# Magnetic field produced by a finite conductive wire vs. superconducting wire

As I learn more about the amazing properties and possibilities of superconductors, I wanted to understand and briefly compare the differences of when a copper wire of finite length($L_f$) creates a magnetic field($B_w$) to a superconductor.

1) Does the Biot-Savart Law apply equally to both? Or would the formula change for a superconductor? In calculating the magnitude of $B_w$ at any point away from the conductor.

2) The distribution of the magnetic field($B_w$) produced by a copper conductor would somewhat look like this:

There is a magnetic field also within the wire, in contrast, a superconductor would expel $B_w$ within, would the result be a stronger/denser field lines at certain regions?

3) Would Type I and Type II superconductors greatly differ with respect to the two questions above?

## 1 Answer

The Biot-Savart law applies. The 'perfect diamagnetism' of a superconductor means that only the outer skin of the superconductor carries current. This may, or may not, be a significant difference; there is a similar effect in metals, which is frequency-dependent: at 2 MHz, for instance, only the outer 50 microns of a copper wire is conducting (this is called the 'skin effect'). At low frequency, though, a copper wire has nearly uniform current distribution (no concentration at the surface), so some B field is inside the bulk of the wire, and the inductance value for a copper wire is thus slightly higher than that of a superconductor (even though the magnetic field OUTSIDE the wire surface will be the same). The absence of B field inside a superconductor means all the current must be on the surface (to a depth of usually a fraction of one micron).

Type I superconductors, if current gets high enough, will allow some internal magnetic field and become non-superconductors at the same time. Type 2 superconductors, will allow some internal magnetic field while remaining superconducting (but at higher field they also become non-superconducting). So, a type 2 superconductor may, in a limited current range, have some internal B field while remaining a superconductor. As long as the conductor shape is cylindrical, though, field outside the wire is the same for copper, type 1 superconductor, and type 2 superconductor.