Question

Water rises up to height ' x ' in a capillary tube immersed vertically in water. When the whole arrangement is taken to a depth 'd' in a mine, the water level rises height ' Y '. If ' R ' is the radius of earth then the ratio (Y x) is

• A. \( \frac{2R}{(R-d)} \)
• B. \( \frac{R^2}{(R-d)} \)
• C. \( \frac{R}{(R-d)} \)
• D. \( \frac{R}{(R+d)} \)

Hint:

- Capillary rise $\propto \frac{1}{g}$ - Gravity decreases inside earth $\Rightarrow$ rise increases

Answer 1

The Correct Option is C

Concept:
Capillary rise: \[ h = \frac{2T\cos\theta}{\rho g r} \Rightarrow h \propto \frac{1}{g} \] Acceleration due to gravity at depth $d$: \[ g_d = g\left(1 - \frac{d}{R}\right) \]

Step 1: Write proportional relation.
\[ x \propto \frac{1}{g}, \quad Y \propto \frac{1}{g_d} \]

Step 2: Take ratio.
\[ \frac{Y}{x} = \frac{g}{g_d} \]

Step 3: Substitute $g_d$.
\[ \frac{Y}{x} = \frac{g}{g\left(1 - \frac{d}{R}\right)} = \frac{1}{1 - \frac{d}{R}} \]

Step 4: Simplify.
\[ \frac{Y}{x} = \frac{R}{R-d} \]