First slide

Surface tension and surface energy

Question

A glass rod of radius r₁ is inserted symmetrically into a vertical capillary tube of radius r₂ such that their lower ends are at the same level. The arrangement is now dipped
in water. The height to which water will rise into the tube will be ( $σ$ - surface tension of water $ρ$ - density of water)

Difficult

A

$\frac{σ}{(r_{2} - r_{1}) ρ g}$
B

$\frac{2 σ}{(r_{2} - r_{1}) ρ g}$
C

$\frac{2 σ}{(r_{2} + r_{1}) ρ g}$
D

$\frac{2 σ}{(r_{2}^{2} + r_{1}^{2}) ρ g}$

Solution

Total upward force due to surface tension $= σ (2 π r_{1} + 2 π r_{2})$ . This supports the weight of the liquid column of height h.
Weight of liquid column $= h [π r_{2}^{2} - π r_{1}^{2}] ρ g$
$Equating, we get h π (r_{2} + r_{1}) (r_{2} - r_{1}) ρ g = 2 π σ (r_{1} + r_{2})$
$⇒ h (r_{2} - r_{1}) ρ g = 2 σ \Rightarrow h = \frac{2 σ}{(r_{2} - r_{1}) ρ g}$

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