Pascal's law — pressure transmits equally

WHAT is Pascal's Law?

Two ideas hide inside this one sentence:

1. Same pressure everywhere on a level — in static fluid at one height, pressure is equal in all directions and at all points on that level.
2. Changes transmit fully — if you raise the pressure at one point by $\Delta p$, every point rises by the same $\Delta p$.

WHY is it true? (Derivation from first principles)

Setup: Take a tiny wedge-shaped fluid element (a triangular prism) of thickness, inside a static fluid. Let the three faces have pressures $p_x$ (on the vertical face), $p_y$ (on the horizontal face), $p_n$ (on the slanted face). The element is so small we can ignore gravity (volume $\to 0$ faster than area).

Let the slanted face make angle $\theta$, with areas:

• vertical face area $= dA_x$
• horizontal face area $= dA_y$
• slant face area $= dA_n$

Geometry of the wedge gives: $dA_x = dA_n \sin\theta, \qquad dA_y = dA_n \cos\theta$

Horizontal balance (x): pressure force on vertical face = horizontal component of force on slant face $p_x\,dA_x = p_n\,dA_n \sin\theta$

Why this step? Pressure pushes perpendicular into a surface; only the horizontal component of the slant-face force opposes the vertical-face force.

Substitute $dA_x = dA_n\sin\theta$: $p_x\,dA_n\sin\theta = p_n\,dA_n\sin\theta \;\Rightarrow\; \boxed{p_x = p_n}$

Vertical balance (y): similarly $p_y\,dA_y = p_n\,dA_n\cos\theta \;\Rightarrow\; p_y = p_n$

Therefore: $\boxed{p_x = p_y = p_n}$

Now the transmission part. Recall the hydrostatic relation (from a fluid column of height $h$): $p = p_0 + \rho g h$

If we increase the external/applied pressure $p_0 \to p_0 + \Delta p$, then at every depth $h$: $p_{\text{new}} = (p_0 + \Delta p) + \rho g h = p_{\text{old}} + \Delta p$

The $\rho g h$ term is unchanged (same fluid, same heights), so the increase $\Delta p$ is identical at every point. That is Pascal's law.

HOW to use it — worked examples

Common mistakes (steel-manned)

Flashcards

Connections

• Hydrostatic pressure — p = p0 + ρgh — supplies the $\rho g h$ term we held constant.
• Pressure — force per unit area — basic definition $p=F/A$ used in the press.
• Hydraulic systems — brakes, lifts, jacks — direct engineering application.
• Archimedes' principle — buoyancy — also relies on isotropic fluid pressure.
• Conservation of energy in machines — explains why the press isn't a free-energy device.
• Incompressibility & continuity equation — why $A_1 d_1 = A_2 d_2$.

Concept Map

Hinglish (regional understanding)