Case 1: $B^{(0)}(\mathbf{x}) = (0,0,1)$

 

Using $G(\mathbf{x}') = \frac{-1}{4\pi r'}$ where $r' = \Vert \mathbf{x}' \Vert$ gives

$$F(\mathbf{x}';\mathbf{x}) = \iiint \frac{\partial^2 G}{\partial {z'}^2} \, dx' \, dy' \, dz'$$

$$= \iint \frac{\partial G}{\partial z'} \, dx' \, dy'$$

$$= \iint \frac{z'}{4\pi {r'}^3 } \, dx' \, dy'$$

$$= \frac{1}{4\pi} \mathrm{atan}\left(\frac{x'y'}{z'r'}\right)$$ (22)

where equation 30 from the Appendix was used in the last step.

By substituting this into equations 14 and 21, the field $B^{(1)}_z$ can be calculated for this case. Note that provided the $z$ axis corresponds to the main static (constant) $B_0$ field, then this case is all that is required for the field calculation of a non-conducting object.