Question

In a capillary rise experiment with a capillary tube of length $l_1$, water rises to a height $h$ such that $h<l_1$. If the capillary tube is cut to a length $l_2$ such that $l_2<h$, and the experiment is repeated, which of the following statements is/are CORRECT?

• A. Water overflows from the top of the tube.
• B. Water does not overflow from the top of the tube.
• C. At equilibrium, radius of curvature of meniscus are same in both the experiments.
• D. At equilibrium, radius of curvature of meniscus are different in both the experiments.

Hint:

If a capillary tube is shorter than the theoretical rise height, the liquid rises only up to the tube top. Equilibrium is achieved by changing the \textbf{meniscus curvature} (hence capillary pressure), so overflow is not necessary.

Answer 1

The Correct Option is B

Step 1: Recall the capillary rise relation.
For a capillary tube of radius $r$, the equilibrium capillary rise is: \[ h = \frac{2\sigma \cos\theta}{\rho g r} \] This formula is obtained by balancing:
Upward force due to surface tension $\Rightarrow$ downward weight of the risen liquid column.
Step 2: What happens if the tube is shorter than the rise height?
Given the original tube length $l_1$ satisfies $h<l_1$, so equilibrium is reached inside the tube.
Now the tube is cut to $l_2$ such that $l_2<h$.
If the liquid were to rise to $h$, the column would extend beyond the tube, which is not possible.
Therefore, the liquid rises only up to the top of the tube, i.e., height $= l_2$, and equilibrium must occur with a different pressure drop across the meniscus than in the original case.
Step 3: Check overflow condition (A) vs (B).
At the tube top, the meniscus adjusts (its curvature changes) such that the Laplace pressure drop balances the hydrostatic head corresponding to height $l_2$.
The system can reach equilibrium without spilling because the capillary pressure can reduce by changing curvature.
Hence, water does not necessarily overflow.
So, (B) is correct and (A) is incorrect.
Step 4: Compare meniscus curvature in both experiments (C) vs (D).
Capillary pressure (Laplace pressure) is: \[ \Delta P = \sigma\left(\frac{1}{R_1} + \frac{1}{R_2}\right) \] In a circular capillary, this is commonly written as: \[ \Delta P = \frac{2\sigma \cos\theta}{r_{\text{tube}}} \] when the meniscus has its equilibrium shape corresponding to rise $h$.
In the shortened tube, the hydrostatic head is only $\rho g l_2$ (since rise stops at $l_2$), which is smaller than $\rho g h$.
Thus, the required pressure drop across the meniscus is smaller, so the meniscus must have a different (less curved) radius of curvature.
Therefore, the radius of curvature is different in the two experiments.
So, (D) is correct and (C) is incorrect.
Step 5: Conclusion.
The correct statements are: \[ \boxed{(B)\ \text{and}\ (D)} \]