Deep Dive: Boundary Conditions
Discontinuities in $\mathbf{B}$ and the continuity of $\mathbf{A}$
Just as the electric field suffers a discontinuity at a surface charge, the magnetic field is discontinuous at a surface current. However, while the normal component of $\mathbf{E}$ is discontinuous, for $\mathbf{B}$ it is the tangential component that changes.
1. The Magnetostatic Triangle
To understand boundary conditions, we first review the connections between current density $\mathbf{J}$, magnetic field $\mathbf{B}$, and vector potential $\mathbf{A}$. Hover over the arrows to see the equations.
This diagram reveals the "missing link" in magnetostatics. While we often calculate $\mathbf{B}$ from $\mathbf{A}$ via curl, calculating $\mathbf{A}$ from $\mathbf{B}$ is rarely required in standard problems.
2. Boundary Conditions for $\mathbf{B}$
Consider a surface carrying a current density $\mathbf{K}$. We apply Maxwell's equations in integral form to a small pillbox and an Amperian loop straddling the surface.
Normal Component
Since $\nabla \cdot \mathbf{B} = 0$, the net flux through a wafer-thin pillbox is zero: $$\oint \mathbf{B} \cdot d\mathbf{a} = 0 \implies B_{above}^\perp - B_{below}^\perp = 0$$
The normal component of the magnetic field is always continuous.
Tangential Component
Ampère's law ($\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$) applied to a rectangular loop perpendicular to the current yields: $$(B_{above}^\parallel - B_{below}^\parallel)l = \mu_0 K l$$
The tangential component is discontinuous by an amount proportional to the surface current.
Combined into a single vector formula, the discontinuity is: $$\mathbf{B}_{above} - \mathbf{B}_{below} = \mu_0 (\mathbf{K} \times \mathbf{\hat{n}})$$
3. Boundary Conditions for $\mathbf{A}$
Like the scalar potential $V$ in electrostatics, the vector potential $\mathbf{A}$ is continuous across any boundary:
This continuity is guaranteed by $\nabla \cdot \mathbf{A} = 0$. However, the derivative of $\mathbf{A}$ inherits the discontinuity of $\mathbf{B}$: