Context
My problem is related to [1]. I do not dispute the solution in [1], however, it is not helping me to understand the problem that I face. I am working through Example 10.5 in Modern Electrodynamics by Zangwill. This problem pertains to the double-curl equation.
In this example, we integrate $\boldsymbol{\nabla}\times\left(\boldsymbol{\nabla}\times \mathbf{A}\right)$ to find $\mathbf{A}$ for a long straight cylindrical wire with radius $a$ which carries a uniform current density $\mathbf{j} = j\,\hat{\mathbf{z}}$. The problem progresses until we arrive to $\mathbf{A} = A_z(\rho)\,\hat{\mathbf{z}}$, where is given by the case structure in the following equation in terms of an as yet to be determined constant $C$. \begin{align} A_z(\rho) = \begin{cases} -\frac{1}{4}\,\mu_0\,j\,\rho^2, &\rho\leq a;~\text{and} \\ -\frac{1}{4}\,\mu_0\,j\,a^2 +C \,\ln{\frac{\rho}{a}}, &\rho\geq a. \end{cases} \end{align} Zangwill states that, the matching condition $$\hat{ \mathbf{n}}_2 \times \left[ \mathbf{B}_1 - \mathbf{B}_2 \right] = \mu_0\,\mathbf{K}(\mathbf{r}_S) . \tag{10} $$ applied at $\rho = a$ fixes $C$. Since, in general, $\mathbf{B}(\mathbf{r}) = \boldsymbol{\nabla} \times \mathbf{A}(\mathbf{r}) $, therefore, in this case, $$ \mathbf{B}(\mathbf{r}) = \begin{cases} + \frac{\mu_0\,j}{2}\,\rho \,\hat{ \boldsymbol{\varphi}} , & \rho \leq a;~\text{and} \\ - \frac{ C }{ \rho} \,\hat{ \boldsymbol{\varphi} } , & \rho \geq a. \end{cases} $$ Then, since $\mathbf{K}(\mathbf{r}_S) = K(a,\varphi,z)\,\hat{\mathbf{z}} $, therefore Eq. 10 evaluated at $\rho = a$ results in $$ \hat{ \boldsymbol{\rho} } \times \left[ - \frac{ C }{ a} - \frac{\mu_0\,j\,a }{2 } \right]\hat{ \boldsymbol{\varphi} } = \frac{ - 2\,C - \mu_0\,j\,a^2 }{2\, a} \,\hat{ \mathbf{z} } = \mu_0\,K(a,\varphi,z)\,\hat{\mathbf{z}} \tag{20}. $$ From Eq. 20, we find that the matching condition gives the following equation for $C$: $$ C = - \frac{1}{2} \,\mu_0\,j\,a^2 - a \mu_0\,K(a,\varphi,z) . \tag{30} $$ Now, Zangwill claims that $$ C = - \frac{1}{2} \,\mu_0\,j\,a^2 . \tag{40} $$ This means that either the surface-current density is identically zero, or that I have made a mistake along the way.
Questions
(1) Smaller question: Have I properly applied the matching conditions so to obtain the correct formula for $C$ (e.g., perhaps I have a sign error).
(2) Larger question: How does one obtain the surface-current density, $\mathbf{K}$, from the current density, $\mathbf{j}$?
My attempts
Option 1 (the correct option): On the one hand, I would say that the existence of a current density in the bulk of a material does not necessarily imply the existence of surface-current density.
Option 2:
Since $I = \int d\mathbf{S}\cdot \mathbf{j}$ therefore $$I = \int_{0}^{2\pi}\int_0^\infty \delta(\rho-a)\,\rho \,d\rho\,d\varphi \,\hat{\mathbf{z}}\cdot j\,\hat{\mathbf{z}}= 2\,\pi\, a \, j. $$ This has the outstanding issue that the units are inconsistent on the left- and right-hand side of the equation. Bearing this in mind I come to option 3.
Option 3:
$$K = \int_{0}^{2\pi}\int_0^\infty \delta(\rho-a)\,\rho \,d\rho\,d\varphi \,\hat{\mathbf{z}}\cdot j\,\hat{\mathbf{z}}= 2\,\pi\, a \, j. $$ This has its own outstanding issue. Namely, upon substitution into Eq. 30, I find that $$ C = \mu_0\,j\,a^2\left[ - \frac{1}{2} - 2\,\pi\, \right], $$ which is a result that is inconsistent with Zangwill's answer (i.e., Eq. 40).
Option 4:
I adopt from joshphysics in [1], that
\begin{align}
\mathbf j = \delta{\left(\rho -a\right)} \,\mathbf K .
\end{align}
However this leaves me scratching my head and asking for relief.
Bibliography
[1] Surface current and current density
[2] Zangwill, Modern Electrodynamics, pp. 31-2, 310, 323-4.
