Pascal Law - Definition, Derivation, Proof and Applications

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

• Δp is the hydrostatic pressure, which is the difference in measured pressure between two points on the fluid. This experienced pressure is a result of the fluid's total weight (in pascals Pa)
• ρ is the density of the fluid (in kilograms per cubic meter kg/ m 3 )
• g is the acceleration due to gravity (g = 9.8 m/ s 2 )
• Δh is the difference in the height of the measured points in the fluid container

Alternatively, Pascal's law can also be expressed as follows.

F = PA

where

• F is the force applied to the fluid
• P is the pressure transmitted throughout the fluid due to the force acting on it
• A is the area of cross-section of the container in which the fluid is stored

Derivation and Proof of Pascal’s Law

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 F₁, F_2, and F₃ can be given as follows.

F₁ = P_x × area of ABCD = P_x ad

F₂ = P_y × area of ABFE = P_y bd

F₃ = 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, 

F₁ sin θ = F₂

F₁ cos θ = F₃

Hence, from the equations of F₁, F₂ and F₃

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.

Applications of Pascal’s Law

• The hydraulic press is an equipment that provides a compressive force and was developed by the English inventor Joseph Bramah in 1795. The principle of working of a hydraulic press is analogous to that of a mechanical lever. A hydraulic press consists of two columns with different areas of cross-section connected to pistons. Since, according to Pascal's law, the force inside a contained system is constant, when a considerable mechanical force is applied on the smaller piston, the pressure is transmitted without any loss to the other larger piston, allowing us to lift very heavy objects. The same principle is found in the working of a hydraulic jack.
• Pascal's law can also be seen in the braking systems of a vehicle. In cars, when the brake is stepped on, the applied force is translated to a uniform pressure on the braking pads, which is in contact with the tire. The pressure on the pads further applies a greater magnitude of force on the tire, bringing the car to a stop.