narrower-tube-rises-higher - Fluid Mechanics and Surface Tension Step by Step

Because the rise height scales as 1 / r, halving the tube radius doubles how far the column climbs, so a narrower tube lifts the liquid higher. That inverse scaling is exact only for stated surface tension, contact angle, tube radius, fluid density, gravity, wetting assumption, and static equilibrium. Real tubes add contamination, meniscus shape, hysteresis, radius measurement, vibration, and uncertainty.

Keeping gamma, density, and gravity fixed, the narrower tube has larger pressure and height. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Capillary geometry supplies radius and height

The radius controls the upward surface-pressure ledger. The height controls the downward column-pressure ledger.

r=\tfrac{1}{2}\ \text{m}\qquad h=2\ \text{m}

Only equal pressures are accepted

The accepted bit is not a judgment word. It is one exactly when the capillary pressure and column pressure match.

\begin{array}{c|c|c|c|c}r&P_{\text{cap}}&h&P_{\text{col}}&\text{accepted}\\1\ \text{m}&20\ \text{Pa}&1\ \text{m}&20\ \text{Pa}&1\\\tfrac{1}{2}\ \text{m}&40\ \text{Pa}&2\ \text{m}&40\ \text{Pa}&1\\\tfrac{1}{4}\ \text{m}&80\ \text{Pa}&4\ \text{m}&80\ \text{Pa}&1\\\end{array}

Capillary rise is accepted only when both pressures match

The tube radius and column height are geometry-checked, and the accepted bit follows the pressure equality.

{2\gamma\over r}=2\cdot10\ \text{N/m}/\tfrac{1}{2}\ \text{m}=40\ \text{Pa}\quad\text{and}\quad \rho g h=2\ \text{kg}/\text{m}^{3}\cdot10\ \text{m}/\text{s}^{2}\cdot2\ \text{m}=40\ \text{Pa}