A and B are two wires. The radius of A is twice that of B. They are stretched by the same load. Then what is the stress on B ?

Options

A. Equal to that on A
B. Four times that on A
C. Two times that on A
D. Half that on A

Explanation

To solve the problem, we need to understand the relationship between stress, force, and the cross-sectional area of the wires.

### Step-by-Step Solution:

1. **Define Stress**:
Stress (σ) is defined as the force (F) applied per unit area (A) of the material. Mathematically, it is expressed as:
\[ \sigma = \frac{F}{A} \]

2. **Determine the Cross-Sectional Area**:
The cross-sectional area (A) of a wire with radius (r) is given by the formula:
\[ A = \pi r^2 \]
Let the radius of wire B be \( r \). Therefore, the radius of wire A, which is twice that of B, will be:
\[ r_A = 2r \]

3. **Calculate the Cross-Sectional Areas**:
For wire A:
\[ A_A = \pi (r_A)^2 = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2 \]
For wire B:
\[ A_B = \pi (r_B)^2 = \pi r^2 \]

4. **Apply the Same Load**:
Since both wires are stretched by the same load (F), we can now calculate the stress for both wires.

5. **Calculate Stress on Wire A**:
The stress on wire A (σ_A) is:
\[ \sigma_A = \frac{F}{A_A} = \frac{F}{4\pi r^2} \]

6. **Calculate Stress on Wire B**:
The stress on wire B (σ_B) is:
\[ \sigma_B = \frac{F}{A_B} = \frac{F}{\pi r^2} \]

7. **Relate the Stresses**:
Now, we can relate the stress on wire B to that on wire A:
\[ \sigma_B = \frac{F}{\pi r^2} = 4 \cdot \frac{F}{4\pi r^2} = 4 \sigma_A \]

8. **Conclusion**:
Therefore, the stress on wire B is four times that of wire A:
\[ \sigma_B = 4 \sigma_A \]

### Final Answer:
The stress on wire B is four times the stress on wire A.