Consider two wires of same material having their ratio of radii to be 2 : 1. If these two wires are stretched by equal force, the ratio of stress produced in them is :

Options

A. `1/4`
B. `1/2`
C. `3/4`
D. 1

Explanation

To solve the problem, we need to determine the ratio of stress produced in two wires of the same material with a given ratio of their radii.

### Step-by-Step Solution:

1. **Understanding Stress**:
Stress is defined as the force applied per unit area. Mathematically, it is given by:
\[ \text{Stress} = \frac{F}{A} \]
where \( F \) is the force applied and \( A \) is the cross-sectional area of the wire.

2. **Cross-Sectional Area**:
The cross-sectional area \( A \) of a wire with radius \( r \) is given by:
\[ A = \pi r^2 \]

3. **Given Information**:
Let the radii of the two wires be \( r_1 \) and \( r_2 \) such that:
\[ \frac{r_1}{r_2} = \frac{2}{1} \]
This implies \( r_1 = 2r \) and \( r_2 = r \) for some radius \( r \).

4. **Calculating Areas**:
The areas of the two wires can be calculated as follows:
- For wire 1:
\[ A_1 = \pi (r_1)^2 = \pi (2r)^2 = 4\pi r^2 \]
- For wire 2:
\[ A_2 = \pi (r_2)^2 = \pi (r)^2 = \pi r^2 \]

5. **Calculating Stress**:
Since both wires are stretched by the same force \( F \):
- Stress in wire 1:
\[ \text{Stress}_1 = \frac{F}{A_1} = \frac{F}{4\pi r^2} \]
- Stress in wire 2:
\[ \text{Stress}_2 = \frac{F}{A_2} = \frac{F}{\pi r^2} \]

6. **Finding the Ratio of Stresses**:
To find the ratio of the stresses produced in the two wires, we calculate:
\[ \frac{\text{Stress}_1}{\text{Stress}_2} = \frac{\frac{F}{4\pi r^2}}{\frac{F}{\pi r^2}} = \frac{1}{4} \]

7. **Final Result**:
Therefore, the ratio of stress produced in the two wires is:
\[ \frac{\text{Stress}_1}{\text{Stress}_2} = \frac{1}{4} \]

### Conclusion:
The ratio of stress produced in the two wires is \( 1:4 \).