The breaking stress for a wire of radius r of given material is F `N//m^(2)`. The breaking stress for the wire of same material of radius 2r is:

Options

A. F/4
B. F/2
C. F
D. 2F

Answer

To solve the problem of determining the breaking stress for a wire of radius 2r given that the breaking stress for a wire of radius r is F N/m², we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Breaking Stress**:
Breaking stress (σ) is defined as the force (F) applied per unit area (A) of the material. The formula for breaking stress is:
\[ \sigma = \frac{F}{A} \]

2. **Calculate the Area of the Wire with Radius r**:
For a wire of radius r, the cross-sectional area (A) is given by:
\[ A = \pi r^2 \]
Therefore, the breaking stress for this wire can be expressed as:
\[ \sigma = \frac{F}{\pi r^2} \]
Given that this breaking stress is F N/m², we can denote this as:
\[ \sigma = F \]

3. **Calculate the Area of the Wire with Radius 2r**:
Now, we need to find the breaking stress for a wire of radius 2r. The cross-sectional area (A') for this wire is:
\[ A' = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2 \]

4. **Calculate the Breaking Stress for the Wire with Radius 2r**:
The breaking stress (σ') for the wire of radius 2r can be expressed as:
\[ \sigma' = \frac{F'}{A'} = \frac{F'}{4\pi r^2} \]
Here, F' is the force applied to the wire of radius 2r.

5. **Relate the Forces**:
Since the material is the same, the breaking stress remains constant. Thus, we can set the breaking stress of the first wire equal to that of the second wire:
\[ \frac{F}{\pi r^2} = \frac{F'}{4\pi r^2} \]

6. **Solving for F'**:
From the above equation, we can simplify:
\[ F' = \frac{F}{4} \]

7. **Calculate the Breaking Stress for Radius 2r**:
Substituting F' back into the equation for σ':
\[ \sigma' = \frac{F/4}{4\pi r^2} = \frac{F}{16\pi r^2} \]

8. **Final Expression for Breaking Stress**:
Since we know that the breaking stress for the wire of radius r is F, we can express the breaking stress for the wire of radius 2r as:
\[ \sigma' = \frac{F}{4} \]

### Conclusion:
Thus, the breaking stress for the wire of radius 2r is:
\[ \frac{F}{4} \text{ N/m}^2 \]