A soap bubble, having radius of 1 mm, is blown from a detergent solution having a surface tension of `2.5xx10^(-2)N//m`. The pressure inside the bubble equals at a point `Z_(0)` below the free surface of water in a container. Taking `g=10 m//s^(2)`, density of water `=10^(3) kg//m^(3)`, the value of `Z_(0)` is :

Options

A. 0.5cm
B. 100 cm
C. 10 cm
D. 1 cm

Explanation

To solve the problem, we need to find the height \( Z_0 \) below the free surface of water where the pressure inside the soap bubble equals the pressure at that point in the water.

### Step-by-Step Solution:

1. **Understand the Pressure Inside the Soap Bubble**:
The excess pressure inside a soap bubble is given by the formula:
\[ \Delta P = \frac{4T}{R} \]
where \( T \) is the surface tension and \( R \) is the radius of the bubble.

2. **Given Values**:
- Surface tension \( T = 2.5 \times 10^{-2} \, \text{N/m} \)
- Radius of the bubble \( R = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)

3. **Calculate the Excess Pressure**:
Substituting the values into the excess pressure formula:
\[ \Delta P = \frac{4 \times (2.5 \times 10^{-2})}{1 \times 10^{-3}} = \frac{1 \times 10^{-1}}{1 \times 10^{-3}} = 100 \, \text{N/m}^2 \]

4. **Pressure at Depth \( Z_0 \)**:
The pressure at a depth \( Z_0 \) in a fluid is given by:
\[ P = P_A + \rho g Z_0 \]
where \( P_A \) is the atmospheric pressure (which will cancel out later), \( \rho \) is the density of the fluid (water), \( g \) is the acceleration due to gravity, and \( Z_0 \) is the depth.

5. **Given Values for Water**:
- Density of water \( \rho = 10^3 \, \text{kg/m}^3 \)
- \( g = 10 \, \text{m/s}^2 \)

6. **Equate the Pressures**:
Since the pressure inside the bubble equals the pressure at depth \( Z_0 \):
\[ \Delta P = \rho g Z_0 \]
Substituting the known values:
\[ 100 = (10^3)(10)Z_0 \]

7. **Solve for \( Z_0 \)**:
Rearranging the equation gives:
\[ Z_0 = \frac{100}{10^3 \times 10} = \frac{100}{10^4} = 0.01 \, \text{m} \]
Converting to centimeters:
\[ Z_0 = 1 \, \text{cm} \]

### Final Answer:
The value of \( Z_0 \) is \( 1 \, \text{cm} \).