Animated Solution for Physics - Properties of Solids and Liquids: A cylindrical capillary tube of $0.2\text{ mm}$ radius is made by joining two capillaries $T_1$ and $T_2$ of different materials having water contact angles of $0^{\circ }$ and $60^{\circ }$, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct ? (Surface tension of water = $0.075\text{ N/m}$, density of water = $1000{\text{ kg/m}}^3$, take $g=10{\text{ m/s}}^2$)

Imagine you are looking at a very special capillary tube. It's not just a single piece of glass; it's a composite tube made by seamlessly joining two different materials, $T_1$ and $T_2$.

The fascinating part is how water interacts with these materials. Material $T_1$ is highly hydrophilic, boasting a contact angle of ${\theta }_1=0^{\circ }$. Material $T_2$, on the other hand, is less welcoming to water, with a contact angle of ${\theta }_2=60^{\circ }$. Because the contact angles dictate the upward pull of surface tension, water will behave drastically differently depending on which section of the tube it finds itself in.

To predict the behavior of the water, we must rely on the fundamental equation for capillary rise. The height $h$ to which a liquid rises in a capillary tube is given by:

Here, $T$ 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 tube. Notice that the height is directly proportional to the cosine of the contact angle. This is the key to unlocking the entire problem.

Before we look at the complex joined tube, let's ask a simple question: What if the entire tube was made of just one material?

If the tube was entirely made of $T_1$ (where ${\theta }_1=0^{\circ }$), the theoretical maximum height $h_1$ would be:

Now, what if the tube was entirely made of $T_2$ (where ${\theta }_2=60^{\circ }$)? Since $\cos 60^{\circ }=0.5$, the upward pull is exactly half as strong. The theoretical maximum height $h_2$ would be:

These two numbers, $7.5\text{ cm}$ and $3.75\text{ cm}$, are the absolute limits of what each material can achieve.

Let's evaluate the first option. When we calculate capillary rise, we often ignore the small amount of water contained within the curved meniscus itself. However, for precise calculations, this weight must be accounted for.

The volume of this meniscus depends entirely on its shape, which is dictated by the contact angle $\theta$. For $T_1$ ($\theta =0^{\circ }$), the meniscus is a perfect hemisphere. For $T_2$ ($\theta =60^{\circ }$), the meniscus is much flatter. Because the shapes are geometrically different, the volume of water they hold is different, leading to a different weight correction. Therefore, Option (A) is correct.

Case I (Options B & C)

In Case I, the highly hydrophilic material $T_1$ is at the bottom. Let's test Option B, where the joint between $T_1$ and $T_2$ is located at $5\text{ cm}$.

Water enters $T_1$ and feels a strong upward pull, aiming for its theoretical maximum of $7.5\text{ cm}$. It easily rushes past the $1\text{ cm}$, $2\text{ cm}$, and $4\text{ cm}$ marks. But at $5\text{ cm}$, it hits the joint and touches material $T_2$.

Suddenly, the rules change. Material $T_2$ can only support a water column of $3.75\text{ cm}$. But the water is already at $5\text{ cm}$! The upward force from $T_2$ is simply too weak to pull the heavy $5\text{ cm}$ column any higher. Does the water fall back down? No. It stops exactly at the joint. To maintain static equilibrium, the water surface flattens out, adjusting its contact angle to a new value between $0^{\circ }$ and $60^{\circ }$ so that the upward force perfectly balances the $5\text{ cm}$ column. Thus, the water will not rise more than $8.75\text{ cm}$; it stops dead at $5\text{ cm}$. Option (B) is incorrect.

Now consider Option C, where the joint is at $8\text{ cm}$. Water enters $T_1$ aiming for $7.5\text{ cm}$. It rises, reaches $7.5\text{ cm}$, and stops. It never even gets close to the joint at $8\text{ cm}$. It behaves exactly as if the entire tube was made of $T_1$. Therefore, the height is indeed $7.5\text{ cm}$. Option (C) is correct.

In Case II, the less hydrophilic material $T_2$ is at the bottom, and the joint is at $5\text{ cm}$.

Water enters $T_2$ and feels a weaker upward pull, aiming for its theoretical maximum of $3.75\text{ cm}$. It rises, reaches $3.75\text{ cm}$, and stops. Because $3.75\text{ cm}$ is well below the $5\text{ cm}$ joint, the water never interacts with material $T_1$. It is completely oblivious to the fact that a better material exists higher up. The height remains $3.75\text{ cm}$. Option (D) is correct.

By carefully analyzing the theoretical limits of each material and understanding the physics of the boundary, we can confidently conclude that the correct statements are (A), (C), and (D).