Consider that a force $\text{P}$ is acting on the surface of a half-space (Boussinesq's problem). The expression for the vertical stress $\left(\sigma_{z}\right)$ at any point $(r, z)$, within the half-space is given as,
\[\sigma_{z}=\frac{3 P}{2 \pi} \frac{z^{3}}{\left(r^{2}+z^{2}\right)^{\frac{5}{2}}}\]
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 $\sigma_{z}$ ?
- $z^{2}=\frac{3}{2} r^{2}$
- $z^{3}=\frac{3}{2} r^{2}$
- $z^{2}=\frac{5}{2} r^{2}$
- $z^{3}=\frac{5}{2} r^{2}$