Sinusoidal voltage applied to zero resistance conductor

[b.shahvir]

Hi Guys,

The following query would sound a bit ridiculous and abstract but it suddenly popped up in my head.

What would happen if I were to apply a purely sinusoidal AC voltage across a zero resistance conductor (theoretically, a super conductor) ? Zero resistance would mean the conductor is assumed to carry infinite current thru it (at least theoretically).

However, it would be interesting to note the behaviour of the AC current waveform since the current cannot be limited by way of 'frictional resistance' as the lattice structure inside the super conductor is considered to be absent.

Also the super conductor is assumed to possesses zero inductance!

Kind Regards,
Shahvir

 

[Phrak]

Hi Guys,

The following query would sound a bit ridiculous and abstract but it suddenly popped up in my head.

What would happen if I were to apply a purely sinusoidal AC voltage across a zero resistance conductor (theoretically, a super conductor) ? Zero resistance would mean the conductor is assumed to carry infinite current thru it (at least theoretically).

First, you're assuming there's no load in the circuit. Or is it a circuit?

However, it would be interesting to note the behaviour of the AC current waveform since the current cannot be limited by way of 'frictional resistance' as the lattice structure inside the super conductor is considered to be absent.

Also the super conductor is assumed to possesses zero inductance!

Kind Regards,
Shahvir

Associated with a changing electric field is a changing magnetic field. There must be inductance.

 

[b.shahvir]

First, you're assuming there's no load in the circuit. Or is it a circuit?

The circuit load is the super conductor itself.

Associated with a changing electric field is a changing magnetic field. There must be inductance.

True, but just for this abstract case please consider it to be zero.

 

[Phrak]

You might consider either a circuit with closed loops, or an infinitely long conductor.

For the infinitely long superconductor in free space the electric field will radiate outward perpendicular to the wire, filling all space. It will alternate in direction at the applied frequency. The wave will travel down the wire at c. For a step change in voltage of 1 volt the current will be 1/330 amperes. And by the way, there is an associated magnetic field looping around the wire.

 

[Dale]

You cannot apply a voltage to a superconductor. You have to heat it up until it becomes resistive in order to apply a voltage.

 

[Phrak]

How do you figure, Dale? It doesn't have to be resistive to be reactive to an applied voltage.

(BTW, that 330 ohms should have been 377)

 

[b.shahvir]

You cannot apply a voltage to a superconductor. You have to heat it up until it becomes resistive in order to apply a voltage.

I repeat, this is not a practical experiment. It is an abstract thought….and hence I request you guys to please assume, under all circumstances, ideal conditions only.

Also, I must concur with Phrak on this one. Thanx

 

[Dale]

No, even in abstract thought it is impossible, and certainly practically. On the abstract side there is no finite current which, when multiplied by 0 Ohms, will give you a non-zero voltage. On the practical side, even an infinite current won't work because it will exceed the maximum superconducting current density.

 

[f95toli]

The problem with this thought experiments is that it makes a number of unphysical assumptions meaning there is no way to come up with a sensible answer.
Also, the "zero resistance conductor" you are describing is NOT a superconductor.
Even ideal superconductors are reactive (have an inductance), not only because of the geometry but also because of their kinetic inductance (and the kinetic inductance is large BECAUSE the superconductor is lossless at dc).

Hence, this experiment would be impossible even if you assumed an ideal superconductor with infinite Jc at 0K.

 

[Bob S]

If a resistive conductor is carrying an ac current, at high frequencies the ciurrent is forced to the outer surface of the conductor by eddy currents inside the conductor. Read about skin depth, which is the penetration of ac currents into the conductor. As the resistance decreases, so does the skin depth (but only as the square root of resistivity), so at nearly zero resistance the skin depth is nearly zero.

 

[cabraham]

The resistance is 0, so the entire impedance consists of the inductive reactance, Xl. The shape of the circuit determines the inductance, L. Xl = 2*pi*f*L. Thus I = V/Xl.

Claude

 

[b.shahvir]

The problem with this thought experiments is that it makes a number of unphysical assumptions meaning there is no way to come up with a sensible answer.
Also, the "zero resistance conductor" you are describing is NOT a superconductor.
Even ideal superconductors are reactive (have an inductance), not only because of the geometry but also because of their kinetic inductance (and the kinetic inductance is large BECAUSE the superconductor is lossless at dc).

Hence, this experiment would be impossible even if you assumed an ideal superconductor with infinite Jc at 0K.

Ok then for God's sake please do not consider it as a super conductor. Just consider it as a plain zero resistance conductor without inductance. Thanx.

 

[Bob S]

Use the skin depth formula for the surface of a circular conductor with radius R; the current flows in a cross sectional area equal to 2 pi R x, where x is the skin depth thickness. Use the conductivity for copper: 59 x 10⁶ per ohm-meter, or 1.67 x 10^-8 ohm meters.
Then take the limit as the resistance goes to zero.

The skin depth d is given by d = sqrt[2 rho/(w u₀)]

where w = 2 pi frequency [units sec^-1] and
u₀ = 4 pi x 10^-7 [units: henrys/meter]

Note that 1 ohm = 1 Henry/sec, so d has units of meters.

 

[f95toli]

Ok then for God's sake please do not consider it as a super conductor. Just consider it as a plain zero resistance conductor without inductance. Thanx.

But then what ARE we suppose to consider it to be?
You are essentially asking something akin to "what is one divided by zero?"...
It might be an interesting philosophical question but no one can give you an answer based on physics.

 

[Phrak]

Which physical laws do you want to throw out, and which do you want to keep?

