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:

Capillary height $\uparrow$ when radius $\downarrow$ and density $\downarrow$.

Answer 1

The Correct Option is C

Capillary rise is given by $h = \dfrac{2 \gamma \cos\theta}{r \rho g}$ where $h$ is inversely related to radius $r$ and density $\rho$ of water. Surface tension is directly related to rise, not inverse.

Answer 2

Alternate approach — Force-balance derivation:
Capillary rise occurs because the upward force of surface tension acting around the tube's circumference balances the downward weight of the raised water column.
Upward force: \( F_{up} = 2\pi r \gamma \cos\theta \)
Weight of water column: \( F_{down} = \pi r^2 h \rho g \)
Equating the two and solving for \( h \):
\[ h = \frac{2\gamma \cos\theta}{r \rho g} \]
From this derived expression, \( h \) is inversely proportional to the tube radius \( r \) (statement A, true) and inversely proportional to the density \( \rho \) (statement B, true), while it is directly, not inversely, proportional to surface tension \( \gamma \), so statement D is false, and statement C, which claims a direct proportionality with radius, contradicts the derivation and is also false.
Hence only statements (A) and (B) are correct.