A capillary tube when immersed vertically in water, the rise of water column is upto height $h_1$ on earth's surface. When this arrangement is taken into a mine of depth ' d ', below earth's surface, the height of the water column is $h_2$. If $R$ is the radius of the earth, the ratio $\frac{h_2}{h_1}$ is

Capillary Height Variation with Gravity Change

The capillary rise height formula gives $h = \frac{2T \cos\theta}{\rho g r}$, where $T$ is surface tension, $\theta$ is contact angle, $\rho$ is density, $g$ is gravitational acceleration, and $r$ is tube radius.

At Earth's surface with gravity $g_s$, the height is $h_1 = \frac{2T \cos\theta}{\rho g_s r}$.

At depth $d$ below the surface, gravity becomes $g_d = g_s \left(1 - \frac{d}{R}\right)$, where $R$ is Earth's radius, giving height $h_2 = \frac{2T \cos\theta}{\rho g_d r}$.

The height ratio is calculated by division:

$\frac{h_2}{h_1} = \frac{g_s}{g_d} = \frac{g_s}{g_s \left(1 - d/R\right)} = \frac{1}{1 - d/R} = \frac{R}{R - d}$.

The ratio $\frac{h_2}{h_1} = \frac{R}{R-d}$, which corresponds to option D.