Two wires of copper having the length in the ratio `4:1` and their radii ratio as `1:4` are stretched by the same force. The ratio of longitudinal strain in the two will be

Options

A. `1:16`
B. `16:1`
C. `1:64`
D. `64:1`

Explanation

To solve the problem, we need to find the ratio of longitudinal strain in two copper wires that are stretched by the same force. We will use the formula for longitudinal strain and the relationship between stress, strain, and Young's modulus.

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
- Length ratio of the two wires (L1:L2) = 4:1
- Radius ratio of the two wires (r1:r2) = 1:4

2. **Calculate the Cross-Sectional Areas**:
The cross-sectional area (A) of a wire is given by the formula:
\[ A = \pi r^2 \]
- For wire 1 (with radius r1):
\[ A_1 = \pi (r_1)^2 \]
- For wire 2 (with radius r2):
\[ A_2 = \pi (r_2)^2 \]
- Given the radius ratio r1:r2 = 1:4, we can express r2 in terms of r1:
\[ r_2 = 4r_1 \]
- Therefore, the areas become:
\[ A_1 = \pi (r_1)^2 \]
\[ A_2 = \pi (4r_1)^2 = 16\pi (r_1)^2 \]

3. **Calculate the Stress in Each Wire**:
Stress (σ) is defined as the force (F) applied per unit area (A):
\[ \sigma = \frac{F}{A} \]
- For wire 1:
\[ \sigma_1 = \frac{F}{A_1} = \frac{F}{\pi (r_1)^2} \]
- For wire 2:
\[ \sigma_2 = \frac{F}{A_2} = \frac{F}{16\pi (r_1)^2} \]

4. **Calculate the Longitudinal Strain**:
Longitudinal strain (ε) is defined as the change in length (ΔL) per unit original length (L):
\[ \epsilon = \frac{\Delta L}{L} \]
Using Hooke's Law, we know that:
\[ \sigma = Y \cdot \epsilon \]
where Y is Young's modulus. Rearranging gives:
\[ \epsilon = \frac{\sigma}{Y} \]
- For wire 1:
\[ \epsilon_1 = \frac{\sigma_1}{Y} = \frac{\frac{F}{\pi (r_1)^2}}{Y} \]
- For wire 2:
\[ \epsilon_2 = \frac{\sigma_2}{Y} = \frac{\frac{F}{16\pi (r_1)^2}}{Y} \]

5. **Calculate the Ratio of Longitudinal Strain**:
Now, we can find the ratio of longitudinal strains:
\[ \frac{\epsilon_1}{\epsilon_2} = \frac{\frac{F}{\pi (r_1)^2}}{\frac{F}{16\pi (r_1)^2}} = \frac{16}{1} = 16 \]

### Final Answer:
The ratio of longitudinal strain in the two wires is **16:1**.