Question

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).

• A. $\frac{2S}{b\rho g}\cos (\theta -\alpha )$
• B. $\frac{2S}{b\rho g}\cos (\theta +\alpha )$
• C. $\frac{2S}{b\rho g}\cos (\theta -\frac{\alpha }{2})$
• D. $\frac{2S}{b\rho g}\cos (\theta +\frac{\alpha }{2})$

Answer 1

The correct option is D $\frac{2S}{b\rho g}\cos (\theta +\frac{\alpha }{2})$


Let $r$ be the radius of curvature of the meniscus.

$\Rightarrow b=r\cos (\theta +\frac{\alpha }{2})$

Excess pressure on concave side of the meniscus

$\Delta P=\frac{2S}{r}$

$\Rightarrow (P_0+h\rho g)-P_0=\frac{2S}{r}$

$\Rightarrow h\rho g=\frac{2S\cos (\theta +\frac{\alpha }{2})}{b}$

$\Rightarrow h=\frac{2S}{b\rho g}\cos (\theta +\frac{\alpha }{2})$

Hence, option $\left(A\right)$ is correct.