Question #43421

A toroid with a rectangular cross-section has an inner radius $a = 0.1 \text{ m}$, an outer radius $b = 0.2 \text{ m}$, and a height $h = 0.05 \text{ m}$. It is uniformly wound with $N = 1000$ turns of wire carrying a steady current $I = 2 \text{ A}$. The core of the toroid is filled with a non-linear ferromagnetic material whose magnetic permeability varies radially as $\mu(r) = \mu_0 \left( \frac{b}{r} \right)$, where $r$ is the distance from the central axis of the toroid and $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$. Calculate the total magnetic flux $\Phi$ passing through the cross-section of the toroid.