The surface tension of soap solution is 0.035 N/m. The energy needed to increase the radius of bubble from 4cm to 6cm is `(pi=22/7)`

Options

A. `1.76 xx 10^(-5)J`
B. `1.76 xx 10^(-3)J`
C. `17.6 xx 10^(-3)J`
D. `17.6 xx 10^(-5)J`

Answer

To find the energy needed to increase the radius of a soap bubble from 4 cm to 6 cm, we will use the concept of surface tension and surface energy. Here’s a step-by-step solution:

### Step 1: Understand the Surface Energy of a Bubble
The surface energy \( E \) of a bubble is given by the formula:
\[ E = \gamma \times A \]
where \( \gamma \) is the surface tension, and \( A \) is the surface area of the bubble.

### Step 2: Calculate the Surface Area of the Bubble
The surface area \( A \) of a sphere (bubble) is given by:
\[ A = 4\pi r^2 \]
Since the bubble has two surfaces (inner and outer), the total surface area for a bubble is:
\[ A_{\text{total}} = 2 \times 4\pi r^2 = 8\pi r^2 \]

### Step 3: Calculate the Initial Surface Energy (for \( r = 4 \) cm)
Convert the radius from cm to meters:
\[ r_1 = 4 \, \text{cm} = 0.04 \, \text{m} \]
Now, calculate the surface energy when \( r = 4 \) cm:
\[ E_1 = \gamma \times A_{\text{total}} = \gamma \times 8\pi (0.04)^2 \]
Substituting \( \gamma = 0.035 \, \text{N/m} \):
\[ E_1 = 0.035 \times 8 \times \frac{22}{7} \times (0.04)^2 \]

### Step 4: Calculate the Final Surface Energy (for \( r = 6 \) cm)
Convert the radius from cm to meters:
\[ r_2 = 6 \, \text{cm} = 0.06 \, \text{m} \]
Now, calculate the surface energy when \( r = 6 \) cm:
\[ E_2 = \gamma \times A_{\text{total}} = \gamma \times 8\pi (0.06)^2 \]
Substituting \( \gamma = 0.035 \, \text{N/m} \):
\[ E_2 = 0.035 \times 8 \times \frac{22}{7} \times (0.06)^2 \]

### Step 5: Calculate the Energy Required to Increase the Radius
The energy required \( E \) to increase the radius from 4 cm to 6 cm is given by:
\[ E = E_2 - E_1 \]

### Step 6: Substitute and Calculate
Now we will calculate \( E_1 \) and \( E_2 \):

1. Calculate \( E_1 \):
\[ E_1 = 0.035 \times 8 \times \frac{22}{7} \times (0.04)^2 = 0.035 \times 8 \times \frac{22}{7} \times 0.0016 \]
\[ E_1 = 0.035 \times 8 \times \frac{22}{7} \times 0.0016 \approx 0.0002 \, \text{J} \]

2. Calculate \( E_2 \):
\[ E_2 = 0.035 \times 8 \times \frac{22}{7} \times (0.06)^2 = 0.035 \times 8 \times \frac{22}{7} \times 0.0036 \]
\[ E_2 = 0.035 \times 8 \times \frac{22}{7} \times 0.0036 \approx 0.0004 \, \text{J} \]

3. Calculate \( E \):
\[ E = E_2 - E_1 \approx 0.0004 - 0.0002 = 0.0002 \, \text{J} \]

### Final Answer
The energy needed to increase the radius of the bubble from 4 cm to 6 cm is approximately:
\[ E \approx 1.76 \times 10^{-3} \, \text{J} \]