Two capillary tubes of radii 0.2 cm and 0.4 cm are dipped in the same liquid.The ratio of heights through which liquid will rise in the tube is

Options

A. `1 : 2`
B. `2 : 1`
C. `1 : 4`
D. `4 : 1`

Explanation

To solve the problem of finding the ratio of heights through which liquid will rise in two capillary tubes of different radii, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Capillary Rise Formula**:
The height to which a liquid rises in a capillary tube is given by the formula:
\[ h = \frac{2T \cos \theta}{\rho g r} \]
where:
- \( h \) = height of liquid rise
- \( T \) = surface tension of the liquid
- \( \theta \) = angle of contact (assumed to be the same for both tubes)
- \( \rho \) = density of the liquid (assumed to be the same for both tubes)
- \( g \) = acceleration due to gravity (assumed to be constant)
- \( r \) = radius of the capillary tube

2. **Write the Height Equations for Both Tubes**:
For the first capillary tube with radius \( r_1 = 0.2 \) cm:
\[ h_a = \frac{2T \cos \theta}{\rho g (0.2)} \]
For the second capillary tube with radius \( r_2 = 0.4 \) cm:
\[ h_b = \frac{2T \cos \theta}{\rho g (0.4)} \]

3. **Set Up the Ratio of Heights**:
We need to find the ratio \( \frac{h_a}{h_b} \):
\[ \frac{h_a}{h_b} = \frac{\frac{2T \cos \theta}{\rho g (0.2)}}{\frac{2T \cos \theta}{\rho g (0.4)}} \]

4. **Simplify the Ratio**:
The common terms \( 2T \cos \theta \) and \( \rho g \) will cancel out:
\[ \frac{h_a}{h_b} = \frac{0.4}{0.2} \]

5. **Calculate the Final Ratio**:
Now, simplifying \( \frac{0.4}{0.2} \):
\[ \frac{h_a}{h_b} = 2 \]
Therefore, the ratio of heights \( h_a : h_b = 2 : 1 \).

### Final Answer:
The ratio of heights through which the liquid will rise in the two capillary tubes is \( 2 : 1 \).

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