Capillary rise of water in a clean glass capillary is governed by which proportionalities? (Assume contact angle near 0° for water–glass.)

Introduction / Context:
Capillary rise explains why water climbs in small tubes, soils, and porous media—vital for geotechnics, hydrology, and materials. It balances surface forces against weight of the liquid column.

Given Data / Assumptions:

• Water in a clean glass tube (contact angle θ ≈ 0°, cos θ ≈ 1).
• Steady meniscus at equilibrium.
• Isothermal conditions; negligible evaporation and contamination.

Concept / Approach:

Equilibrium height h results from vertical surface-tension force around the tube circumference balancing the weight of the liquid column. The standard formula is h = 4 * σ * cos θ / (ρ * g * d), where σ is surface tension, ρ is density, g is gravity, and d is tube diameter.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: σ has N/m, dividing by ρ g d (N/m^3) yields meters—consistent. Empirical data for small diameters confirms large rises, illustrating inverse dependence on d.

Why Other Options Are Wrong:

Options (a), (b), and (c) are each individually true; therefore the most complete correct statement is (d) “All of these.” (e) contradicts the derived relation.

Common Pitfalls:

Forgetting cos θ; using wrong diameter (radius versus diameter); applying the formula to contaminated or non-wetting systems where θ differs significantly from 0°.

Final Answer:

All of these