A soap bubble of radius 3cm is chaged with 9 nC (nanocoulomb) Find the excess pressure inside the bubble surface tension of soap solution =`3×10^(-3)` `m^(-1)`

Explanation

To find the excess pressure inside a soap bubble that is charged, we can follow these steps:

### Step 1: Identify the given values
- Radius of the soap bubble, \( r = 3 \, \text{cm} = 3 \times 10^{-2} \, \text{m} \)
- Charge on the bubble, \( Q = 9 \, \text{nC} = 9 \times 10^{-9} \, \text{C} \)
- Surface tension of the soap solution, \( \sigma = 3 \times 10^{-3} \, \text{N/m} \)

### Step 2: Calculate the excess pressure due to surface tension
The formula for excess pressure inside a soap bubble due to surface tension is given by:
\[ \Delta P_{\text{surface}} = \frac{4\sigma}{r} \]
Substituting the values:
\[ \Delta P_{\text{surface}} = \frac{4 \times (3 \times 10^{-3})}{3 \times 10^{-2}} \]
Calculating this gives:
\[ \Delta P_{\text{surface}} = \frac{12 \times 10^{-3}}{3 \times 10^{-2}} = 0.4 \, \text{N/m}^2 \]

### Step 3: Calculate the excess pressure due to electric field
The formula for excess pressure due to the electric field is:
\[ \Delta P_{\text{electric}} = \frac{Q^2}{2 \epsilon_0 A} \]
Where:
- \( A = 4\pi r^2 \) (surface area of the bubble)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)

First, calculate the area:
\[ A = 4\pi (3 \times 10^{-2})^2 = 4\pi (9 \times 10^{-4}) = 36\pi \times 10^{-4} \, \text{m}^2 \]

Now substituting into the pressure formula:
\[ \Delta P_{\text{electric}} = \frac{(9 \times 10^{-9})^2}{2 \times (8.85 \times 10^{-12}) \times (36\pi \times 10^{-4})} \]
Calculating \( (9 \times 10^{-9})^2 = 81 \times 10^{-18} \):
\[ \Delta P_{\text{electric}} = \frac{81 \times 10^{-18}}{2 \times (8.85 \times 10^{-12}) \times (36\pi \times 10^{-4})} \]
Calculating the denominator:
\[ 2 \times (8.85 \times 10^{-12}) \times (36\pi \times 10^{-4}) \approx 2 \times (8.85 \times 10^{-12}) \times (113.097 \times 10^{-4}) \approx 2 \times (1.000 \times 10^{-15}) \approx 2.000 \times 10^{-15} \]
Thus:
\[ \Delta P_{\text{electric}} \approx \frac{81 \times 10^{-18}}{2.000 \times 10^{-15}} \approx 0.0405 \, \text{N/m}^2 \]

### Step 4: Calculate the net excess pressure
The net excess pressure inside the bubble is given by:
\[ \Delta P_{\text{net}} = \Delta P_{\text{surface}} - \Delta P_{\text{electric}} \]
Substituting the values:
\[ \Delta P_{\text{net}} = 0.4 - 0.0405 = 0.3595 \, \text{N/m}^2 \approx 0.36 \, \text{N/m}^2 \]

### Final Answer
The excess pressure inside the bubble is approximately \( 0.36 \, \text{N/m}^2 \).

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