Capillarity in a Narrow Tube — Rise/Fall Height Formula A clean glass tube of small inside diameter d is dipped vertically into a liquid. What is the expression for the capillary rise (or fall) height h in terms of specific weight w, angle of contact alpha, and surface tension sigma?

Introduction:
Capillarity causes liquids to rise or fall in narrow tubes depending on wetting characteristics. This phenomenon is central to soil suction, ink flow in pens, medical diagnostics, and porous media flow.

Given Data / Assumptions:

• Tube is narrow, clean, and circular with internal diameter d.
• Liquid has surface tension sigma and angle of contact alpha with glass.
• Specific weight w = rho * g is constant; temperature effects neglected.

Concept / Approach:
At equilibrium, upward surface tension component along the contact line balances the weight of the liquid column displaced relative to the free surface. For a circular tube, the contact line length equals the tube circumference, and geometry gives the standard expression for h.

Step-by-Step Solution:
Upward force due to surface tension: F_up = sigma * (perimeter) * cos(alpha) = sigma * (pi * d) * cos(alpha).Weight of liquid column of height h: W = (area) * h * w = (pi * d^2 / 4) * h * w.Set equilibrium F_up = W to obtain: sigma * pi * d * cos(alpha) = (pi * d^2 / 4) * h * w.Solve for h: h = (4 * sigma * cos(alpha)) / (w * d).

Verification / Alternative check:
For water–glass, alpha ≈ 0, so cos(alpha) ≈ 1 and rise is positive; for mercury–glass, alpha > 90°, cos(alpha) negative, predicting a depression, which matches observations.

Why Other Options Are Wrong:
The factor 4 is essential from geometry; dividing by d^2 is incorrect because perimeter/area ratio yields 1/d; the reciprocal form in option D inverts the physics.

Common Pitfalls:
Using surface tension units inconsistently; forgetting that contamination changes alpha; applying the formula to oversized tubes where meniscus curvature is negligible.

Final Answer:
h = (4 * sigma * cos(alpha)) / (w * d)