Pre-Quantum Electrodynamics

Electrostatic fields: discontinuous at location of suface charge.
Magnetostatic fields: discontinuous at location of surface current.

Equation \ref{eq:DivBisZero}: \(\oint {\bf B} \cdot d{\bf a} = 0\) applied to wafer-thin pillbox straddling surface: normal component \[ B^{\perp}_{above} = B^{\perp}_{below}. \label{Gr(5.72)} \] Tangential component: amperian loop of side length \(l\) perpendicular to surface current: \[ \oint {\bf B} \cdot d{\bf l} = (B^{\parallel}_{above} - B^{\parallel}_{below}) l = \mu_0 I_{enc} = \mu_0 K l, \] and therefore \[ B^{\parallel}_{above} - B^{\parallel}_{below} = \mu_0 K \label{Gr(5.73)} \] So: component of \({\bf B}\) that is parallel to surface but perpendicular to current flow is discontinuous, whereas the one parallel to the flow is continuous. In vector notation:

where \(\hat{\bf n}\) points upwards. For the vector potential, the relations are

This can be seen first from the condition \({\boldsymbol \nabla} \cdot {\bf A} = 0\), which guarantees that the normal component is continuous. Second, \({\boldsymbol \nabla} \times {\bf A} = {\bf B}\) leads to \[ \oint {\bf A} \cdot d{\bf l} = \int {\bf B} \cdot d{\bf a} = \Phi, \] where the loop is vanishingly small and straddles the surface. Since the flux then goes to zero, the tangential components of \({\bf A}\) are also continuous.

However, the derivative of \({\bf A}\) inherits the discontinuity of \({\bf B}\): explicitly,

\begin{align} {\bf B}_{above} - {\bf B}_{below} &= {\boldsymbol \nabla} \times {\bf A}_{above} - {\boldsymbol \nabla} \times {\bf A}_{below} \nonumber \\ &= \left| \begin{array}{ccc} \hat{\bf x} & \hat{\bf y} & \hat{\bf z} \\ \partial_x & \partial_y & \partial_z \\ A_{x, above} & A_{y, above} & A_{z, above} \end{array} \right| - \left| \begin{array}{ccc} \hat{\bf x} & \hat{\bf y} & \hat{\bf z} \\ \partial_x & \partial_y & \partial_z \\ A_{x, below} & A_{y, below} & A_{z, below} \end{array} \right| \end{align}

We put the normal direction along \(\hat{\bf z}\) and the current along \(\hat{\bf x}\). Since the normal component is continuous, \(\partial_{x,y} A_z\) is the same above and below, and we can drop these terms. The \(\hat{\bf z}\) term vanishes since it just involves the difference of \(B^{\perp}\), which is the same above and below. Similary, the \(\hat{\bf x}\) term is the magnetic field parallel to the surface current, which also isn't discontinuous. What is left is \[ %\hat{\bf x} (-\partial_z A_{y,above} + \partial_z A_{y, below}) + \hat{\bf y} (\partial_z A_{x, above} - \partial_z A_{x, below}) %+ \hat{\bf z} (\partial_x A_{y, above} - \partial_x A_{y below} - \partial_y A_{x, above} + \partial_y A_{x, below}) = \mu_0 K \hat{\bf x} \] Therefore, reidentifying the normal component, we get