Question

A capillary glass tube records a rise of 20 cm when dipped in water. When the area of cross-section of the tube is reduced to half of the former value, water will rise to a height of:

• A. $10\sqrt{2}cm$
• B. 10cm
• C. 20cm
• D. $20\sqrt{2}cm$

Answer 1

The correct option is D $20\sqrt{2}cm$

If cross section area is reduced to half, then radius of capillary tube becomes $\frac{1}{\sqrt{2}}$ times of the previous one. i.e $r^{\prime }=\frac{r}{\sqrt{2}}$
The height $h$ through which a liquid will rise in a capillary tube of radius $r$ is given by $h=\frac{2S\cos \theta }{r\rho g}$
where $S$ is the surface tension, $\rho$ is the density of the liquid and $\theta$ is the angle of contact.
We see that $h\propto \frac{1}{r}$, So we have
$\frac{h^{\prime }}{h}=\frac{r}{r^{\prime }}$
$\Rightarrow \frac{h^{\prime }}{h}=\sqrt{2}\Rightarrow h^{\prime }=\sqrt{2}h=\sqrt{2}\times 20=20\sqrt{2}cm$