Question

Two capillary tubes of same diameter are kept vertically in two liquids whose densities are in the ratio \( 4:3 \). If their surface tensions are in the ratio \( 6:5 \), the ratio of heights \( \left( \frac{h_1}{h_2} \right) \) of liquids in the two capillary tubes is (Their angle of contacts are same)

• A. \( 10 : 7 \)
• B. \( 8 : 7 \)
• C. \( 2 : 1 \)
• D. \( 3 : 2 \)

Hint:

The height of the liquid in the capillary tube is inversely proportional to the density of the liquid and directly proportional to the surface tension.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The height of liquid in a capillary tube depends on surface tension, density, tube radius, and contact angle.
Step 2: Key Formula or Approach:
Capillary rise formula: \( h = \frac{2T \cos \theta}{r \rho g} \).
For constant \( r, \theta, \) and \( g \): \( h \propto \frac{T}{\rho} \).
Step 3: Detailed Explanation:
Given:
Density ratio \( \frac{\rho_1}{\rho_2} = \frac{4}{3} \).
Surface tension ratio \( \frac{T_1}{T_2} = \frac{6}{5} \).
Calculate the ratio of heights:
\[ \frac{h_1}{h_2} = \left( \frac{T_1}{T_2} \right) \cdot \left( \frac{\rho_2}{\rho_1} \right) \]
\[ \frac{h_1}{h_2} = \left( \frac{6}{5} \right) \cdot \left( \frac{3}{4} \right) = \frac{18}{20} = \frac{9}{10} \]
Step 4: Final Answer:
The ratio of heights is \( 9/10 \).