# A superconducting ring in a magnetic field

**Electricity and Magnetism**Level 4

A ring of radius \(a=50 mm\) is made from a thin wire with circular cross section of radius \(b=1 mm\). The ring was placed in a homogeneous magnetic field of induction \(B=0.5 mT \) and perpendicular to the ring's plane. The ring was cooled down until it became superconducting ( zero resistivity). Then the magnetic field was switched off. Find the final electric current **in Amps** in the ring. In order to solve this problem you need to consider the magnetic flux created by the current in the ring (self inductance). Thus in addition to the magnetic flux created by the external magnetic field, we have
\[ \Phi_{self}= L I \]
the magnitude L is called inductance and for a ring, it can be determined by means of the formula
\[ L=\mu_{0} a (\ln(\frac{8 a}{b})-2).\]

**Details and assumptions**

\[\frac{\mu_{0}}{4\pi}= 10^{-7} H/m\]