Figure 3.

Two modes of the exact solution of the PE equation for the nonspherical geometries of Fig. 2 expressed as a function of $\xi =r_{ij}^2∕R_i{R}_j$. The source charge q_i is always at the midpoint of the interior: at (x = 0, z₀ = a/2) for the plates, and at (θ = 0, z₀ = a/2) for the spheres. In the longitudinal mode, the test charge q_j moves along the positive z direction, the other coordinate being fixed at x = 0 or θ = 0, respectively. In the transverse mode, the z coordinate of the test charge is kept constant, z = z₀ = a/2, while x (or θ for the spheres) increases. The solutions are expressed via dimensionless Ψ variable defined in Eq. 7. Note that Ψ = 1 corresponds to the exact solution for a perfect sphere, while Ψ = exp ( − 0.25ξ) (dashed green line) is the canonical GB.