Two-coil magnetic circuit, current for 1.2 T

The mmf of the two coils acts together on one core. The net mmf must equal the mmf needed for the iron and the gap at 1.2 T.

Given: $A = 16$ cm$^2$, $l_g = 0.06$ cm, $l_c = 40$ cm, $\mu_r = 6000$, $B = 1.2$ T, $N_1 = 6000$ (current $I$, unknown), $N_2 = 500$ (current 2 A).

Mmf needed

$$\begin{aligned} F_c &= \frac{B l_c}{\mu_0\mu_r} = \frac{1.2\times0.40}{4\pi\times10^{-7}\times6000} = 63.66\ \text{AT} \\ F_g &= \frac{B l_g}{\mu_0} = \frac{1.2\times6\times10^{-4}}{4\pi\times10^{-7}} = 572.96\ \text{AT} \\ F &= 636.6\ \text{AT} \end{aligned}$$

The flux is $\Phi = BA = 1.2\times16\times10^{-4} = 1.92$ mWb.

Net mmf of the coils

The current directions are drawn in the figure, which is not available. Coil 2 supplies $N_2I_2 = 500\times2 = 1000$ AT, which already exceeds the 636.6 AT needed. A positive $I$ in coil 1 must therefore oppose coil 2. Taking the coils as opposing:

$$\begin{aligned} N_1I - N_2I_2 &= 636.6 \\ 6000\,I &= 636.6 + 1000 = 1636.6 \\ I &= 0.2728\ \text{A} \end{aligned}$$

(If the coils aided each other, the required current would be negative with magnitude 0.0606 A, i.e. coil 1 would have to be reversed.)

Answer: $I \approx 0.273$ A