converge with different speeds. For similar modeling projects, it may be worthwhile to explore these different bases. However, for our purposes it was sufficient to express equation (1) in simple Cartesian coordinates and employ the resultant basis. As usual in solving separable PDE’s, we write $A_y$ and $A_z$ as a product of three single-variable functions: $A_x(x,y,z) = H(x)I(y)J(z)$ and $A_z(x,y,z) = K(x)L(y)M(z)$. Substituting these expressions into equation (1) and introducing separation constants, we have that\

$$\begin{aligned}
\frac{H''(x)}{H(x)} = -\lambda_x,  \frac{J''(z)}{J(z)} = \lambda_z,  \frac{I''(y)}{I(y)} = \lambda_x - \lambda_z = \lambda_y \\
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\frac{K''(x)}{K(x)} = -\lambda_x,  \frac{M''(z)}{M(z)} = \lambda_z,  \frac{L''(y)}{L(y)} = \lambda_x - \lambda_z = \lambda_y      \end{aligned}$$

Note that corresponding separation constants are the same for $A_y$ and $A_z$. After deriving the full form of $A_y$ and $A_z$, taking the curl to calculate $\mathbf{B}$, and taking the divergence and curl of $\mathbf{B}$, one can see that this must be the case in order to satisfy Maxwell’s equations. In many instances of solving separable PDE’s, one might be able to appeal to boundary conditions to determine that $\lambda_x$ and $\lambda_z$ are constrained to have a certain sign. However, for the sake of remaining general, we are not able to impose boundary conditions. Indeed, 