Electromagnetic induction > Magnetic flux conservation. Superconductors in magnetic field > Problem 11.5.10

Statement

$11.5.10.$ A superconducting rod of section $σ$ and length $l$ flies at a constant speed through a coil of section $S$ and length $h$ made of superconducting wire. Draw a graph of the dependence of the current in the coil on the position of the rod if the coil is short-circuited and the initial current in it is $I_0$. Consider the cases: a) $l > h$; b) $l < h$. Ignore the edge effects.

Solution

The superconducting rod displaces the magnetic field from itself, which reduces the inductance of the coil (the area of coverage by the magnetic field decreases), and from the law of conservation of flux this leads to an increase in the current in the coil.:

$$L(x) = \frac{\mu_0 N^2S}{h} - \frac{\mu_0 N^2\sigma}{h}\frac{x}{h}$$

$$L_0 = \frac{\mu_0 N^2S}{h}$$

$$l_0 L_0 = LI$$

$$I(x) = \frac{I_0}{1 - \frac{\sigma}{S}\frac{x}{h}}$$

In the case of point a), the rod can fill the entire solenoid, so:

a)
$$I_{max} = \frac{I_0}{1 - \frac{\sigma}{S}}$$

And in point b) it cannot:

b)

$$I_{max} = \frac{I_0}{1 - \frac{\sigma}{S} \frac{l}{h}}$$

Answer

a)
$$I_{max} = \frac{I_0}{1 - \frac{\sigma}{S}}$$

b)

$$I_{max} = \frac{I_0}{1 - \frac{\sigma}{S} \frac{l}{h}}$$