Question

A glass capillary tube is of the shape of a 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. $\frac{2S}{b\rho g}cos\left(\theta -\alpha \right)$
• B. $\frac{2S}{b\rho g}cos\left(\theta +\alpha \right)$
• C. $\frac{2S}{b\rho g}cos\left(\theta -\alpha /2\right)$
• D. $\frac{2S}{b\rho g}cos\left(\theta +\alpha /2\right)$

Answer 1

Answer: (D)

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