A thin wire is bent in the form of a ring of diameter `3.0cm`.The ring is placed horizontally on the surface of soap solution and then raised up slowly. Upward force necessary to break the vertical film formed between the ring and the solution is

Options

A. `6 pi T "dyne"`
B. `2 pi T "dyne"`
C. `4 pi T "dyne"`
D. `3 pi T "dyne"`

Explanation

To solve the problem of finding the upward force necessary to break the vertical film formed between the ring and the soap solution, we can follow these steps:

### Step 1: Understand the Problem
We have a thin wire bent into a ring with a diameter of 3.0 cm, which is placed on the surface of a soap solution. When we raise the ring, we need to calculate the upward force required to break the surface tension film that forms around the ring.

### Step 2: Identify the Relevant Concepts
The upward force required to break the film is related to the surface tension of the liquid. Surface tension (T) acts along the perimeter of the ring, and since the film has two surfaces (inside and outside), we will consider both.

### Step 3: Calculate the Circumference of the Ring
The circumference (C) of the ring can be calculated using the formula:
\[ C = \pi d \]
where \( d \) is the diameter of the ring. Given that the diameter is 3.0 cm:
\[ C = \pi \times 3.0 \text{ cm} = 3\pi \text{ cm} \]

### Step 4: Determine the Total Length of the Film
Since there are two surfaces of the film acting on the ring, the effective length (L) that contributes to the surface tension force is:
\[ L = 2 \times C = 2 \times (3\pi) = 6\pi \text{ cm} \]

### Step 5: Calculate the Upward Force
The upward force (F) necessary to break the film can be calculated using the formula:
\[ F = T \times L \]
Substituting the value of L:
\[ F = T \times (6\pi) \]

### Step 6: Finalize the Expression for the Upward Force
Thus, the upward force required to break the vertical film is:
\[ F = 6\pi T \text{ dynes} \]

### Step 7: Conclusion
The upward force necessary to break the vertical film formed between the ring and the soap solution is \( 6\pi T \) dynes.