Magnetic Circuit Analysis

6. The magnetic flux density in the air gap is $B_g = 0.8 \text{ T}$, and the current $I_2 = 8 \text{ A}$. Determine the current $I_1$. The given parameters are: $\ell_1 = 150 \text{ cm}$, $\ell_2 = 20 \text{ cm}$, $\ell_g = 250 \text{ cm}$, $\delta = 3 \text{ mm}$, $N_1 = 4000$, $N_2 = 1000$, and the cross-sectional area of the core is uniform and equal to $S = 25 \text{ cm}^2$. Assume that the magnetic permeability of the core is constant and equal to $\mu_r = 2000$. Note that the length of the middle section is $2\ell_2 + \delta$.

This question includes visual content: The image shows a schematic of a rectangular magnetic core with a center vertical branch containing an air gap. The core has two windings labeled $N_1$ and $N_2$ at the top and bottom of the left rectangular section, carrying currents $I_1$ and $I_2$ respectively. The lengths of various core sections are labeled: $\ell_1$ for the far-left outer vertical section, $\ell_3$ for the far-right outer vertical section, $\ell_g$ for the middle section (including the air gap), and $\delta$ for the width of the air gap itself.