Problem 59 - Master Resource Book in JEE Main Physics

Question

Two capillary tubes of radii \(0.2 \mathrm{~cm}\) and \(0.4 \mathrm{~cm}\) are dipped in the same liquid. The ratio of heights through which liquid will rise in the tubes is (a) \(1: 2\) (b) \(2: 1\) (c) \(1: 4\) (d) \(4: 1\)

Step-by-Step Solution

Answer

The ratio of heights is \(2:1\).

Step 1: Understand the Concept of Capillary Action

When a capillary tube is dipped into a liquid, the liquid rises or falls in the tube due to surface tension. The height of the liquid is inversely proportional to the radius of the tube.

Step 2: Use the Formula for Capillary Rise

The formula for capillary rise is given by: \[ h = \frac{2T \cos \theta}{\rho g r} \]where \( h \) is the height of liquid rise, \( T \) is the surface tension of the liquid, \( \theta \) 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.

Step 3: Express Height in Terms of Radius

From the formula, notice that the height \( h \) through which the liquid rises is inversely proportional to the radius \( r \) of the tube, i.e., \( h \propto \frac{1}{r} \).

Step 4: Calculate the Ratio of Heights

Let \( h_1 \) and \( h_2 \) be the heights of the liquid in tubes of radii \( r_1 \) and \( r_2 \) respectively. Then, the ratio is given by:\[ \frac{h_1}{h_2} = \frac{r_2}{r_1} \]Substitute \( r_1 = 0.2 \) cm and \( r_2 = 0.4 \) cm:\[ \frac{h_1}{h_2} = \frac{0.4}{0.2} = 2 \]

Step 5: Interpret the Result

Thus, the liquid rises twice as high in the smaller radius tube compared to the larger radius tube. Therefore, the ratio of heights is 2:1.

Key Concepts

Surface Tension

Surface tension is a fascinating property of liquids that enables them to resist an external force. Simply put, it is the tendency of liquids to shrink into the smallest possible surface area. This property results because the molecules at the surface of a liquid have a stronger attraction to each other than to the surrounding air.
This cohesive force causes liquid surfaces to behave like a stretched elastic membrane. It explains why water droplets form a shape like a bead on surfaces.
In the context of capillary action, surface tension leads to the liquid climbing up the walls of a tube. The steeper the slope, the higher the liquid will rise. The height to which the liquid rises is determined by the balance between the cohesive forces and the adhesive forces of the liquid with the tube walls.

Capillary Rise Formula

Capillary rise occurs when a liquid moves through or up a narrow space, such as inside a thin tube. This phenomenon is due to the combination of surface tension and the adhesive force between the liquid and the tube.
The height to which the liquid rises can be calculated using the capillary rise formula:

• \( h = \frac{2T \cos \theta}{\rho g r} \)

In this formula:

• \( h \) is the height the liquid rises to
• \( T \) is the surface tension of the liquid
• \( \theta \) is the contact angle, indicating how the liquid interacts with the surface
• \( \rho \) is the density of the liquid
• \( g \) is the gravitational acceleration
• \( r \) is the radius of the tube

The formula shows that the height of capillary action is inversely proportional to the radius of the tube. That means smaller tubes have higher capillary action.

Radius of Tube

The radius of a capillary tube plays a crucial role in determining the height to which a liquid will rise due to capillary action. This phenomenon occurs because the adhesive forces (between the liquid and the tube) become more significant in smaller tubes.
Mathematically, the relationship is given by the inverse proportionality of the height of the liquid rise to the tube radius:
If the radius is halved, the height of the liquid column is expected to double, assuming all other variables remain constant.
This is why, in exercises like the one provided, you can see how understanding the radius lets us calculate the liquid rise height ratio between different tubes. Smaller tubes lead to higher columns of liquid, due to the increased effects of capillary action.