Question

The height of the capillary rise of water in the 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 the tube
(D). 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) and (B) only.
• D. (D) only.

Hint:

Remember Jurin's equation: $h \propto \frac{1}{r}$ and $h \propto \frac{1}{\rho}$. Finer-textured soils (like clay) have smaller pore radii ($r$) and therefore exhibit a much higher capillary rise than coarse-textured soils (like sand).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Capillary rise is the upward movement of water in narrow pores against the force of gravity, driven by the cohesive and adhesive forces of surface tension.
Key Formula or Approach:
The height of capillary rise ($h$) is given by Jurin's Law: \[ h = \frac{2 T \cos \theta}{r \rho g} \] Where:
- $T$ = surface tension of water
- $\theta$ = contact angle (typically assumed to be zero for pure water on clean silica)
- $r$ = radius of the capillary tube
- $\rho$ = density of the liquid (water)
- $g$ = acceleration due to gravity

Step 2: Detailed Explanation:
From the formula, we can evaluate each relationship:
- Height ($h$) vs. Radius ($r$): Since $r$ is in the denominator, $h \propto \frac{1}{r}$. Therefore, capillary rise is inversely proportional to the radius of the pore (Statement A is correct, Statement C is incorrect).
- Height ($h$) vs. Density ($\rho$): Since $\rho$ is in the denominator, $h \propto \frac{1}{\rho}$. Therefore, it is inversely proportional to the density of water (Statement B is correct).
- Height ($h$) vs. Surface Tension ($T$): Since $T$ is in the numerator, $h \propto T$. Therefore, height is directly proportional to surface tension, making Statement D incorrect.
Thus, only statements (A) and (B) are correct.

Step 3: Final Answer:
The correct option is (C).