A metallic square plate is subjected to a uniform hydrostatic pressure (P). Choose the correct Mohr’s circle representing the state of stress at any point in the plate from the options given below. For the Mohr’s circle, normal stress is positive towards the right and shear stress is positive in the upward direction.

The correct option is the second option, which represents the Mohr's circle as a single point at $-P$ on the normal stress axis.

Here is the detailed step-by-step explanation:

1. Stress State Under Hydrostatic Pressure
When a square plate is subjected to uniform hydrostatic pressure $P$, the stress acting at any point on the plate is compressive and equal in all directions. Under standard sign conventions, tensile stress is considered positive (plotted to the right of the origin $O$) and compressive stress is considered negative (plotted to the left of the origin $O$).
Therefore, the normal stresses in the $x$ and $y$ directions are:
${\sigma }_x=-P$
${\sigma }_y=-P$
Since the pressure is uniform in all directions, there are no shear stresses acting on these planes:
${\tau }_{xy}=0$

2. Calculating the Center of Mohr's Circle
The center of Mohr's circle lies on the horizontal normal stress axis ($\sigma$-axis) at coordinates $({\sigma }_{\mathrm{avg}},0)$:
${\sigma }_{\mathrm{avg}}=\frac{{\sigma }_x+{\sigma }_y}{2}$
Substituting the values of ${\sigma }_x$ and ${\sigma }_y$:
${\sigma }_{\mathrm{avg}}=\frac{-P+(-P)}{2}=-P$
Thus, the center of Mohr's circle is located at $(-P,0)$.

3. Calculating the Radius of Mohr's Circle
The radius $R$ of Mohr's circle is calculated using the following formula:
$R=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$
Substituting the values of the stress components:
$R=\sqrt{{\left(\frac{-P-(-P)}{2}\right)}^2+0^2}=0$

4. Conclusion
Because the radius of the circle is $R=0$, the Mohr's circle degenerates into a single point on the horizontal axis.
Since the stress state is compressive ($\sigma =-P$), this point lies on the negative side of the normal stress axis (to the left of the origin $O$).
This corresponds to the diagram in the second option, showing a single point plotted at $-P$ on the horizontal axis.