Magnet: 1m iron ring with 1cm gap

assume that the fringing fields around the gap can be neglected then from the continuity of the flux you have $Phi=BA$ through every cross-section $A$, be it in the gap or in the core.

Since $B = mu_gmu_0 H_g = mu_cmu_0 H_c$ where $mu_c$ is the core's permeability and for air gap $mu_g=1$; also from Ampere's law we get $NI=H_cell_c + H_g ell_g$. Now let $mu^*$ be an equivalent or average permeability that characterizes the core+gap so that $NI = frac{B}{mu^*mu_0 }(ell_c+ell_g)$. Then $$frac{B}{mu_c mu_0}ell_c + frac{B}{mu_g mu_0}ell_g = frac{B}{mu^*mu_0 }(ell_c+ell_g) frac{1}{mu_c }ell_c + frac{1}{mu_g }ell_g = frac{1}{mu^*}(ell_c+ell_g) mu^* = frac{ell_c+ell_g}{frac{ell_c}{mu_c } + frac{ell_g}{mu_g }}$$ Also $$ B=frac{mu^*mu_0 NI}{ell_c+ell_g} = frac{mu_0 NI}{frac{ell_c}{mu_c } + frac{ell_g}{mu_g }}$$

This is your formula for an air gap, $mu_g=1$