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Pressure Difference Across a Liquid      Surface:

All the molecules lying within the surface film of a liquid are pulled downwards due to the cohesive force between the molecules. This downward force exerted per unit area of a liquid surface is called cohesion pressure.

Plane Liquid Surface:

If the free surface of a liquid is plane, a molecule on the surface is attracted by other molecules in all direction. Hence the resultant force on a molecule due to surface tension is zero & the cohesion pressure is negligible (pressure above equals to pressure below).

Concave Liquid Surface:

If the free surface of a liquid is concave, it is attracted by other molecules in tangentially upward direction, so resultant force acts in upward direction. So force above the liquid surface is greater than that below of it. To balance this, an excess of pressure acting downward on the concave surface & the cohesion pressure is decreased.

Convex Liquid Surface:

If the free surface of a liquid is convex, it is attracted by other molecules in tangentially downward direction, so resultant force acts in downward direction (into the liquid). So force below the liquid surface is greater than that above of it. To balance this, an excess of pressure acting upward on the convex surface & the cohesion pressure is increased. 

Derivation:

If there is a curved liquid surface at rest, the inward pressure on it due to surface tension is balanced by an equal pressure acting outwards. To find the value of excess pressure let us consider a curvilinear surface ABCD. 

Let AD=BC=dx           

      CD=AB=dy

So the initial area A1=dxdy

Let pressure below the surface =P2

Pressure above the surface=P1

Excess pressure, P=P2-P1

Force due to this excess pressure, F=Pdxdy

Let the liquid surface is expanded to the new position A'B'C'D' having a displacement dz. So workdone    w=Fdx

                    w=Pdxdydz ....................(a)

If T is the surface tension, then work-done due to surface tension is w=TdA ...................(b)

where dA=Increase in surface area.

From the fig.    

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