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

Answer 1

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

Let r = radius of curvature of meniscus
$b=rcos(\theta +\frac{\alpha }{2})$
Excess pressure on concave side of meniscus $\frac{2S}{r}$
$\Rightarrow (P_0+h\rho g)-P_0=\frac{2S}{r}$

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

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