Question #43390

A toroidal solenoid of $N$ turns carries a steady current $I \text{ (A)}$. The toroid has a rectangular cross-section with inner radius $a \text{ (m)}$ and outer radius $b \text{ (m)}$. It contains a magnetic core with a radially varying relative permeability $\mu_r(r) = \beta r$, where $\beta \text{ (m}^{-1}\text{)}$ is a positive constant and $r \text{ (m)}$ is the radial distance from the axis of symmetry. A narrow radial air gap of uniform width $g \text{ (m)}$ is cut through the entire cross-section of the core. Assuming fringing fields are negligible, which of the following expressions gives the exact magnetic flux density $B(r) \text{ (T)}$ within the air gap?