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  4. A glass capillary tube is of the shape of truncated cone with an apex angle α so that its two ends have cross sections of different radii. When dipped in water vertically, water rises in it to a height h, where the radius of its cross section is b. If the surface tension of water is S, its density is ρ, and its contact angle with glass is θ, the value of h will be ( g is the acceleration due to gravity)

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Q. A glass capillary tube is of the shape of truncated cone with an apex angle $\alpha$ so that its two ends have cross sections of different radii. When dipped in water vertically, water rises in it to a height $h$, where the radius of its cross section is $b$. If the surface tension of water is $S$, its density is $\rho$, and its contact angle with glass is $\theta$, the value of $h$ will be ( $g$ is the acceleration due to gravity)Physics Question Image

JEE AdvancedJEE Advanced 2014

Solution:

If $R$ be the meniscus radius
$R \cos (\theta+\alpha / 2)=b$
Excess pressure on concave side of meniscus $=\frac{2 S }{ R }$
$h \rho g =\frac{2 S }{ R }=\frac{2 S }{ b } \cos \left(\theta+\frac{\alpha}{2}\right) $
$\Rightarrow h =\frac{2 S }{ b \rho g } \cos \left(\theta+\frac{\alpha}{2}\right)$

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