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Question

A capillary tube with inner cross-section in the form of a square of side a is dipped vertically in a liquid of density ρ and surface tension σ which wets the surface of the capillary tube with an angle of contact θ. The approximate height to which the liquid will rise in the tube is: (Neglect the effect of surface tension at the corners of the capillary tube)

A
2σcos θaρg
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B
4σcos θaρg
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C
8σcos θaρg
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D
4σsecθaρg
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Solution

The correct option is B 4σcos θaρg
Upward force due to surface tension on the top surface of the liquid is,
F=4aσcosθ
The force due to surface tension will be acting on the liquid surface along the perimeter of the square(4a).

The component of surface tension force along vertical direction (4aσcosθ) will balance the weight of the liquid column.

If liquid is raised to a height h, then
F=mg
4aσcosθ=(A×h×ρ)g
4aσcosθ=a2hρg
h=4σcosθaρg

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