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Pascal's principle state that if an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount.

Is there a mathematical derivation or conceptual explanation as to why this is the case, or has it just been experimentally observed and accepted?

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  • $\begingroup$ In Thermodynamics it is possible to show, that for an ideal gas in thermal equilibrium Pascal's principle is valid. Depending on your background, you might want to look into a book like Reif: "Fundamentals of statistical and thermal physics". However, you are asking about a fluid. Although a fluid might be a gas, I suppose you are most interested in liquids. I have never studied liquids. So if you are interested in liquids, you have to dig deeper. $\endgroup$
    – Semoi
    Apr 9, 2020 at 11:22

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We know that pascal's law is true from observation.

However, from a classical perspective it's easy to illustrate why Pascal's principle is true using:

  1. The fact that the fluid is in equilibrium (a = 0).
  2. Newton's second law, $F = ma$.
  3. The definition of pressure ($P={F/A}$)

We can make use of a simple example. Consider a simple prism of fluid (We'll use a prism because it has sides of unequal area):

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Because the fluid is in equilibrium, we can use Newton's 2nd law and the definition of pressure to say

$P_{{1}}A_{{1}}=P_{{3}}A_{{3}}\sin \left( \theta \right)$

$P_{{2}}A_{{2}}=P_{{3}}A_{{3}}\cos \left( \theta \right) $

Simply re-writing the trig functions & areas in terms of the prism side lengths yields

$P_{{1}} = P_{{2}} = P_{{3}}$

Which is one way to state Pascal's principle.


This is a simple classical example, and not a proof. For a real 'proof' using more fundamental principles, Pascal's principle is technically a specific case of the Navier-Stokes equation from fluid mechanics.

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Experimental observation and therefore acception. In physics , unlike math we start with the results of an experiment and then we try to write down those results in math language(equations and mathematical symbols). This is the experiment Pascal made :

Pascal's barrel:

Pascal's barrel is the name of a hydrostatics experiment allegedly performed by Blaise Pascal in 1646. In the experiment, Pascal supposedly inserted a long vertical tube into a barrel filled with water. When water was poured into the vertical tube, the increase in hydrostatic pressure caused the barrel to burst

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