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, we need to find the upward force necessary to break the vertical film formed between the ring and the soap solution. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a thin wire bent into a ring with a diameter of 3.0 cm. When this ring is placed on the surface of a soap solution, a film of soap forms around the ring. To lift the ring from the soap solution, we need to apply an upward force that overcomes the surface tension of the soap film.

### Step 2: Determine the Radius of the Ring
The diameter of the ring is given as 3.0 cm. To find the radius (r), we use the formula:
\[ r = \frac{\text{diameter}}{2} = \frac{3.0 \, \text{cm}}{2} = 1.5 \, \text{cm} \]

### Step 3: Calculate the Circumference of the Ring
The circumference (C) of the ring can be calculated using the formula:
\[ C = 2\pi r \]
Substituting the value of the radius:
\[ C = 2\pi \times 1.5 \, \text{cm} = 3\pi \, \text{cm} \]

### Step 4: Understand the Surface Tension Force
The upward force needed to break the soap film is related to the surface tension (T) of the soap solution. Since there are two surfaces of the film (one on the top and one on the bottom of the ring), the total force (F) required to break the film is given by:
\[ F = 2 \times T \times \text{length of the ring} \]
Here, the length of the ring is the circumference we calculated.

### Step 5: Substitute the Values
Now substituting the values into the force equation:
\[ F = 2 \times T \times (3\pi \, \text{cm}) \]
This simplifies to:
\[ F = 6\pi T \, \text{dynes} \]
where T is the surface tension in dynes/cm.

### Step 6: Final Expression
Thus, the upward force necessary to break the vertical film formed between the ring and the soap solution is:
\[ F = 6\pi T \, \text{dynes} \]

### Summary
To summarize, the upward force required to break the film is directly proportional to the surface tension of the soap solution and the circumference of the ring. The final answer is expressed in terms of the surface tension.

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