Naked Science Forum
General Science => General Science => Topic started by: paul cotter on 21/11/2023 21:58:00
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Imagine a simple circuit of a battery, a switch, a lamp and a piece of superconducting wire in a loop. What happens when the switch is closed is the question and is a little more complex than it initially appears. Since any piece of wire has both inductance and resistance it therefore has a time constant, l/r which defines the time it takes for the current to rise to ~63% of the value v/r, where v is the applied voltage. In the case of the superconducting wire this time constant will be infinite and hence no current will flow! This scenario will also involve a potential across the piece of superconducting wire which is obviously impossible. Ok, you say maybe we cannot model a superconductor as having a time constant but we can look at limits: as the resistance is artificially lowered the time constant will be valid and each time it is lowered the time constant will increase until we approach zero resistance where the current will flow but at such a slow rate that it appears to be non-conducting. In this situation I am just looking at the piece of superconducting wire in isolation. There has to be some error in this analysis but I can't find it.
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This scenario will also involve a potential across the piece of superconducting wire which is obviously impossible
It's allowed because it's the voltage across an inductor.
(which is proportional to the rate of change of current.)
And the R for calculating the time constant is the total resistance in the circuit.
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One would think that in the shop super conductor the resistance is 0, the voltage drop is 0 and the current used is 0. If you connected a super conductor in a short circuit to a battery the resistance would be in the internal resistance of the battery, which would theoretically get hot and burst into flames, theoretically of course.
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I was very tired when I posted that last night and I see through the apparent fallacy now. I am working at the moment and will explain later.
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Hi BC, you are quite correct in what you say but my question was from a slightly different angle, namely the behaviour of the superconductor with an attempt to apply a voltage, the rest of the circuit only there to supply said voltage. An individual who normally is not known for bs stated that because a superconductor has an infinite time constant the current flow when a voltage is applied would be infinitely slow, ie no current. This seemed absurd to me and in a half asleep state I could not get my head around it and I posted it here. After a night's sleep it is all clear: as r→0 it will take increasing longer to reach 63% of v/r and v/r will be increasing at the same rate. If r=0 it will take an infinite length of time to reach an infinite current. I don't think a true superconductor has a valid time constant as that would involve the taboo of divide by zero.
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One thing they do with superconductors is make electromagnets.
These often have large inductances.
Let's pick a number out of a hat and say it's 10 henries.
Imagine I connect a power supply to that coil and it has an output voltage of 10 volts.
Initially no current is flowing.
When I make the connection the inductance of the coil limits the rate of change of current.
The current will rise (from zero) initially at 1 amp per second.
After 10 seconds the current flowing will be 10 amps.
Imagine that I have things set up so that the coil is short circuited by a piece of "superconducting stuff" which is too warm to super-conduct.
Initially it makes no real difference. Some current flows in it but, if I'm clever, not much.
But when the current reaches 10 amps, I cool that short circuit down and it becomes a superconductor.
The power supply is short circuited and so it blows a fuse or trips out or whatever.
But the current of 10 amps continues to flow in the coil and through the link (which is now cold enough to be a superconductor).
Equivalently, you can make a switch out of superconductive stuff and close the loop with that.
Interestingly, (almost) all the modeling of inductors and capacitors that you see in high school text books assumes that they have zero resistance.
All the maths you saw was designed to work with superconductors :-)
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Which is, kind of, how we charge MRI magnets.
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Following my skirmish with covid in the summer '22 I had strange sinus problems and my doctor sent me for an mri, my first such experience. I was stunned by how loud the gradient magnets were. Previously I had thought radio waves and magnets, should be quiet enough to hear a pin drop! The machine in question had a 1.5T regular magnet.
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I used to own and sell 0.6 T open and upright MRI machines. These are MUCH quieter!
The gradient coils are carrying hundreds of amps so actually undergo detectable magnetostriction. A conventional solenoid MRI is essentially a plastic tube with the gradient coils encased in a fiberglass former, so there's nothing to damp the acoustic noise.
Our machines used a steel frame weighing between 30 and 150 tons, with the gradient coils embedded in the steel pole pieces - you could bash it with a hammer and not wake the patient, which was actually a problem: patients frequently fell asleep and instead of sitting or standing still, would twitch randomly during the procedure. We put a TV screen in the room but instead of playing the usual soothing DVDs of aquarium scenes, fireplaces or SwanLake, we had to run old comedies and episodes of Top Gear to keep people alert enough to not move!
