Question

In a capillary rise experiment with a capillary tube of length \(l_1\), water rises to a height \(h\) such that \(h \lt l_1\).

If the capillary tube is cut to a length \(l_2\) such that \(l_2 \lt h\), 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:

Capillary rise height depends only on fluid properties and tube radius. If tube length is less than capillary height, water will overflow, but the meniscus curvature remains unchanged.

Answer 1

The Correct Option is A, C

Step 1: Recall capillary rise relation.
The height of capillary rise is given by: \[ h = \frac{2T \cos \theta}{\rho g r} \] where \(T\) = surface tension, \(\theta\) = contact angle, \(\rho\) = density, \(g\) = acceleration due to gravity, and \(r\) = radius of tube. This height \(h\) depends only on liquid properties and tube radius, not on tube length. 

Step 2: Case 1 (long tube).
For \(l_1 > h\), the liquid rises to height \(h\) and stops. The meniscus curvature is determined by tube radius \(r\). 

Step 3: Case 2 (short tube).
If \(l_2 < h\), then the liquid tries to rise to \(h\) but the tube length is insufficient. Hence the liquid will overflow from the top of the tube. So, statement (A) is correct and (B) is incorrect. 

Step 4: Radius of curvature of meniscus.
The radius of curvature of meniscus is a function of tube radius \(r\) and does not change between the two experiments. Thus, statement (C) is correct and (D) is incorrect. 

Final Answer: \[ \boxed{(A) \text{ and } (C)} \]