Capillarity in a narrow glass tube dipped in a liquid: the rise or fall of the liquid column is

Introduction / Context:
Capillary rise or depression is essential in groundwater flow, soil mechanics, and microfluidics. It depends on interfacial forces, liquid properties, and the tube geometry—especially the diameter and contact angle with the wall.

Given Data / Assumptions:

• Cylindrical, clean glass tube with small diameter.
• Static equilibrium of a liquid column in air.
• Surface tension σ and contact angle θ are well-defined.

Concept / Approach:

The standard relation for capillary rise is h = (4 σ cos θ) / (w * d), where w is specific weight of the liquid and d is tube diameter. Thus, h increases with surface tension and cos θ and decreases with specific weight and diameter. For water in clean glass, θ ≈ 0, so cos θ ≈ 1 and rise is pronounced in narrower tubes.

Step-by-Step Solution:

Verification / Alternative check:

Check limiting cases: as d → large, h → 0 (no observable capillarity); as σ → 0 or cos θ → 0, rise vanishes, confirming the relationship.

Why Other Options Are Wrong:

(a) Ignores dependence on cos θ, w, and diameter. (b) Uses sine instead of cosine; the correct factor is cos θ. (c) Rise is inversely, not directly, proportional to specific weight. (e) Directly contradicts the formula.

Common Pitfalls:

Using radius instead of diameter without adjusting constants; neglecting contamination that changes θ; assuming the same rise for different liquids.

Final Answer:

Inversely proportional to the diameter of the glass tube