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Funny thing is that although the noise was initially disconcerting I was actually starting to doze just at the end of the scan, another 5mins and I would have been asleep. I also noted the distinct smell of hot electrical insulation at the end, similar to the smell of a heavily loaded transformer.
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I also noted the distinct smell of hot electrical insulation at the end, similar to the smell of a heavily loaded transformer.
I can probably identify the manufacturer from that information!
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Well Alan, some further information may allow you to confirm your suspicions. The company providing the service is Affidea, located in the old meath hospital building. I asked the operator several questions about his toy but all he could tell me was the magnetic field strength and the fact that it was not superconducting. I wanted to know what frequency was used- I suppose I could work it out, assuming it is only proton resonance(no gadolinium injection).
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I've never heard of a 1.5T non-superconducting MRI magnet! And just to add to the fun, Affidea's website shows a photograph of a conventional x-ray unit under "MRI".
The most powerful resistive (room temperature) unit we built was 1T. It weighed over 200 tons and required 200 kW electrical input and liquid cooling. I left the company about 10 years ago and AFAIK we never managed to sell one!
The gyromagnetic ratio of a proton is about 42 MHz/T so your 1.5T machine was working at around 63 MHz.
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Alan, I am only reporting what I was told, he could well have been in error. Would not surprise me as he did not seem knowledgeable. The whole machine was not very big and if made of solid steel would not have weighed more than ~40-50 tons.A tube job, not one of the later fancy designs
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The air-cored solenoids rarely work out at more than 40 tons including all the electronics - we can put the whole kit plus an office and a generator (but sadly never a toilet - and I always end up testing them in a freezing truck park) in a single road trailer. Some 1.5 T units weigh less than 10.
I would worry about an operator who didn't know he was using a supercon. There's an oxygen monitor in view of the operator just in case the helium starts to boil off!
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I was stunned by how loud the gradient magnets were.
Not just noisy, some of the ones I've been in felt like the kid on the bus kicking the back of your seat. On one occasion they offered me the music of my choice so I decided to time how long I was in there by counting at the rate of 3 minutes per record, but I couldn't even hear when one track ended and another started.
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I was offered music too but being of an ever curious mentality I declined as I wanted to listen to the workings of the machine, not expecting to be deafened.
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The amount of noise depends on the imaging sequence being used, which determines the strength and slew rate of the gradient fields.
Echoplanar (EPI) and Diffusionweighted (DWI) sequences are the noisiest by tens of dB. Wikipedia will I am sure explain these modes in more space than I have time for!
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I've been pondering this circuit with a superconducting wire. Closing the switch should mean no current flows, thanks to the infinite time constant of the superconductor. Tried tweaking the resistance idea, but the current just slows down to a crawl. Any thoughts?
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I was stunned by how loud the gradient magnets were.
The one time I went for an MRI, the operator was concerned that I would be bothered by the noise.
I had been warned that MRIs were noisy, so I took the earplugs, but declined any music.
After the MRI, the operator asked if I was ok after the noise; I responded that I was interested by all the different tones and rhythms.
The only problem I had with the noise was when they told me they were finished, and would take me out, I couldn't understand what they said (earplugs, but no headphones). Anyway, the intent became self-evident a few seconds later...
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Closing the switch should mean no current flows, thanks to the infinite time constant of the superconductor
A characteristic of Superconductivity is that it has zero DC resistance.
If your circuit has a single turn, it will have minimal inductance, and current will rise extremely quickly (assuming that your power source has zero impedance). This is not pure DC.
Superconductors do not work so well with rapidly-changing magnetic fields, as flux vortices dissipate some of the energy.
- That's why the various superconductive power distribution systems tested around the world tend to deliver DC rather than AC.
https://encyclopedia.pub/entry/9252
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Closing the switch should mean no current flows, thanks to the infinite time constant of the superconductor.
If we assume the inductance is around 100nH (for a circuit with length 4 inches), and the inductance is the main impediment to current flow...
The current can be described as V=L dI/dt
- Where V is the applied voltage. We'll use 1V DC in this example
- L is the inductance in Henries (but we will use 100nH in this example = 10-7 H)
- I is the current flowing through the circuit
- dI/dt is the rate of increase of current with time, in Amps per Second.
So V=L dI/dt
1 = 10-7 dI/dt
or dI/dt = 107 Amps/second = 10 million Amps after 1 second, which is a huge amount of current!
This is enough to turn most thin superconductors into normal conductors, due to exceeding the critical current.
So the assumption that "no current flows" is a fallacy.