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

The correct answer is 2RT.

Step-by-Step Explanation:

When a soap film is pierced inside a loop of thread, the soap film outside the loop pulls the thread radially outwards in all directions due to surface tension. Because of this outward force, the thread stretches out to form a circular loop of radius $R$.

To find the tension in the thread, let us analyze a very small section of the thread that subtends a small angle $d\theta$ at the center of the circular loop.

1. Outward Force due to Surface Tension:
The arc length of this small section of the thread is:
$dl=Rd\theta$
Since a soap film has two free surfaces (top and bottom in contact with air), the total force per unit length exerted by the film is $2T$, where $T$ is the surface tension of the soap solution.
Therefore, the radially outward force $d{F}_{\text{out}}$ acting on this element is:
$d{F}_{\text{out}}=2Tdl=2TRd\theta$

2. Inward Balancing Force due to Tension in the Thread:
Let $T_t$ be the tension acting along the thread. At the ends of our small element, the tension forces act tangentially. The components of these tension forces pointing toward the center of the loop balance the outward radial force.
The net inward force $d{F}_{\text{in}}$ along the radial direction is given by:
$d{F}_{\text{in}}=2T_t\sin \left(\frac{d\theta }{2}\right)$

3. Applying Small Angle Approximation:
Since $d\theta$ is extremely small, we can use the approximation $\sin \left(\frac{d\theta }{2}\right)\approx \frac{d\theta }{2}$:
$d{F}_{\text{in}}\approx 2T_t\left(\frac{d\theta }{2}\right)=T_{t}d\theta$

4. Equating the Forces:
For the element of the thread to be in equilibrium, the outward force must equal the inward force:
$T_{t}d\theta =2TRd\theta$

Canceling $d\theta$ from both sides, we get the tension in the thread:
$T_t=2RT$