The ratio of radii of two wires of same material is `2 : 1`. If these wires are stretched by equal forces, then the ratio of stresses produced in them will be

Options

A. `2:1`
B. `1:2`
C. `1:4`
D. `4:1`

Answer

To solve the problem, we need to find the ratio of stresses produced in two wires of the same material, given that the ratio of their radii is 2:1 and they are stretched by equal forces.

### Step-by-Step Solution:

1. **Understand the Definition of Stress**:
Stress (σ) is defined as the force (F) applied per unit area (A) over which the force acts. Mathematically, it can be expressed as:
\[ \sigma = \frac{F}{A} \]

2. **Determine the Cross-Sectional Area**:
Since the wires are circular in cross-section, the area (A) can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
where \( r \) is the radius of the wire.

3. **Calculate the Stress for Each Wire**:
For wire 1 with radius \( r_1 \):
\[ \sigma_1 = \frac{F}{\pi r_1^2} \]
For wire 2 with radius \( r_2 \):
\[ \sigma_2 = \frac{F}{\pi r_2^2} \]

4. **Find the Ratio of Stresses**:
To find the ratio of stresses, we can write:
\[ \frac{\sigma_1}{\sigma_2} = \frac{\frac{F}{\pi r_1^2}}{\frac{F}{\pi r_2^2}} = \frac{r_2^2}{r_1^2} \]
Here, the \( F \) and \( \pi \) cancel out since they are the same for both wires.

5. **Substitute the Given Ratio of Radii**:
We are given that the ratio of the radii \( r_1 : r_2 = 2 : 1 \). This means:
\[ r_1 = 2k \quad \text{and} \quad r_2 = k \quad \text{for some constant } k. \]
Now substituting these values into the ratio of stresses:
\[ \frac{\sigma_1}{\sigma_2} = \frac{(k)^2}{(2k)^2} = \frac{k^2}{4k^2} = \frac{1}{4} \]

6. **Final Result**:
Therefore, the ratio of stresses produced in the two wires is:
\[ \sigma_1 : \sigma_2 = 1 : 4 \]

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