# What determines the direction of current in a superconductor

Type I superconductors have no electric field nor magnetic field inside of them, when they are in the superconducting state. This means no voltage difference across any two points or regions inside of them. Yet they carry a current. This means the Cooper pairs (or the electrons responsible for the current) are moving in a particular direction.

My question is, what determine this particular direction, if there's no voltage involved?

Edit: If we have to apply a voltage initially, to get the Cooper pairs moving in a particular direction and then we remove that voltage and the current will still persist, then it would mean that the superconductor has a sort of "memory" in that it's possible to retrieve where (and how strong?) the voltage was applied? Does it also mean that the superconductor behaves the same way with and without the applied voltage? If so, that would be very strange and I'd like some clarifications.

• There is no electric field in a superconductor, but there can be a voltage across it. Recall that $\vec E=-\nabla\phi-\frac{\partial\vec A}{\partial t}$. – Chris Feb 3 '18 at 13:11
• Oh right I forgot about the vector potential. But still, according to wikipedia, even in the absence of voltage there's a current. Maybe originally someone has to put some voltage to "push" the Cooper pairs and then when the voltage is removed, the current persists. If that's true, it's very strange. It's like the superconductor has a memory of how the original voltage was applied. – AccidentalBismuthTransform Feb 3 '18 at 13:13
• Think of it more as an inertia. The tendency of things is to stay in motion unless something is stopping them, right? So the fact that there is no resistance means there's nothing to stop the current once it gets going. – Chris Feb 3 '18 at 13:15
• Great, so this answers the question. What sets the current direction is the initial applied voltage (there must be one, apparently, in order for a current to establish). Feel free to write an answer. Edit: Does it also mean that if we cool down a superconductor below Tc, there should be no current unless we apply a voltage difference? Or would microscopic voltage fluctuations take over and establish a long lasting current? – AccidentalBismuthTransform Feb 3 '18 at 13:19
• Added as an answer. Voltage fluctuations won't do much, since by their nature they are random, so they're just as likely to reduce any current as increase it. You can get persistent currents by other means, though, like bringing a permanent magnet nearby. See the Meissner effect. – Chris Feb 3 '18 at 13:27

There is no electric field in a superconductor, but there can be a voltage across it. Recall that:

$$\vec E = -\nabla\phi-\frac{\partial\vec A}{\partial t}$$

so the voltage need not be zero for the electric field to be zero.

Any superconducting loop has some inductance, so this voltage is required to get a current going. Since it has no resistance, a voltage is not required to keep the current going: in fact, keeping a voltage applied will continuously increase the current, as:

$$V=L\frac{dI}{dt}$$

(Eventually, if the voltage is not removed, the superconductor will reach a critical current. At this point, it becomes normally conductive. This can be bad, since an awful lot of heat is released all at once.)

Once the current is going, there's nothing to resist it, and so it can keep flowing even if the voltage is removed.

• In order for the 1st sentence to be true, you'd have to show that $\frac{\partial \vec A}{\partial t}$ could be different from $\vec 0$ I think. But even if the sentence is true, there can also be no voltage and a current (something not possible in an ordinary conductor). – AccidentalBismuthTransform Feb 3 '18 at 13:32
• @no_choice99 $\frac{\partial\vec A}{\partial t}$ is nonzero because of the changing magnetic field created by the current. It's really more the other way around- you apply a voltage and the magnetic potential changes in the right way to keep an electric field from forming in the conductor. And yes, there can be a persistent current without a voltage. That is only impossible in a normal conductor because all normal conductors have some resistance. You can think of it as like a damped spring (ordinary conductor) vs. an undamped one (superconductor). – Chris Feb 3 '18 at 13:36
• @no_choice99, consider this analogy: A block is moving across a frictionless plane. What direction is it moving? No force is required to keep it moving so you say you can't determine the direction. Actual answer: It's moving whatever direction it was pushed initially to start it moving. – The Photon Feb 3 '18 at 16:56