Strong magnetic field and critical current

A superconductor stays superconducting only while the magnetic field at every point is below the critical field $H_c$. A strong field destroys superconductivity, and the critical field depends on temperature:

$$H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right]$$

where $H_0$ is the critical field at 0 K.

Effect of a strong magnetic field

• For $H < H_c$: flux is expelled (Meissner effect), resistance is zero.
• For $H \ge H_c$ (type-I): flux penetrates completely and the material becomes normal, with normal resistance.
• For type-II: flux starts entering at $H_{c1}$ (mixed state), and superconductivity is fully lost at $H_{c2}$.
• A stronger field means a lower temperature is needed to stay superconducting, so $T_c$ falls as $H$ rises.

  H
 H0 |*
    |   *
    |  Super-   *      Normal
    |  conducting  *
    |                *
    +-----------------*---- T
    0                 Tc

Critical current (Silsbee's rule)

A current in the wire itself produces a magnetic field. If this self-field at the surface reaches $H_c$, superconductivity is destroyed. The current at which this happens is the critical current $I_c$.

Consider a long straight superconducting wire of radius $r$ carrying current $I$:

        H (circles around wire)
      .-~~~~~-.
     /  .---.  \
    |  ( I ⊙ )  |   radius r
     \  '---'  /
      '-~~~~~-'

By Ampere's circuital law, the field at the surface is

$$\oint H\,dl = I \;\Rightarrow\; H \cdot 2\pi r = I \;\Rightarrow\; H = \frac{I}{2\pi r}$$

Superconductivity is lost when $H = H_c$, so

$$\begin{aligned} I_c &= 2\pi r H_c \\ &= 2\pi r H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] \end{aligned}$$

The critical current density is

$$J_c = \frac{I_c}{\pi r^2} = \frac{2H_c}{r}$$

If an external field $H_a$ is also applied (along the same direction as the self-field at the surface), the total must stay below $H_c$:

$$H_a + \frac{I}{2\pi r} \le H_c \;\Rightarrow\; I_c = 2\pi r\,(H_c - H_a)$$

So $I_c$ falls linearly as the applied field increases, and falls as temperature rises. This sets the maximum current a superconducting cable or magnet winding can carry.