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`

Answer

To solve the problem, we need to find the ratio of longitudinal strain in two copper wires given their lengths and radii ratios. Let's break it down step by step.

### Step 1: Understand the Given Ratios
- The lengths of the two wires are in the ratio \( L_1 : L_2 = 4 : 1 \).
- The radii of the two wires are in the ratio \( R_1 : R_2 = 1 : 4 \).

### Step 2: Define Longitudinal Strain
Longitudinal strain (\( \epsilon \)) is defined as the change in length per unit length, which can be expressed as:
\[ \epsilon = \frac{\text{Stress}}{Y} \]
where \( Y \) is the Young's modulus of the material.

### Step 3: Define Stress
Stress (\( \sigma \)) is defined as the force applied per unit area. For a wire with a circular cross-section, the area \( A \) is given by:
\[ A = \pi R^2 \]
Thus, the stress can be expressed as:
\[ \sigma = \frac{F}{A} = \frac{F}{\pi R^2} \]

### Step 4: Write the Expression for Longitudinal Strain
Substituting the expression for stress into the equation for longitudinal strain, we have:
\[ \epsilon = \frac{F}{\pi R^2 Y} \]

### Step 5: Calculate the Ratio of Longitudinal Strain
Since both wires are made of the same material (copper), their Young's modulus \( Y \) will be the same. Therefore, we can write the ratio of the strains in the two wires as:
\[ \frac{\epsilon_1}{\epsilon_2} = \frac{\frac{F}{\pi R_1^2 Y}}{\frac{F}{\pi R_2^2 Y}} = \frac{R_2^2}{R_1^2} \]

### Step 6: Substitute the Values of Radii
From the given ratio \( R_1 : R_2 = 1 : 4 \), we can express \( R_2 \) in terms of \( R_1 \):
\[ R_2 = 4R_1 \]
Now substituting this into the ratio of strains:
\[ \frac{\epsilon_1}{\epsilon_2} = \frac{(4R_1)^2}{(R_1)^2} = \frac{16R_1^2}{R_1^2} = 16 \]

### Conclusion
Thus, the ratio of longitudinal strain in the two wires is:
\[ \frac{\epsilon_1}{\epsilon_2} = 16 : 1 \]