Problem 1009 - JEE Physics

Question

In a capillary tube water rises by \(1.2 \mathrm{~mm}\). The height of water that will rise in another capillary tube having half the radius of the first is (A) \(1.2 \mathrm{~mm}\) (B) \(2.4 \mathrm{~mm}\) (C) \(0.6 \mathrm{~mm}\) (D) \(0.4 \mathrm{~mm}\)

Step-by-Step Solution

Answer

The height of water that will rise in the second capillary tube with half the radius of the first tube is 2.4 mm. Therefore, the correct answer is (B) \(2.4 \mathrm{~mm}\).

Step 1: Determine the formula for the height of water in the first tube

Using the equation for capillary action, the height of water in the capillary tube is: \(h_1 = \dfrac{2 \cdot S \cos \theta}{r_1 \cdot \rho g}\)

Step 2: Find the formula for the height of water in the second tube

Since the second capillary tube has half the radius of the first one, we can write the radius as \(r_2 = \dfrac{r_1}{2}\). Now, using the equation for capillary action, the height of water in the second capillary tube is: \(h_2 = \dfrac{2 \cdot S \cos \theta}{r_2 \cdot \rho g}\)

Step 3: Relate the heights of water in the two tubes

We can write the second equation in terms of the first equation: \(h_2 = \dfrac{2 \cdot S \cos \theta}{\dfrac{r_1}{2} \cdot \rho g}\) Now, multiplying both numerator and denominator by 2: \(h_2 = \dfrac{4 \cdot S \cos \theta}{r_1 \cdot \rho g}\) We can notice that all the physical property terms are the same for both tubes, i.e., \(S\), \(\theta\), \(\rho\), and \(g\). So, we can write the equation in terms of the heights: \(h_2 = 2h_1\)

Step 4: Substitute the original height and find the new height

Now, we know the height of water in the first capillary tube is 1.2 mm. Substitute this value into the equation above: \(h_2 = 2(1.2)\) \(h_2 = 2.4\) So, the height of water that will rise in the second capillary tube with half the radius of the first tube is 2.4 mm. Therefore, the correct answer is (B) \(2.4 \mathrm{~mm}\).

Key Concepts

Surface Tension

Surface tension is a fascinating property of liquids. It makes the surface of a liquid behave somewhat like a flexible membrane. This happens due to the cohesive forces between molecules at the liquid's surface being stronger than those in the liquid's interior. As a result, the surface molecules stick together and create a kind of 'skin'.
In the context of capillary action, surface tension plays a crucial role. It helps the liquid climb up the walls of a thin tube. If you imagine a liquid droplet, surface tension causes it to try and minimize its surface area, leading to the formation of a round shape. But in a tube, surface tension pulls the liquid up along the sides. It's similar to how water can climb a paper towel simply by touching the edge.

• Surface tension is higher in water due to hydrogen bonding.
• The stronger the surface tension, the more liquid can rise in a capillary tube.

This property is essential for various natural and technological processes, from the way plants drink water from their roots to the design of inkjet printers.

Fluid Dynamics

Fluid dynamics is the study of how liquids and gases move. It's an important branch of physics that uses concepts such as pressure, velocity, and flow rates to understand fluid behavior.
Regarding capillary action, fluid dynamics provides a framework to explain the movement of liquid through narrow spaces without external forces. Inside a capillary tube, the movement is affected not only by gravity but also by other forces like surface tension.

• Gravity tends to pull the liquid back down.
• Surface tension and adhesion pull the liquid up.

Normally, in fluid dynamics, you'd deal with flowing water or moving air. But in capillary action, it's more about static conditions and how liquids can defy gravity due to their molecular interactions. Understanding these dynamics helps us grasp how liquids behave in different tubes and conditions.

Radius Effect on Capillary Rise

The radius of a capillary tube has a significant impact on the height to which a liquid can rise. Smaller tubes allow the liquid to rise higher due to the balance of forces acting upon the liquid. This is easier to see when you understand the formula for capillary rise:
\[ h = \dfrac{2 \cdot S \cos \theta}{r \cdot \rho g} \]
In this equation, \(h\) is the height of the liquid column, \(S\) is the surface tension of the liquid, \(\theta\) is the contact angle between the liquid and tube material, \(r\) is the radius of the tube, \(\rho\) is the liquid's density, and \(g\) is gravitational acceleration.

• As the radius decreases, the height \(h\) increases.
• If the radius is halved, the height will double (assuming all other conditions remain constant).

This is why a smaller tube sees a higher rise; the narrow space helps the surface tension exert a stronger pulling force relative to the weight of the liquid. This critical relationship is essential in designing systems that rely on capillary action, from medical diagnostics to plant nutrient delivery systems.