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.