If a section of soap bubble (of radius R) through its centre is considered then force on one half due to surface thension is

Options

A. `2piRT`
B. `4piRT`
C. `piR^(2)T`
D. `(4T)/(R)`

Answer

To solve the problem of finding the force on one half of a soap bubble due to surface tension, we can follow these steps:

### Step 1: Understand the Structure of the Soap Bubble
A soap bubble consists of two layers of liquid soap. When we consider a section of the soap bubble through its center, we are effectively looking at a hemispherical shape.

### Step 2: Identify the Relevant Physical Quantities
- **Radius of the soap bubble**: \( R \)
- **Surface tension of the soap solution**: \( T \)

### Step 3: Calculate the Excess Pressure Inside the Soap Bubble
The excess pressure inside a soap bubble is given by the formula:
\[ P = \frac{4T}{R} \]
where \( T \) is the surface tension and \( R \) is the radius of the bubble.

### Step 4: Relate Pressure to Force and Area
The excess pressure can also be expressed in terms of force and area:
\[ P = \frac{F}{A} \]
where \( F \) is the force acting on the area \( A \).

### Step 5: Determine the Area of the Cross-Section
For a hemispherical section of the soap bubble, the area \( A \) can be calculated as:
\[ A = \pi R^2 \]
This is the area of the circular cross-section of the hemisphere.

### Step 6: Set Up the Equation
Equating the two expressions for pressure, we have:
\[ \frac{4T}{R} = \frac{F}{\pi R^2} \]

### Step 7: Solve for the Force \( F \)
Rearranging the equation to solve for \( F \):
\[ F = \frac{4T}{R} \cdot \pi R^2 \]
This simplifies to:
\[ F = 4\pi R T \]

### Step 8: Conclusion
Thus, the force on one half of the soap bubble due to surface tension is given by:
\[ F = 4\pi R T \]

### Final Answer
The force on one half due to surface tension is \( 4\pi R T \).

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