What is Pascal’s Law? Explain its utilities

Pascal’s Law : It states that intensity of pressure for a fluid at rest is equal in all directions. This is proved as :

The fluid element is of very small dimensions i.e. dx, dy, ds.
Consider a arbitrary fluid element of the wedge shape in a fluid mass at rest. Let the width of the element perpendicular to the plane of paper is unity and
px, py , pz are the pressure intensity acting on the face AB, AC, BC respectively.
Then the forces acting on the element are :
i. Pressure forces normal to the surfaces ,
ii. Weight of element in the vertical direction ,
The forces on the faces are :
Forces on the face AB = px × Area of face AB = px × dy × 1
Forces on the face AC = py × area of face AC = py × dx × 1
Forces on the face BC = pz × area of face BC = pz × ds × 1
Weight of element = ( Mass of element ) × g = ( Volume × rho ) × g
= [ ( AB × AC ) / 2 × 1 ] × rho × g
where, rho = density of fluid
Resolving the forces in x – direction, we have
px × dy × 1 – pz.ds × 1 cos ( thita ) = 0
px × dy × 1 – pz × dy × 1 = 0
px = pz ……(i)
{ ds cos ( thita ) = AB = dy }
Similarly, Resolving the forces in y – direction, we get
py × dx × 1 – pz × ds × 1 cos ( 90 ° – thita ) – ( dx × dy ) / 2 × 1 × rho × g = 0
py × dx – pz ds sin ( thita ) – ( dx . dy ) / 2 × rho × g = 0
But [ ds . Sin ( thita ) ] = dx and also the element is very small and hence the weight is negligible.
Therefore, py.dx – pz × dx = 0
py = pz ……(ii)
From (i) and (ii) ; we have
px = py = pz
This equation shows that the pressure at any point in x, y and z directions is equal.