Problem 3.338

A thin ring made of a magnetic has a mean diameter $d=30 \mathrm{~cm}$ and supports a winding of $N=800$ turns. The crosssectional area of the ring is equal to $S=5.0 \mathrm{~cm}^{2}$. The ring has a cross-cut of width $b=2.0 \mathrm{~mm}$. When the winding carries a certain current, the permeability of the magnetic equals $\mu=1400 .$ Neglecting the dissipation of magnetic flux at the gap edges, find: (a) the ratio of magnetic energies in the gap and in the magnetic; (b) the inductance of the system; do it in two ways: using the flux and using the energy of the field.