Question

Water rises to a height $h$ in a capillary tube of area of cross-section $a$. To what height will the water rise in a capillary tube of area of cross-section $4a$ ?

• A. $\frac{h}{4}$
• B. $\frac{h}{2}$
• C. $2h$
• D. $4h$

Answer 1

The correct option is B $\frac{h}{2}$

Area of cross-section $a=\pi r^2$.
$\Rightarrow r=\sqrt{\frac{a}{\pi }}$
The height to which a liquid rises in a capillary tube is given by,
$h=\frac{2T\text{cos}\theta }{r\rho g}=\frac{2\sqrt{\pi }T\text{cos}\theta }{\sqrt{a}\rho g}$
Keeping the liquid and material of tube to be fixed,
$\Rightarrow h\propto \frac{1}{\sqrt{a}}$
If $a$ is increased $4$ times,
$\Rightarrow \frac{a_2}{a_1}=4$
From Eq.$\left(i\right)$,
$\Rightarrow \frac{h_2}{h_1}=\sqrt{\frac{a_1}{a_2}}$
$\Rightarrow \frac{h_2}{h_1}=\frac{1}{2}$
$\because h_1=h$, given
$\Rightarrow h_2=\frac{h_1}{2}$
Hence, $h$ will decrease by a factor of $2$.
$\therefore$The correct choice is $\left(b\right)$