How would you go about calculating the amount of Coulombs at a specific place on a superconducting wire?

As long as it's not right at the critical temperature, I would use a sufficiently low voltage to be able to use a garden variety 20A ammeter.

You can use a meter with an electromagnetic sensor(a loop), no actual contact with circuit is made. The current in the superconductor will create a magnetic field which will induce a current in the pick-up coil.

I would approximate the wire as infinitely thin, put a handy Gaussian surface around it to get the total charge, multiply by the area of the region of interest, and then divide by the total surface area of the wire (at this point abandoning the infinitely thin approximation).

The "number of Coulombs" at a "specific place"? What are you talking about? Are you asking how to get the rate at which the charge flows (current), or are you talking about obtaining the amount of charge on a certain length of wire? At any rate, if you're trying to get it theoretically, I don't think the fact that it's superconducting would matter. I'll just comment on the theoretical part, since if you know the theory, you can pretty much work an experimental portion around it to actually measure what you need.

I'll assume you're looking for the charge, not the current. Then, imagine the wire frozen in time, with all the particles in it frozen where they are. Assume the wire has a net charge, and the charges aren't moving.

I wouldn't "approximate the wire as infinitely thin" so much as assuming the entire wire to be infinitely long compared to the cross-sectional area. If you're looking for the charge on a specific length of rod, use Gauss' law and do the surface integral to find how much charge is, well, there. 'Surface integral' sounds scary, but it's easy.

Imagine an L long portion of the wire encircled by a Gaussian surface that's a cylinder with radius R. (Imagine the straight wire, with an L long portion of it with a cylinder wrapped around it, so that the axis of the wire and the axis of rotation of the cylinder is the same. Or, for the L long portion of the wire in question, the set of all the points in space that are exactly a perpendicular distance R from the wire's central axis , and just sort of cover up the two holes at either end with more area so that it looks like a cylinder. Fuck the forum software for not letting me draw this with text; fuck me for being so lazy and not uploading a picture.)

Anyway. Once you've got the drawing, really, all you need to do is do the surface integral, which is easy. If the electric field is denoted E ...

The electric flux becomes {integral sign}E dot dA, which then becomes
{integral sign}EdA * cos(theta). Working out all the thetas for each "portion" of the surface integral, you will then realize that the two circles on either end don't matter, since the electric field is perpendicular to the "dA" (careful on the direction of dA, it's perpendicular to where the "area" is facing). So the result is 2(pi)RLE for the electric flux. Multiply this by the Permittivity constant to get the desired total charge along the length L in question.

However. I doubt if you want that. So next, I'll assume you want the current.

Now imagine the charges flowing again. This time you'll need the Ampere-Maxwell law. Since this integral is a surface integral with the dot product of B and a ds (not a dA), you'll be drawing a loop, not a surface. So draw the loop around the wire. Look up the Ampere-Maxwell law. If the current stays steady and the electric flux stays steady ...

{surface integral sign}B dot ds = permeability constant * permittivity constant * time derivative of the electric flux + permability constant * enclosed current

... this big ugly thing will simplify to

{surface integral sign}B dot ds = permeability constant * enclosed current.

Since the electric flux is constant, and the time derivative of that is obviously zero (haha .. funny). So from there on, it's simple to just work out the surface integral (again, like before, draw a CIRCULAR loop, so that the B's and the ds's are perpendicular to one another and the integral is so simplified as to not even be funny .. make life easier for yourself) and obtain the current, if you know the magnetic field at the particular points in question, and obviously since the permeability constant is already known. Look it up in a book.

If you want to used experimental means to get this, I'd suggest you get the theoretical part down before you attempt it. That way, you'll know exactly what to measure so that you can complete your equation. Yay. Don't forget significant figures or your answer might go majorly off whack.

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