem to make sense. All you are saying is that the resistance varies with temperature. Whatever the resistance is, Ohms Law should still hold.

 

[sophiecentaur]

ohms law fails when the resistance of the conductor increases as it gets hot by electron friction

You obviously haven't read what Ohm's Law says.

 

[Nevertamed]

in Ohm's law as resistance is proportional to voltage, then: temperature is a limitation, resistance varies with temperature, and depending on materials and room temp, it can vary a lot. unless we state that temp is constant we can secure the lineality of the ecuation

 

[vk6kro]

It depends on the definition of Ohm's Law.

There is the one we all saw on day 1 of Electricity at school. Graph voltage vs current and if it is a straight line then you can work out the resistance.

According to that version, even a rheostat would be regarded as non ohmic because it can change.

It leads to the strange situation where a lamp filament is "non-ohmic" if you measure its voltage and current curve slowly (and give it time for the temperature to change) but "ohmic" if you change the voltage rapidly. The current in a lamp filament with 60 Hz AC on it is close to sinusoidal, meaning the resistance is close to constant because the temperature is fairly constant and the resistance is also stable.

Much more useful is the incremental one which assumes that Ohm's Law always applies and you can work out the resistance at any point on a V vs I curve by taking a small change in voltage and observing the small change in current that results.

This small signal resistance is very real and it can be measured as the actual impedance of a circuit. It may be quite different to the ratio of DC voltage to DC current that applies at that point.

 

[sophiecentaur]

Ohm's Law is its own definition. The condition for V/I being constant is for the temperature to be constant. Measurement with a 'probe', low level AC of a filaments lamp will reduce the temperature fluctuation and give a value of dV/dI that is not resistance. The resistance is still V/I for the particular point on the temperature dependent VI curve that you will get using DC.

The only time that Ohm's Law truly fails for a metal is for enormous current flux when there are just no more conduction electrons available and the conductor becomes .non linear.

 

[es1]

I wonder though: It is Ohm's Law that is failing or that the device (or circuit) in question is no longer well modeled by an ideal resistor in the examples given? I would argue Ohm's law isn't breaking, the ideal resistor model is.

If one replaced the ideal model in Ohm's law with a new model that also took temp, time, etc., into consideration, wouldn't Ohm's law still hold? I think it does.

I agree with Sophie and the only thing that really breaks it is current density and possibly arc'ing across the geometry of the element. Because if the function for R had to take current or voltage as an independent variable then equation we would be left with would be self referencing and I think this would count as a new beast.

 

[sophiecentaur]

I wonder though: It is Ohm's Law that is failing or that the device (or circuit) in question is no longer well modeled by an ideal resistor in the examples given? I would argue Ohm's law isn't breaking, the ideal resistor model is.

If one replaced the ideal model in Ohm's law with a new model that also took temp, time, etc., into consideration, wouldn't Ohm's law still hold? I think it does.

I agree with Sophie and the only thing that really breaks it is current density and possibly arc'ing across the geometry of the element. Because if the function for R had to take current or voltage as an independent variable then equation we would be left with would be self referencing and I think this would count as a new beast.

But, as with many so called laws of nature, Ohm's Law has no 'authority' to make things behave a certain way. It doesn't punish them if they don't conform. Ohm's law is merely a description of what happens under a particular set of circumstances. If one reads the wording carefully (actually, it's blindingly obvious that you should always do that) then there isn't much to disagree with and there's certainly no reason to hop about saying "Ohm got it wrong". Add some extra factors and metals will behave differently - so what?

I would say that the only time a metal stops obeying Ohm's law is when it is under such extreme conditions that it can almost not be classed as a metal because its very structure will have changed (metallic bonding won't even be the same). George O. can't really be blamed for not having put a codacil on his Law to take account of this.

 

[Nevertamed]

But, as with many so called laws of nature, Ohm's Law has no 'authority' to make things behave a certain way. It doesn't punish them if they don't conform. Ohm's law is merely a description of what happens under a particular set of circumstances. If one reads the wording carefully (actually, it's blindingly obvious that you should always do that) then there isn't much to disagree with and there's certainly no reason to hop about saying "Ohm got it wrong". Add some extra factors and metals will behave differently - so what?

I would say that the only time a metal stops obeying Ohm's law is when it is under such extreme conditions that it can almost not be classed as a metal because its very structure will have changed (metallic bonding won't even be the same). George O. can't really be blamed for not having put a codacil on his Law to take account of this.

wonderful answer

 

[Kholdstare]

Can anyone quote the exact statement given by Ohm with web reference. Just being curious.

Given a constant temperature current will be equal to voltage multiplied by resistance. - correct
Given a constant temperature current will be proportional to voltage. - wrong

 

[vk6kro]

Can anyone quote the exact statement given by Ohm with web reference. Just being curious.

Given a constant temperature current will be equal to voltage multiplied by resistance. - wrong
Given a constant temperature current will be proportional to voltage. - correct

The history of Ohm's Law is covered in the Wikipedia article:
http://en.wikipedia.org/wiki/Ohms_law

 

[sophiecentaur]

Can anyone quote the exact statement given by Ohm with web reference. Just being curious.

Given a constant temperature current will be equal to voltage multiplied by resistance. - correct
Given a constant temperature current will be proportional to voltage. - wrong

?
If you get your history right, you will realize that Ohm only introduced Resistance as the constant of proportionality for a linear relationshipbetween two quantities that he could actually MEASURE. They didn't sell Ohmmeters at the time!

 

[Nevertamed]

Can anyone quote the exact statement given by Ohm with web reference. Just being curious.

Given a constant temperature current will be equal to voltage multiplied by resistance. - correct
Given a constant temperature current will be proportional to voltage. - wrong

dude could u explain wtf are u talking about? under that condition voltage follows current proportionally: CORRECT

I= V R: WRONG

 

[Kholdstare]

?
If you get your history right, you will realize that Ohm only introduced Resistance as the constant of proportionality for a linear relationshipbetween two quantities that he could actually MEASURE. They didn't sell Ohmmeters at the time!

So Ohm did introduce resistance as a constant of proportionality. I think the statement of Ohm's law will be something very similar to this:
"Given a constant temperature current through a material will be proportional to the voltage drop across it (in steady state)."

Now, Ohm's law is indeed true if we do not take into consideration the quantum effects.
In fact if you are able to measure I-V curve for a long conductor with very high accuracy or say you measured it exactly, you will discover that the I-V curve is never linear. The variation in resistance is laughably tiny, but it exists and due to QM effects. In nano-scale structures the variation is tremendous. In those cases the resistance depends on the applied bias.

Now, as Ohm's law is only a model to predict the behavior of real materials (just like QM is a model to predict the behavior of electrons - we don't even understand what electrons actually are, only know their mass, charge, observables etc.), the idea that Ohm's law only predicts the behavior of ideal conductors is bogus. If it did that we won't be needing it cause there is no ideal conductor.

My conclusion is that when it comes to accuracy, the I-V relationship derived from QM beats Ohm's law (both in large and small materials) proving Ohm's law does have its limits.