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

Step 1: Remember the basic capillary rise formula. 
For a narrow capillary tube with radius $r$, the height to which a liquid rises due to surface tension is given by:

\[ h = \frac{2\sigma \cos\theta}{\rho g r} \]

This expression comes from balancing two effects:

• The upward pull caused by surface tension at the liquid–air interface.
• The downward force due to the weight of the liquid column.

 

Step 2: Effect of shortening the tube.
Initially, the tube length $l_1$ is greater than the capillary rise height, so the liquid reaches its natural equilibrium inside the tube.
Now the tube is cut to a shorter length $l_2$ such that $l_2 < h$.
Since the tube is not long enough to accommodate the full rise height $h$, the liquid can only rise up to the top of the tube, i.e., up to height $l_2$.
Equilibrium is therefore achieved with a modified pressure balance at the meniscus.

Step 3: Does the liquid overflow?
At the open end of the shortened tube, the meniscus changes its curvature so that the capillary pressure matches the hydrostatic pressure corresponding to height $l_2$.
Because the meniscus can adjust its shape, the system can reach equilibrium without any spillage.
Hence, overflow is not compulsory.
So, statement (B) is correct, while statement (A) is incorrect.

Step 4: Comparing meniscus curvature in both cases.
The pressure difference across a curved interface is given by the Laplace relation:

\[ \Delta P = \sigma\left(\frac{1}{R_1} + \frac{1}{R_2}\right) \]

For a circular capillary, this is often written as:

\[ \Delta P = \frac{2\sigma \cos\theta}{r} \]

In the original tube, the pressure difference balances a hydrostatic head of $\rho g h$.
In the shortened tube, the hydrostatic head is only $\rho g l_2$, which is smaller.
As a result, a smaller pressure difference is required, meaning the meniscus must be less curved.
Therefore, the radius of curvature of the meniscus is not the same in the two experiments.

So, statement (D) is correct and statement (C) is incorrect.

Step 5: Final result.
The correct answers are:

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