Simple U - tube Manometer

Equating the pressure at the level XX'(pressure at the same level in a continuous body of fluid is equal),

For the left hand side:

P_x = P₁ + \(\rho\)g(a+h)

For the right hand side:

P_x' = P₂ + \(\rho\)ga + \(\rho\)_mgh

Since P_x = P_x'

P₁ + \(\rho\)g(a+h) = P₂ + \(\rho\)ga + \(\rho\)_mgh

P₁ - P₂ = \(\rho\)_mgh - \(\rho\)gh

i.e. P₁ - P₂ = (\(\rho\)_m - \(\rho\))gh.
The maximum value of P₁ - P₂ is limited by the height of the manometer. To measure larger pressure differences we can choose a manometer with heigher density, and to measure smaller pressure differences with accuracy we can choose a manometer fluid which is having a density closer to the fluid density.