Question:
A liquid rises to a height of $4 \text{ cm}$ in a capillary tube of radius $r$. If another capillary tube of radius $r/2$ is dipped in the same liquid, the height of liquid rise will be
A liquid rises to a height of $4 \text{ cm}$ in a capillary tube of radius $r$. If another capillary tube of radius $r/2$ is dipped in the same liquid, the height of liquid rise will be
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Remember, for a given liquid and contact angle, the height of the liquid rise in a capillary tube is inversely proportional to its radius.
Updated On: Jun 3, 2026
- $8 \text{ cm}$
- $2 \text{ cm}$
- $4 \text{ cm}$
- $16 \text{ cm}$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The height to which a liquid rises in a capillary tube is given by the formula: \[h = \frac{2T\cos{\theta}}{\rho gr}\] where \( T \) is the surface tension of the liquid, \( \theta \) is the contact angle between the liquid and the tube material, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( r \) is the radius of the capillary tube.
Step 2: Meaning
This formula shows that the height of the liquid rise (\( h \)) in a capillary tube depends inversely on the radius (\( r \)) of the tube. This means if the radius decreases, the height increases, and vice versa.
Step 3: Analysis
Given: In the first case, the height \( h_1 = 4 \text{ cm} \) with radius \( r \). In the second case, the radius is halved to \( r/2 \). Using the capillary rise formula for both cases: For the first tube: \[h_1 = \frac{2T\cos{\theta}}{\rho gr}\] For the second tube with half the radius: \[h_2 = \frac{2T\cos{\theta}}{\rho g(r/2)} = \frac{4T\cos{\theta}}{\rho gr} = 2h_1\] Since \( h_1 = 4 \text{ cm} \): \[h_2 = 2 \times 4 \text{ cm} = 8 \text{ cm}\]
Step 4: Conclusion
The height of the liquid rise in the second capillary tube with radius \( r/2 \) is twice that of the first tube.
Final Answer: (A)
The height to which a liquid rises in a capillary tube is given by the formula: \[h = \frac{2T\cos{\theta}}{\rho gr}\] where \( T \) is the surface tension of the liquid, \( \theta \) is the contact angle between the liquid and the tube material, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( r \) is the radius of the capillary tube.
Step 2: Meaning
This formula shows that the height of the liquid rise (\( h \)) in a capillary tube depends inversely on the radius (\( r \)) of the tube. This means if the radius decreases, the height increases, and vice versa.
Step 3: Analysis
Given: In the first case, the height \( h_1 = 4 \text{ cm} \) with radius \( r \). In the second case, the radius is halved to \( r/2 \). Using the capillary rise formula for both cases: For the first tube: \[h_1 = \frac{2T\cos{\theta}}{\rho gr}\] For the second tube with half the radius: \[h_2 = \frac{2T\cos{\theta}}{\rho g(r/2)} = \frac{4T\cos{\theta}}{\rho gr} = 2h_1\] Since \( h_1 = 4 \text{ cm} \): \[h_2 = 2 \times 4 \text{ cm} = 8 \text{ cm}\]
Step 4: Conclusion
The height of the liquid rise in the second capillary tube with radius \( r/2 \) is twice that of the first tube.
Final Answer: (A)
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