Calculate the height of liquid column required to balance the excess pressure inside a soap bubble of radius `3 xx 10^(-3) m` .The density of liquid is `900 kgm ^(-3)` and the surface tension of soap solution is `30 xx 10^(-3) Nm^(-1)`.

Explanation

To solve the problem of calculating the height of the liquid column required to balance the excess pressure inside a soap bubble, we can follow these steps:

### Step 1: Understand the relationship between surface tension, radius, and excess pressure
The excess pressure \( P \) inside a soap bubble is given by the formula:
\[ P = \frac{4T}{R} \]
where \( T \) is the surface tension and \( R \) is the radius of the bubble.

### Step 2: Substitute the known values into the formula
Given:
- Radius \( R = 3 \times 10^{-3} \, \text{m} \)
- Surface tension \( T = 30 \times 10^{-3} \, \text{N/m} \)

Substituting these values into the excess pressure formula:
\[ P = \frac{4 \times (30 \times 10^{-3})}{3 \times 10^{-3}} \]

### Step 3: Calculate the excess pressure
Calculating the excess pressure:
\[ P = \frac{120 \times 10^{-3}}{3 \times 10^{-3}} = 40 \, \text{N/m}^2 \]

### Step 4: Relate excess pressure to the height of the liquid column
The excess pressure can also be expressed in terms of the height \( h \) of the liquid column using the formula:
\[ P = \rho g h \]
where \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid column.

### Step 5: Rearrange the formula to solve for height \( h \)
Rearranging the formula gives:
\[ h = \frac{P}{\rho g} \]

### Step 6: Substitute the known values for density and gravity
Given:
- Density \( \rho = 900 \, \text{kg/m}^3 \)
- Acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \)

Substituting the values into the height formula:
\[ h = \frac{40}{900 \times 9.8} \]

### Step 7: Calculate the height \( h \)
Calculating the height:
\[ h = \frac{40}{8820} \approx 0.00453 \, \text{m} = 4.53 \times 10^{-3} \, \text{m} \]

### Final Result
The height of the liquid column required to balance the excess pressure inside the soap bubble is approximately:
\[ h \approx 4.53 \times 10^{-3} \, \text{m} \]