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.

Answer 1

The Correct Option is B, D

To understand the problem presented in the capillary rise experiment, let's analyze the given situation step by step: 

1. Initially, a capillary tube of length \( l_1 \) is used, and water rises to a height \( h \) such that \( h < l_1 \). This indicates that the water does not reach the top of the tube and stays within the confines of the tube's length. This rise occurs due to capillarity, where the adhesive forces between the tube and the liquid and the surface tension of the liquid drive it to rise.
2. Next, the tube is cut to a length \( l_2 \) such that \( l_2 < h \), and the experiment is repeated. Because the tube is now shorter, there's a possibility that the water could reach the top of the tube.
3. The question asks what happens under these new conditions and the state of the meniscus. Let's analyze the statements provided:
   • Water overflows from the top of the tube: This would imply that the water level reaches beyond the tube's end and spills over. However, in a capillary action experiment, water rises only to the extent allowed by the surface tension and the adhesive forces, forming a meniscus due to these forces, and stops at that limit.
   • Water does not overflow from the top of the tube: Simplifying further, if the tube was longer or equal to or shorter than \( h \), the water tends to rise only until the height dictated by the capillary action. Therefore, in this case, water simply rises to the original height \( h \) and stays below the tube's edge.
   • At equilibrium, the radius of curvature of meniscus are same in both experiments: The radius of curvature is dependent on both the adhesive forces and the geometry of the system. However, changes like the length of the tube do not directly affect these unless they alter the contact angle or surface tension.
   • At equilibrium, the radius of curvature of meniscus are different in both experiments: When the tube is cut to a smaller length than the original rise, it changes the balance of forces at play, especially with respect to the constraint imposed by the new tube length. Thus, the radius of curvature could indeed be altered.
4. Given this analysis, the correct statements are:
   • Water does not overflow from the top of the tube: The capillary rise limits the height to \( h \), not to overflow beyond it.
   • At equilibrium, radius of curvature of meniscus are different in both experiments: The shorter length effectively changes the conditions under which equilibrium of forces is reached, altering the meniscus' curvature.

Answer 2

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: \[ \boxed{(B)\ \text{and}\ (D)} \]