Exercise 4.3.1: Toroid: reluctance and inductance

$\Phi =NI/Z$ is linear in $I$ because $Z$ does not depend on the current (linear material). The flux linkage is $\lambda =N\Phi =N^{2}I/Z$, so

$W=\frac{1}{2}L{I}^2=\frac{1}{2}\times 0.0402\times 1=20.1\mathrm{mJ}$. With ${\mu }_r=80$ both $Z$ and $L$ change tenfold: $L=4.02\mathrm{mH}$, $W=2.0\mathrm{mJ}$, one tenth.

With the field: $w=B^2/(2{\mu }_0{\mu }_r)=1.005^2/(2\times 4\pi \times 10^{-7}\times 800)=503J/{\mathrm{m}}^3$ times the core volume $Sl=4\times 10^{-5}{\mathrm{m}}^3$ gives $20.1\mathrm{mJ}$, the value of (c). Note the counter-intuitive energy result: a core with smaller ${\mu }_r$ stores less energy at fixed current (it has a smaller $L$), while at fixed $B$ the opposite is true ($w=B^2/2\mu$), which is why energy-storing inductors are gapped (Exercise 4.3.2).