Exploring the Conservation Law for Magnetic Flux in Superconducting Circuits

[Drew Drowden]

There is a conservation law for superconductors " the flux through a superconducting ring cannot change". This can be shown using Faraday's law: -
E.M.F=-d/dt (BA) where B is the measured magnetic field passing through the ring and A is the area enclosed by the ring

therefore for a ring -d/dt(BA)= iR+Ldi/dt where L is the inductance of the ring and i is the current flowing through the ring

since R=0 in a superconductor, integrating we have BA+Li = a constant.

So the flux through a superconducting ring can't change.

In the above figure we have some superconducting material in its superconducting state. Magnetic flux from a permanent magnet or a coil is directed through the ring R. As the section P is rotated the flux through R is increased because the integration path wraps around R. What I don't understand is how the conservation law for magnetic flux applies to this diagram.

 

[Hesch]

What I don't understand is how the conservation law for magnetic flux applies to this diagram.

The conservation law does not apply, because you are disturbing the balance:

I don't quite understand the diagram, but when rotating the section P you will add/remove energy from the system, as an emf in P will be introduced. You will need some positive/negative torque to rotate P, thereby changing the energy in the system, included the magnetic energy.

I think that if there is no rotational friction in P, ( and no brakes ), P will by rotation "unwrap your path", until no magnetic energy resides in the system, and P will be rotating with a tremendous speed: All the magnetic energy has been converted to mechanical energy.

And I think that P will overshoot and will change rotational direction, that an oscillation in the system will be started so that the sum of mechanical energy and magnetic energy will be constant at all time.

( I've never tried it out ).

Compare it with a synchronous motor supplied with some dc-current. Turn it ( by hand ) 90 electical degrees, and let it go: It will start the same oscillation as above.

 

[Drew Drowden]

What you have said is perfectly correct. If P is left to move freely and S is frictionless then it would behave just as you have said. But if P is physically moved the conservation law must still apply even thought the balance is being disturbed. BA + Li= a constant means that if i is increased B must decrease etc. Of course you can increase i or B too much and the superconductor will cease to be a superconductor. The integration contour in this case is a big circle through C completed by a spiral through R. As we move P (as you have pointed out) an emf is generated and the current increases. The conservation law tells us that B through R must decrease but by following the dotted lines indicating the integration contour the current will flow the opposite direction through C therefore the flux through C must increase. I've only just noticed this about the current flows.

 

[Hesch]

BA + Li= a constant

I don't quite understand the diagram, but . . . . .

In the figure there are two rings/loops, C and R. I thought that both C and R were solid rings, and that P were connected to R through some brushes, but now you suggest that R actually is a coil/spiral with many turns?

Now, if S is rotated, you must physically/mechanically unwrap the coil. This will not change A ( in C or R ), but it will change L in R as L is proportional to N², where N is the number of turns in R.

Is BA regarded as the flux through C or R?

I think that the equation: BA + Li= a constant must be formulated in more details, the system as a whole being divided into two subsystems with a mutual inductance M.

 

[Drew Drowden]

R is a solid loop and flux (BA) flows through R. C is a second loop which branches off from R, loops around and is connected to P. P is free to rotate so the the point S can move around on the surface of R. R is a superconducting ring. What we have is a superconducting surface with two holes in it. The integration contour will form a helix as P is rotated around. That means we integrate around the ring for each turn. The integration contour is confined to the superconducting material and is represented by the dotted line in the diagram. As P is rotated the integration contour will be wound up. It's the integration contour, a mathematical construct , that takes the form of one big loop connected by a helix at R. Now the integration along this contour has many possible answers because you could integrate around any of the circles as many times as you like. So as P is rotated the flux through R increases and the flux through C decreases ( following the directions of the dotted lines).

Ultimately this superconducting surface will be morphed into a different shape (not shown) to create a device called a flux pump. By moving P you'll be able to pump magnetic flux into a superconducting coil represented by C. Such an invention would be very valuable because you could energise a superconducting coil without any power leads. Power leads would conduct heat into your coil compounding the difficulties of keeping it cold ( at the superconducting temperatures).

There is quite a lot I don't understand about the circuit. I don't understand how to apply the conservation law or how the currents are related to the integration contour.

 

[rumborak]

I'm not really understanding the setup, but I figured I'd drop the comment that Faraday's Law is not as universal as it's sometimes made out to be. It rests on certain assumption that can easily violated given the right scenario.

 

[Hesch]

I don't understand how to apply the conservation law or how the currents are related to the integration contour.

So as P is turned N rounds, you choose an integration contour ±N passes through R, but the current i will not follow this contour ( why should it, how will the current know about these N turns? ). Thus the equation: BA + Li= a constant will not be valid, and the conservative law will not apply.

As said, the energy in the system will apply so that Δ(magnetic energy) = Δ(mechanical energy), added by turning P. The magnetic energy can be calculated as

∫_volume (E_magn(x,y,z) ) dV, where E_magn is the magnetic energy density = ½*B(x,y,z)²/μ₀.

B(x,y,z) can be calculated by means of Biot-Savart law. The emf that arises turning P will change the current → change the flux → change the energy density.

I suggest you make some program, that will demonstrate it based on some model of the system. Let a computer calculate it (numerically) while you are asleep ( or while you are on holydays ).