The pressure in a liquid at a given depth below the surface ………….. .

Explanation

In order to understand the increase in pressure with depth below the water surface, consider a water sample of cross sectional area in the form of a cylinder. Let `h_1 and h_2` be the depths from the air-water interface to level 1 and level 2 of the cylinder, respectively. Let `F_1` be the force acting downwards on level 1 and `F_2` be the force acting upwards on level 2, such that `F_1 = P_1 A and F_2 = P_2A`. Let us assume the mass of the sample to be m and under equilibrium condition, the total upwards force `(F_2)` is balanced by the total downward force `(F_1 + mg)`, in other words, the gravitational force will act downward which is being exactly balanced by the difference between the force `F_2 - F_1`.
`F_2 - F_1 = mg = F_G" " .....(1)`
where m is the mass of the water available in the sample element. Let `rho` be the density of the water then , the mass of water available in the sample element is
`m = rho V = rho A (h_2 - h_1)`
`V = A(h_2 - h_1)`

Hence, gravitational force, `F_G = rho A(h_2 - h_1)g`
On substituting the W value in equation (1) `F_2 = F_1 + mg implies P_2A = P_1A + rhoA(h_2 - h_1) g`
Cencelling out A on both sides,
`P_2 = P_1 + rho(h_2 + h_1)g " "....(2)`
If we choose the level 1 at the surface of the liquid (i.e., air water interface) and the level 2 at a depth .h. below the surface, then the value of atmospheric pressure (say `P_a`). In addition, the pressure `(P_2)` at a depth becomes P. Substituting these values in equation, we get
`P = P_a + rho gh" " .....(3)`

which means, the pressure at a depth h is greater than the pressure on the surface of the liquid, where `P_a` is the atmospheric pressure which is equal to `1.013 xx 10^5 Pa`.
If the atmospheric pressure is neglected or ignored then
`P = rho gh " ".....(4)`
for a given liquid, `rho` is fixed and g is also constant, then pressure due to the fluid column is directly proportional to vertical distance or height of the fluid column. This implies, the height of the fluid column is more important to decides the pressure and not the cross sectional or base area or even the shape of the container.
Let us consider three vessels of different shapes A, B and C as shown in figure. These vessels are connected at the bottom by a horizontal pipe. When they are filled with a liquid (say water), it occupies the same level even though the vessels hold different amounts of water. It is true because the liquid at the bottom of each section of the vessel experiences the same pressure.