Introduced by the French physicist and mathematician Blaise Pascal between 1653 and 1663, Pascal's law states that if a pressure is applied to one part of an incompressible fluid contained in a container, the applied pressure is distributed throughout the fluid so that a uniform change is experienced on all points of the fluid. Pascal's law is also known as Pascal's principle or the principle of transmission of fluid pressure. Moreover, the applied external pressure is transmitted throughout the fluid without any loss in magnitude. The expression of Pascal's law can be given as follows.
Δp = ρg. Δh
where
Alternatively, Pascal's law can also be expressed as follows.
F = PA
where
Let us consider a right-angled prism of very minute dimensions immersed in a fluid with a density value ‘ρ’. Provided that the overall size of the prism is small, all points on the prism can be considered to be of the same depth. And as a result of the prism being in a uniform depth, the force of gravity, given by g, acting on the prism is the same across all points on the prism. Take into consideration the three surfaces ABCD, ABFE, EFCD of the prism. Let
AD = BC = ∂s = a
AE = BF = ∂y = b
ED = FC = ∂x = c
AB = EF = ∂z = d
Thus, let the surface area of these surfaces be given by ad, bd, and cd, respectively. Let the pressure acting on the surfaces ABCD, ABFE, and EFCD be given by P x , P y , P
Due to the pressure acting on the liquid, a force that acts perpendicular to the surface of the prism is observed. Let Px exert force F 1 on the surface ABCD, P y exerts a force F 2 on the surface ABFE, and P z exerts a force F 3 on the surface EFCD.
The value of the forces F_{1}, F_{2,} and F_{3} can be given as follows.
F_{1} = P_{x} × area of ABCD = P_{x} ad
F_{2} = P_{y} × area of ABFE = P_{y} bd
F_{3} = P_{z} × area of EFCD = P_{z} cd
The net force acting on the prism due to all the forces F1, F2, and F3 is zero and will keep the prism in a state of equilibrium.
We know that,
Sin θ= ∂y/ ∂s = b/ a
Cos θ = ∂x/ ∂s = c/ a
Therefore,
F_{1} sin θ = F_{2}
F_{1} cos θ = F_{3}
Hence, from the equations of F_{1}, F_{2} and F_{3}
P_{x} ad (b/ a) = P_{y} bd
P_{x} ad (c/ a) = P_{z} cd
As a result, it can be established that P_{x} = P_{y} and P_{x} = P_{z}
This, in turn, implies that
P_{x} = P_{y} = P_{z}
Hence proving the principle of transmission of fluid-pressure, i.e. Pascal's law.