Question

Height of capillary rise of water in soil is
A. inversely proportional to the radius of the tube
B. inversely proportional to the density of water
C. directly proportional to the radius of water
D. directly proportional to the hydrostatic pressure difference
E. inversely proportional to the surface tension of water
Choose the correct answer from the options given below:

• A. A, B and D only
• B. A only
• C. A, B and E only
• D. D and E only

Hint:

Capillary rise simplified: \( h \approx \frac{0.15}{r} \) in cm for soil water.
Narrower tube = Higher rise.
Stronger Surface Tension = Higher rise.
Denser Liquid = Lower rise (harder to lift).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to evaluate several statements regarding the physics of capillary rise in soil (modeled as a tube) and select the combination of correct proportionalities.
Key Formula or Approach:
The height of capillary rise is governed by Jurin's Law:
\[ h = \frac{2\gamma \cos \theta}{\rho g r} \]
Where:
\( h \) = height of rise
\( \gamma \) = surface tension
\( \theta \) = contact angle
\( \rho \) = density of the liquid
\( g \) = acceleration due to gravity
\( r \) = radius of the capillary tube

Step 2: Detailed Explanation:

• Statement A (Correct): From the formula, \( h \propto 1/r \). Capillary rise is inversely proportional to the radius. Smaller pores (clay) lead to higher rise than large pores (sand).

• Statement B (Correct): From the formula, \( h \propto 1/\rho \). Capillary rise is inversely proportional to the density of the liquid.

• Statement C (Incorrect): Capillary rise relates to the radius of the tube or pore, not a "radius of water." Furthermore, if it meant the same as $r$, it is inversely proportional, not directly.

• Statement D (Correct): The capillary rise is driven by the pressure difference across the curved meniscus ($P_c = 2\gamma/r$). In equilibrium, this equals the hydrostatic pressure head of the liquid column ($P = \rho g h$). Since \( h = P_c / (\rho g) \), the height is directly proportional to the capillary/hydrostatic pressure difference.

• Statement E (Incorrect): The formula shows \( h \propto \gamma \). Height is directly proportional to surface tension, not inversely.

Step 3: Final Answer:
Statements A, B, and D are physically correct based on the laws of capillarity.