Consider that a force P is acting on the surface of a half-space (Boussinesq's problem). The expression for the vertical stress ($\sigma_{z}$) at any point (r, z), within the half-space is given as, where, r is the radial distance, and z is the depth with downward direction taken as positive. At any given r, there is a variation of $\sigma_{z}$ along z, and at a specific z, the value of $\sigma_{z}$ will be maximum. What is the locus of the maximum $\sigma_{z}$?

(A) $z^{2}=\frac{3}{2}r^{2}$
(B) $z^{3}=\frac{3}{2}r^{2}$
(C) $z^{2}=\frac{5}{2}r^{2}$
(D) $z^{3}=\frac{5}{2}r^{2}$