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, the 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 $ , then the value of $h$ will be ( $g$ is the acceleration due to gravity)

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

Answer 1

Answer: (D)

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

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