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, the radius of curvature of meniscus are same in both the experiments.
• D. At equilibrium, the radius of curvature of meniscus are different in both the experiments.

Hint:

In capillary rise experiments, the height of the liquid depends on both the surface tension and the radius of the capillary tube. Changing the tube length or radius of curvature will affect the height and meniscus curvature.

Answer 1

The Correct Option is B, D

In the capillary rise experiment, the height of water is governed by the equation:
\[ h = \frac{2\gamma \cos \theta}{\rho g r} \] where \( \gamma \) is the surface tension, \( \theta \) is the contact angle, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( r \) is the radius of the capillary tube.
- Option (A): Water will not overflow from the top of the tube when the tube is cut to a length \( l_2 \) such that \( l_2<h \). This is because the height \( h \) at equilibrium depends on the length of the tube and the radius of curvature of the meniscus. If the length of the tube is shorter than the rise height, water will not overflow. Hence, option (A) is incorrect.
- Option (B): Since the capillary tube length is shorter than the rise height in the second experiment, water will not overflow from the top. This is the correct statement. Therefore, the correct answer is (B).
- Option (C): The radius of curvature of the meniscus depends on the dimensions of the tube and the height of the liquid. If the tube length changes, the radius of curvature also changes. Therefore, the radius of curvature of the meniscus is not necessarily the same in both experiments, making option (C) incorrect.
- Option (D): Since the height of the liquid is different in both experiments (due to the change in tube length), the radius of curvature of the meniscus will also differ in both experiments. This is the correct statement. Thus, the correct answer is (D).
Thus, the correct answers are (B) and (D).