Question

A container is partially filled with a liquid of density ${\rho }_2$ in which a capillary tube of radius $r$ is vertically inserted. Now another liquid of density ${\rho }_1({\rho }_1<{\rho }_2)$ is slowly poured in the container to a height $h$ as shown. There is only denser liquid in the capillary tube. The rise of denser liquid in the capillary tube is also $h$. Assuming zero contact angle, the surface tension of liquid is

• A. $r{\rho }_{2}gh$
• B. $2\pi r{\rho }_{2}gh$
• C. $\frac{r}{2}({\rho }_2-{\rho }_1)gh$
• D. $2\pi ({\rho }_2-{\rho }_1)gh$

Answer 1

The correct option is C $\frac{r}{2}({\rho }_2-{\rho }_1)gh$

From this diagram
$P_A=P_B$
$P_0-\frac{2T}{r}+{\rho }_{2}gh=P_0+{\rho }_{1}gh$
$\frac{2T}{r}$=excess pressure on the concave side
${\rho }_{2}gh$=pressure due to the liquid that rises in capillary between point A and free surface
$\frac{2T}{r}=({\rho }_2-{\rho }_1)gh$
$T=\frac{r}{2}({\rho }_2-{\rho }_1)gh$