The height of liquid column raised in a capillary tube of certain radius when dipped in liquid A vertically is `5cm`.If the tube is dipped in a similar manner in another liquid `B` of surface tension and density double the values of liquid `A`,the height of liquid column raised in liquid `B` would be......m.

Options

A. 0.2
B. 0.5
C. 0.1
D. 0.05

Answer

To solve the problem, we will use the formula for the height of a liquid column in a capillary tube, which is given by:

\[ h = \frac{2T \cos \theta}{\rho g r} \]

where:
- \( h \) is the height of the liquid column,
- \( T \) 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 capillary tube.

### Step 1: Determine the height of liquid A
Given that the height of the liquid column raised in liquid A is \( h_A = 5 \, \text{cm} \), we can write:

\[ h_A = \frac{2T_A \cos \theta}{\rho_A g r} \]

### Step 2: Analyze liquid B
For liquid B, we know that both the surface tension and the density are double that of liquid A:

- Surface tension of liquid B: \( T_B = 2T_A \)
- Density of liquid B: \( \rho_B = 2\rho_A \)

### Step 3: Write the height equation for liquid B
Using the same formula for liquid B, we have:

\[ h_B = \frac{2T_B \cos \theta}{\rho_B g r} \]

Substituting the values for \( T_B \) and \( \rho_B \):

\[ h_B = \frac{2(2T_A) \cos \theta}{(2\rho_A) g r} \]

### Step 4: Simplify the equation for \( h_B \)
This simplifies to:

\[ h_B = \frac{4T_A \cos \theta}{2\rho_A g r} = \frac{2T_A \cos \theta}{\rho_A g r} \]

### Step 5: Relate \( h_B \) to \( h_A \)
Notice that:

\[ h_B = 2 \left( \frac{2T_A \cos \theta}{\rho_A g r} \right) = 2h_A \]

### Step 6: Substitute the value of \( h_A \)
Since \( h_A = 5 \, \text{cm} \):

\[ h_B = 2 \times 5 \, \text{cm} = 10 \, \text{cm} \]

### Step 7: Convert to meters
Finally, converting centimeters to meters:

\[ h_B = 10 \, \text{cm} = 0.1 \, \text{m} \]

### Final Answer
The height of the liquid column raised in liquid B is \( 0.1 \, \text{m} \).

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