 Solution

For the closed untapered torus, apply the magnetostatic Ampère law to a circular path through the core. It links all $N$ turns, each carrying electric current $I$:Since $L\gg R$, variation of the path length and of the tangential magnetic field intensity across the section is negligible. Thus $H_{\mathrm{in}}L\simeq NI$, giving, within this thin-torus approximation,The units are $\mathrm{A}{\mathrm{m}}^{-1}$. The core magnetic flux density is $B_{\mathrm{in}}={\mu }_0\mu NI/L$ in the linear material model.