Iron ring with air gap: current, inductance, energy

Given: mean diameter $= 15$ cm, $A = 10$ cm$^2 = 10^{-3}$ m$^2$, $N = 200$, $l_g = 2$ mm, $B = 1$ T, $\mu_r = 500$.

Lengths and flux

$$l_i = \pi\times0.15 - 0.002 = 0.4692\ \text{m},\qquad \Phi = BA = 10^{-3}\ \text{Wb}$$

Mmf

$$\begin{aligned} F_i &= \frac{B l_i}{\mu_0\mu_r} = \frac{0.4692}{4\pi\times10^{-7}\times500} = 746.8\ \text{AT} \\ F_g &= \frac{B l_g}{\mu_0} = \frac{0.002}{4\pi\times10^{-7}} = 1591.5\ \text{AT} \\ F &= 2338.4\ \text{AT} \end{aligned}$$

Exciting current

$$I = \frac{F}{N} = \frac{2338.4}{200} = 11.69\ \text{A}$$

Inductance

$$L = \frac{N\Phi}{I} = \frac{200\times10^{-3}}{11.69} = 0.0171\ \text{H}$$

Stored energy

$$W = \tfrac{1}{2}LI^2 = \tfrac{1}{2}\times0.0171\times11.69^2 = 1.169\ \text{J}$$

Check: $W = \tfrac12 F\Phi = \tfrac12\times2338.4\times10^{-3} = 1.169$ J. About 68% of this energy is stored in the 2 mm air gap.

Answer: $I \approx 11.69$ A; $L \approx 17.1$ mH; $W \approx 1.17$ J