Derive an expression for the height of capillary rise between two parallel plates dipping in a liquid of density $σ$ separated by a distance d. The surface tension of the liquid is T.

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Solution

The meniscus between two plates is cylindrical in shape. Pressure at A (the lowest point of the meniscus)
$P_{A} = P_{o} - T . r$
Pressure at B = pressure at C = $p_{0}$ - pressure at A + $ρ g h$
$∴ P_{o} = P_{o} - \frac{T}{r} + ρ g h \times h = \frac{T}{ρ g r} = \frac{2 T}{ρ g d}$
Alternative method:
Force upward = $2 l T c o s θ = 2 l T (∵ θ = 0)$ Gravitational pull = (Volume $\times$ Density)g = lhdp g
$∴ 2 l T = l h d p g \Rightarrow h = \frac{2 T}{d}$

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