Inductance and average induced emf of iron ring coil

Inductance of a toroid: $L = N^2/S = \mu_0\mu_r N^2 A/l$. Average emf = $L \times$ (change of current)/(time).

Data: $l = 0.5$ m, $N = 1000$, $d = 40$ mm, $\mu_r = 5000$.

Cross-sectional area

$$A = \frac{\pi d^2}{4} = \frac{\pi(0.04)^2}{4} = 1.2566\times10^{-3}\ \text{m}^2$$

Inductance

$$\begin{aligned} L &= \frac{\mu_0\mu_r N^2 A}{l} \\ &= \frac{4\pi\times10^{-7}\times5000\times(1000)^2\times1.2566\times10^{-3}}{0.5} \\ &= 15.79\ \text{H} \end{aligned}$$

Average induced emf

$$\begin{aligned} e_{avg} &= L\frac{\Delta I}{\Delta t} = 15.79\times\frac{10 - 2}{5\times10^{-3}} \\ &= 15.79\times1600 = 25266\ \text{V} \end{aligned}$$

Answer: L ≈ 15.79 H; average emf ≈ 25.3 kV (it opposes the rise of current, by Lenz's law). The very large value comes from the high μr and fast change of current; in practice saturation would reduce L.