There is a horizontal film of soap solution. On it a thread is placed in the form of a loop. The film is pierced inside the loop and the thread becomes a circular loop of radius `R`. If the surface tension of the loop be `T`, then what will be the tension in the thread?

Options

A. `pi R^(2)/T`
B. `pi^(2)R^(2)T`
C. `2piRT`
D. `2RT`

Explanation

To solve the problem of finding the tension in the thread when a soap film is pierced, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a horizontal soap film with a thread forming a loop. When the film is pierced, the thread takes the shape of a circular loop with radius \( R \).
- The surface tension of the soap film is denoted as \( T \).

2. **Identify Forces Acting on the Thread**:
- The surface tension acts along the surface of the soap film, pulling the thread inward. Since the soap film has two surfaces (front and back), the effective surface tension acting on the thread will be \( 2T \).

3. **Consider the Geometry of the Circular Loop**:
- Let’s consider a small segment of the circular loop. If we take two points on the loop that subtend an angle \( 2\theta \) at the center, we can analyze the forces acting on this segment.

4. **Resolve the Tension**:
- The tension \( T' \) in the thread will have components acting vertically due to the surface tension. For the two segments of the thread at angle \( \theta \):
- The vertical component of the tension from each side is \( T' \sin(\theta) \).
- Therefore, the total upward force from both sides is \( 2T' \sin(\theta) \).

5. **Calculate the Arc Length**:
- The arc length \( s \) subtended by the angle \( 2\theta \) at radius \( R \) is given by:
\[ s = R \cdot 2\theta \]
- Since there are two surfaces of the soap film, the effective length contributing to the surface tension is \( 2s \).

6. **Set Up the Force Balance**:
- The total upward force due to surface tension must balance the downward force due to the tension in the thread:
\[ 2T' \sin(\theta) = 2T \cdot s \]
- Substituting \( s = R \cdot 2\theta \):
\[ 2T' \sin(\theta) = 2T \cdot (R \cdot 2\theta) \]

7. **Simplify the Equation**:
- Canceling out the common factors of 2:
\[ T' \sin(\theta) = 2T \cdot R \cdot \theta \]

8. **Approximate for Small Angles**:
- For small angles, \( \sin(\theta) \approx \theta \). Thus, we can replace \( \sin(\theta) \) with \( \theta \):
\[ T' \theta = 2T \cdot R \cdot \theta \]

9. **Solve for Tension in the Thread**:
- Dividing both sides by \( \theta \) (assuming \( \theta \neq 0 \)):
\[ T' = 2T \cdot R \]

### Final Answer:
The tension in the thread is given by:
\[ T' = 2TR \]