Question

The height of the water in the capillary tube of radius $2\,cm$ is $4\,cm$. What should be the radius of capillary, if the water rises to $8\,cm$ in the tube?
A. $1$ cm
B. $2$ cm
C. $3$ cm
D. $4$ cm

Answer 1

Hint: When the liquid column rises above level of zero pressure due to a net upward force produced by the attraction of the water molecules to a solid surface, adhesion of water molecules to the walls of the vessel causes an upward force on the liquid causing capillary rise or action.

Complete step by step answer:
The liquid rises due to the forces of adhesion, cohesion and surface tension. Capillary action occurs when adhesive force between the solid and liquid is more than the cohesive force (liquid to itself).Capillarity depends on the nature of the liquid and solid.

When $\theta <{{90}^{\circ }}$, liquid rises up in the tube and concave meniscus is observed.Where $\theta $ is the angle of contact and is defined as the angle between the tangents drawn to the liquid surface at a point of contact of liquid and solid inside the liquid.The narrower the bore of a glass tube, greater the extent of raising or lowering of the liquid. We measure the height of liquid column using the formula:
$h=\dfrac{2\gamma \cos \theta }{\rho gr}$

Therefore, height of light column is inversely proportional to the radius of the tube.
$h\alpha \dfrac{1}{r}$
$\Rightarrow \dfrac{{{h}_{1}}}{{{h}_{2}}}=\dfrac{{{r}_{2}}}{{{r}_{1}}}$
Where ${{h}_{1}}=4$ cm, ${{h}_{2}}=8$ cm, ${{r}_{1}}=2$ cm and we have to find ${{r}_{2}}$
$\dfrac{4}{8}=\dfrac{{{r}_{2}}}{2}$
$\therefore {{r}_{2}}=1$ cm
Radius of capillary should be $1$ cm.

Hence option A is the correct answer.

Note: In capillary rise experiment, the work done by the force of surface tension is almost equal to the rise in gravitational potential energy of the liquid. Capillary action has a lot of importance in daily life. Transfer of water through xylem in plants takes place due to capillary action.