Pascal’s Law

The French scientist Blaise Pascal observed that the pressure in a fluid at rest is the same at all points if they are at the same height. This fact may be demonstrated in a simple way.

Fig. 1 Proof of Pascal’s law. ABC-DEF is an element of the interior of a fluid at rest. This element is in the form of a right-angled prism. The element is small so that the effect of gravity can be ignored, but it has been enlarged for the sake of clarity. 

Fig. 1 shows an element in the interior of a fluid at rest. This element ABC-DEF is in the form of a right-angled prism. In principle, this prismatic element is very small so that every part of it can be considered at the same depth from the liquid surface and therefore, the effect of the gravity is the same at all these points. But for clarity we have enlarged this element. The forces on this element are those exerted by the rest of the fluid and they must be normal to the surfaces of the element as discussed above. 

Thus, the fluid exerts pressures $P_a$, $P_b$ and $P_c$ on this element of area corresponding to the normal forces $F_a$, $F_b$ and $F_c$ as shown in Fig. 1 on the faces BEFC, ADFC and ADEB denoted by $A_a$ , $A_b$ and $A_c$ respectively. Then

F$_b$ sinθ = F$_c$, F$_b$ cosθ = F$_a$ (by equilibrium) 

A$_b$ sinθ = A$_c$, A$_b$ cosθ = A$_a$ (by geometry) Thus,

$\frac{F_a}{A_a}=\frac{F_b}{A_b}=\frac{F_c}{A_c}$

Hence, pressure exerted is same in all directions in a fluid at rest. It again reminds us that like other types of stress, pressure is not a vector quantity. No direction can be assigned to it. The force against any area within (or bounding) a fluid at rest and under pressure is normal to the area, regardless of the orientation of the area.

Now consider a fluid element in the form of a horizontal bar of uniform cross-section. The bar is in equilibrium. The horizontal forces exerted at its two ends must be balanced or the pressure at the two ends should be equal. This proves that for a liquid in equilibrium the pressure is same at all points in a horizontal plane. 

Suppose the pressure were not equal in different parts of the fluid, then there would be a flow as the fluid will have some net force acting on it. Hence in the absence of flow the pressure in the fluid must be same everywhere in a horizontal plane.