Question 3: Capillary Rise Range

Solution

The height of the liquid column depends on the pressure difference supported by the curved surfaces (menisci).

Case 1: Standard Capillary Rise (Tube Dipped)

When the tube is sufficiently immersed, there is only one meniscus at the top. The pressure just below the meniscus is $P_{atm} – \frac{2\sigma}{r}$. Hydrostatic balance gives:

$$ \rho g h_{min} = \frac{2\sigma}{r} \implies h_{min} = \frac{2\sigma}{\rho g r} $$

Case 2: Maximum Height (Tube Lifted)

If the tube is lifted out of the water but stays in contact with the surface, a meniscus forms at the bottom as well. This bottom meniscus can curve downwards (convex from outside) to support additional weight. The maximum pressure difference occurs when the radius of curvature at the bottom is also $r$.

Pressure balance:

$$ P_{top} + \rho g h = P_{bottom} $$ $$ (P_{atm} – \frac{2\sigma}{r}) + \rho g h_{max} = (P_{atm} + \frac{2\sigma}{r}) $$ $$ \rho g h_{max} = \frac{4\sigma}{r} \implies h_{max} = \frac{4\sigma}{\rho g r} $$

Since the tube is “put in contact with the surface”, the height can vary between these limits depending on the exact positioning.