A wire forming a loop is dipped into soap solution and taken out so that a film of soap solution is formed. A loop of 6.28 cm long thread is gently put on the film and the film is pricked with a needle inside the loop. The thread loop takes the shape of a circle. Find the tension in the thread. Surface tension of soap solution `=0.030Nm^-1`.

Explanation

To find the tension in the thread loop formed by the soap film, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Length of the thread loop, \( L = 6.28 \, \text{cm} = 6.28 \times 10^{-2} \, \text{m} \)
- Surface tension of the soap solution, \( \sigma = 0.030 \, \text{N/m} \)

2. **Relate Length of the Thread to Radius**:
The length of the thread loop is equal to the circumference of the circle formed by the thread:
\[ L = 2\pi r \]
Rearranging gives us the radius \( r \):
\[ r = \frac{L}{2\pi} \]

3. **Substitute the Length**:
Substitute \( L \) into the equation to find \( r \):
\[ r = \frac{6.28 \times 10^{-2}}{2\pi} \]
Using \( \pi \approx 3.14 \):
\[ r = \frac{6.28 \times 10^{-2}}{2 \times 3.14} \approx \frac{6.28 \times 10^{-2}}{6.28} = 0.01 \, \text{m} = 10^{-2} \, \text{m} \]

4. **Calculate Tension in the Thread**:
The tension \( T \) in the thread can be calculated using the formula:
\[ T = \sigma \cdot r \]
Substituting the values of \( \sigma \) and \( r \):
\[ T = 0.030 \, \text{N/m} \times 10^{-2} \, \text{m} \]
\[ T = 0.030 \times 10^{-2} = 3 \times 10^{-4} \, \text{N} \]

5. **Final Answer**:
The tension in the thread is:
\[ T = 3 \times 10^{-4} \, \text{N} \]