Two capillary tubes A and B of radii 2\ mm and 4\ mm are dipped in the same liquid. What is the ratio of heights through which liquid rises in the tubes A and B ? a) 1:2 b) 2:1 c) 1:4 d) 4:1

The height $h$ to which a liquid rises in a capillary tube is given by the formula:

$$h=\frac{2T\cos \theta }{\rho gr}$$

where:

• $T$ = surface tension of the liquid
• $\theta$ = contact angle (same for both tubes)
• $\rho$ = density of the liquid (same for both tubes)
• $g$ = acceleration due to gravity (same for both tubes)
• $r$ = radius of the capillary tube

For tube A ($r_A=2\text{ mm}$) and tube B ($r_B=4\text{ mm}$), the ratio of heights is:

$$\frac{h_A}{h_B}=\frac{\frac{2T\cos \theta }{\rho g{r}_A}}{\frac{2T\cos \theta }{\rho g{r}_B}}=\frac{r_B}{r_A}=\frac{4}{2}=2$$

Thus, the ratio of heights is 2:1. The correct answer is (b) 2:1.