Boundary Conditions Magnetostatics

When a magnetic field crosses the interface between two different media, the field components do not change arbitrarily. Specific boundary conditions govern exactly how the normal and tangential components of the magnetic field must behave across the interface. These conditions follow directly from Maxwell's equations and are essential for solving field problems in multi-material electromagnetic devices such as transformers, motors, and magnetic shielding systems.

Normal Component of B: Gauss Law Applied

The normal boundary condition for B states that the normal component of B is continuous across any interface: B_1n = B_2n. This is derived directly from Gauss law for magnetism, which states that divergence of B = 0 everywhere, meaning the total magnetic flux out of any closed surface is zero. Applying this to a thin pillbox-shaped Gaussian surface straddling the interface, and shrinking its height to zero, immediately gives B_1n = B_2n.

In terms of H, this condition becomes mu_1 * H_1n = mu_2 * H_2n, because B = mu * H in each medium. This means the normal component of H is discontinuous at the interface whenever the two permeabilities differ. The ratio H_1n / H_2n = mu_2 / mu_1, so the material with higher permeability has a smaller normal H for the same flux.

Tangential Component of H: Ampere Law Applied

The tangential boundary condition for H is derived from Ampere's law in integral form applied to a thin rectangular path (Amperian loop) straddling the interface. In the static case with no surface current at the interface, the result is H_1t = H_2t: the tangential component of H is continuous. If a surface current density K (in A/m) exists at the boundary, the general condition is n x (H_1 - H_2) = K, or in scalar form H_1t - H_2t = K.

Since B_t = mu * H_t, the tangential component of B is discontinuous: B_1t / mu_1 = B_2t / mu_2, so B_1t / B_2t = mu_1 / mu_2. At an interface between air (mu_r = 1) and iron (mu_r = 5000), the tangential B in iron is 5000 times larger than in air. This flux refraction effect means that field lines bend sharply when crossing from low-permeability to high-permeability media, concentrating flux inside the high-mu material.

Refraction of Field Lines at the Interface

Using the two boundary conditions together, one can derive the law of refraction for magnetic field lines: tan(theta_1) / tan(theta_2) = mu_1 / mu_2, where theta_1 and theta_2 are the angles the field lines make with the interface normal in medium 1 and medium 2 respectively. This is analogous to Snell's law in optics. When a field line moves from a low-permeability medium into a high-permeability medium, it bends toward the interface (theta_2 approaches 90 degrees), meaning the lines tend to stay inside the high-mu material.

Given:
An interface between air (mu_r1 = 1) and silicon steel (mu_r2 = 4000).
In medium 1 (air): B_1 = 0.002 T at angle theta_1 = 60 degrees with interface normal.

Why this formula applies:
B_n1 = B_n2 (normal component of B is continuous)
H_t1 = H_t2 (tangential H is continuous, no surface current)
These give tan(theta_1)/tan(theta_2) = mu_r1/mu_r2

Formula:
B_n1 = B_1 * cos(theta_1) = B_n2
H_t1 = H_t2 --> B_t1/mu_1 = B_t2/mu_2 --> B_t2 = mu_r2/mu_r1 * B_t1

Substitution:
B_n1 = 0.002 * cos(60) = 0.002 * 0.5 = 0.001 T = B_n2
B_t1 = 0.002 * sin(60) = 0.002 * 0.866 = 0.001732 T
B_t2 = (4000/1) * 0.001732 = 6.928 T

Calculation:
|B_2| = sqrt(B_n2^2 + B_t2^2) = sqrt(0.001^2 + 6.928^2) = sqrt(0.000001 + 47.997)
|B_2| = sqrt(47.997) = 6.928 T

Final Answer:
B_n2 = 0.001 T, B_t2 = 6.928 T, |B_2| = 6.928 T
(Field is strongly refracted into the high-mu steel core)

Exam Tip: Normal B is always continuous across any interface (from div B = 0). Tangential H is continuous only when surface current K = 0 (from Ampere law). The tangential B is discontinuous in proportion to the ratio of permeabilities. Remember: B_n continuous; H_t continuous; B_t and H_n are NOT continuous.

• Normal B is always continuous: B_1n = B_2n (from div B = 0 via Gauss pillbox).
• Tangential H is continuous when surface current K = 0: H_1t = H_2t (from Ampere's law).
• General tangential H condition: n x (H_1 - H_2) = K; K is surface current density in A/m.
• Tangential B is NOT continuous: B_1t/mu_1 = B_2t/mu_2, so B_2t = (mu_2/mu_1) * B_1t.
• Refraction law: tan(theta_1)/tan(theta_2) = mu_1/mu_2, analogous to Snell's law in optics.

Quick Revision

• B_n is continuous across any interface (from div B = 0).
• H_t is continuous when no surface current: H_1t = H_2t.
• With surface current K: H_1t - H_2t = K (scalar, component along interface).
• B_t is discontinuous: B_1t/mu_1 = B_2t/mu_2.
• Refraction: tan(theta_1)/tan(theta_2) = mu_1/mu_2; flux lines bend into high-mu material.
• Trap: H_n is not continuous (mu_1 * H_n1 = mu_2 * H_n2); only B_n is continuous.
• In high-mu iron core next to air, nearly all flux stays inside the core due to refraction.