Pascal's Principle

Pascal's principle states that a change in pressure applied to an enclosed fluid at rest is transmitted undiminished to every point in the fluid and to the walls of the container.

Consider an external pressure change $\Delta P$ applied at one location. Pascal's principle states:

$$\Delta P_{\text{transmitted}} = \Delta P$$

This equality holds at every point throughout the enclosed fluid.

Physical Basis

The principle follows from the equilibrium condition for fluids at rest. A fluid cannot sustain shear stresses, so at any point the pressure acts equally in all directions. When an external pressure is applied, the fluid must reach a new equilibrium state. For the fluid to remain static, pressure changes must balance throughout the system.

This principle applies to both incompressible liquids and compressible gases once the system reaches equilibrium. For incompressible fluids, equilibrium is established nearly instantaneously. For compressible fluids, the system adjusts until pressure changes are uniform.

Application to a Piston

For a piston of area $A$ with applied force $F$, the pressure increase at the piston surface is:

$$\Delta P = \frac{F}{A}$$

This same $\Delta P$ acts on the container walls and all surfaces in contact with the fluid, providing the foundation of hydraulic systems.