 

[Phrak]

I repeat, this is not a practical experiment. It is an abstract thought….and hence I request you guys to please assume, under all circumstances, ideal conditions only.

Also, I must concur with Phrak on this one. Thanx

Actually, you don't. Reactance is a result of inductance.

 

[Bob S]

But then what ARE we suppose to consider it to be?
You are essentially asking something akin to "what is one divided by zero?"...
It might be an interesting philosophical question but no one can give you an answer based on physics.

Please work out my post above on skin depth (#13 ?), work out the problem using a finite resistance, and let the resistance go toward zero slowly.

 

[f95toli]

Please work out my post above on skin depth (#13 ?), work out the problem using a finite resistance, and let the resistance go toward zero slowly.

And? That will obviously give a skin depth of zero, but what has that to do with the question?
When doing calculations involving e.g. transmission lines this is usually a very good approximation, but a lossless TL still has an inductance and capacitance per unit length so the impedance is never zero.

 

[b.shahvir]

But then what ARE we suppose to consider it to be?
You are essentially asking something akin to "what is one divided by zero?"...
It might be an interesting philosophical question but no one can give you an answer based on physics.

You guys are absolutely correct and i do not dispute the fact that to assume such a physical phenomenon is next to impossible even theoretically. But if I set practical limitations, then my idea would become pretty distorted! I do not know any other way to compromise on it then

Thanks & Regards,
Shahvir

 

[Dale]

It is not just a practical limitation, it is a theoretical limitation. Just look at Ohm's law. If the resistance is 0 then there is never any voltage regardless of the current. All points in a material with no resistance must be at the same voltage by definition. You cannot apply a voltage to one even theoretically.

 

[Phrak]

If that were true, Dale, transmission lines would transmit signals instantaneously.

 

[b.shahvir]

It is not just a practical limitation, it is a theoretical limitation. Just look at Ohm's law. If the resistance is 0 then there is never any voltage regardless of the current. All points in a material with no resistance must be at the same voltage by definition. You cannot apply a voltage to one even theoretically.

I do not dispute this fact. But the voltage will appear zero only when seen externally, since in absence of conductor resistance or inductance it will be totally used up in driving the infinite current into the zero resistance loop. Hence externally the reflected voltage across it would be zero!

Regards,
Shahvir

 

[Bob S]

And? That will obviously give a skin depth of zero, but what has that to do with the question?
When doing calculations involving e.g. transmission lines this is usually a very good approximation, but a lossless TL still has an inductance and capacitance per unit length so the impedance is never zero.

You forget that if the current is flowing in a circular conductor of resistivity rho = 1.76 x 10^-8 ohm-meters, radius R, and length L, then the effective resistance of the conductor is

resistance = rho L/(2 pi R d), where d is the skin depth.

so rho appears in the equation more than once.

 

[Bob S]

If that were true, Dale, transmission lines would transmit signals instantaneously.

The permeability of free space is u₀ = 4 pi x 10^-7 henrys per meter, and the permittivity of free space is e₀ = 8.85 x 10^-12 farads per meter, so the signal velocity is sqrt(1/(e₀ u₀)) = c (speed of light).

 

[Phrak]

The permeability of free space is u₀ = 4 pi x 10^-7 henrys per meter, and the permittivity of free space is e₀ = 8.85 x 10^-12 farads per meter, so the signal velocity is sqrt(1/(e₀ u₀)) = c (speed of light).

Yeah, well, epsilon of some suitable dialectric, is generally more than free space, epsilon_0.

 

[Dale]

If that were true, Dale, transmission lines would transmit signals instantaneously.

Transmission lines do not have 0 resistance, but in any case Ohm's law is only valid in the usual small-circuit assumption. If the small circuit assumption is violated then you need to use Maxwell's laws instead of circuit theory.

 

[Bob S]

Yeah, well, epsilon of some suitable dialectric, is generally more than free space, epsilon_0.

What are the permeability and permittivity in free space (interstellar vacuum)? What about the velocity of light in free space (= 1/sqrt(e₀ u₀) = 3 x 10⁸ m/s), and the impedance of free space (=sqrt(u₀/e₀) = 377 ohms)? If there is nothing in the interstellar vacuum, then how can it have an impedance?

 

[Born2bwire]

Transmission lines do not have 0 resistance, but in any case Ohm's law is only valid in the usual small-circuit assumption. If the small circuit assumption is violated then you need to use Maxwell's laws instead of circuit theory.

We can still work a transmission line with 0 resistance, we can't work a line with 0 impedance.

There still an inherent capacitance and inductance associated with free-space, so he can make any kind of general assumptions about the conductor in my opinion (and once he's specified the conductivity to be infinite it doesn't matter what he says about its inductance or capacitance). Working a transmission line with zero resistance isn't a problem, the physical geometry of the transmission line and the permittivity and permeability of the surrounding medium provide an overall impedance to the transmission line. If he wants to start saying that he has the conductor embedded in an infinite medium of zero inductance or that the resulting transmission line has zero inductance/impedance, then the problem becomes one that isn't workable.

Would this result in infinite currents, assuming that the medium has a non-zero permittivity and permeability, no.

 

[Dale]

I understand what you are saying, but the OP specified both 0 resistance and 0 inductance (so 0 impedance, real and imaginary). Obviously it is unphysical, but in principle you cannot apply any voltage across it.

 

[Phrak]

I understand what you are saying, but the OP specified both 0 resistance and 0 inductance (so 0 impedance, real and imaginary). Obviously it is unphysical, but in principle you cannot apply any voltage across it.

Since we're addressing an unphysical never-ever land, I think it could, considering an ideal charged fluid constrained to a 'wire', where only Coulomb's forces, constrained to c, are in vogue.