= Solution
At an interface with unit normal $\hat{\mathbf n}$ pointing from medium 1 to medium 2, the magnetic <boundary conditions> are
$$
\boxed{\hat{\mathbf n}\cdot(\mathbf B_2-\mathbf B_1)=0,\qquad\hat{\mathbf n}\times(\mathbf H_2-\mathbf H_1)=\mathbf K_f.}
$$
The first follows from a pillbox applied to $\nabla\cdot\mathbf B=0$. The <tangential magnetic field jump from a free surface current> follows from a shrinking rectangular Ampère loop; $\mathbf K_f$ is the free surface current. With no free sheet current, normal $B$ and tangential $H$ are continuous. In linear isotropic media $\mathbf B=\mu\mathbf H$, so neither all components of $B$ nor all components of $H$ generally remain continuous.
For <magnetic field-line refraction at a permeability interface>, with angles measured from the normal, these conditions give $\tan\theta_2/\tan\theta_1=\mu_2/\mu_1$. They determine the bending of <magnetic flux> at a material interface. In a slender core electromagnet, flux continuity and <Ampère's law> lead to the magnetic-circuit estimate
$$
\Phi\simeq\frac{NI}{\sum_j l_j/(\mu_jA_j)}.
$$
An air gap can dominate the reluctance even when the iron core has high <magnetic permeability>. <Boundary conditions> also determine <demagnetizing fields>; fringing, <saturation> and <hysteresis> limit the simple linear circuit model. A high-<magnetic permeability> material redirects flux, whereas a perfect-conductor boundary is a different physical assumption.<Electromagnetism by David Tong>[Chapter 7: magnetic fields in matter and interface conditions]{ref}
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