There are two wires of same material and same length while the diameter of second wire is 2 times the diameter of first wire, then ratio of extension produced in the wires by applying same load will be

Options

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

Explanation

To solve the problem of finding the ratio of extension produced in two wires of the same material and length but different diameters, we can follow these steps:

### Step 1: Understand the relationship between stress, strain, and extension
According to Hooke's Law, strain (which is the extension per unit length) is directly proportional to stress (which is the force applied per unit area). Mathematically, this is expressed as:
\[ \text{Strain} \propto \text{Stress} \]

### Step 2: Define stress
Stress is defined as the force applied divided by the cross-sectional area of the wire:
\[ \text{Stress} = \frac{F}{A} \]
Where:
- \( F \) is the force applied
- \( A \) is the cross-sectional area of the wire

### Step 3: Calculate the cross-sectional area
The cross-sectional area \( A \) of a wire with diameter \( d \) is given by:
\[ A = \frac{\pi d^2}{4} \]

Let:
- For the first wire, diameter \( d_1 \) and area \( A_1 = \frac{\pi d_1^2}{4} \)
- For the second wire, diameter \( d_2 = 2d_1 \) and area \( A_2 = \frac{\pi (2d_1)^2}{4} = \frac{\pi \cdot 4d_1^2}{4} = \pi d_1^2 \)

### Step 4: Find the ratio of areas
Now we can find the ratio of the areas of the two wires:
\[ \frac{A_2}{A_1} = \frac{\pi d_1^2}{\frac{\pi d_1^2}{4}} = \frac{4}{1} \]

### Step 5: Relate the extension to the stress
Since the same load \( F \) is applied to both wires, we can write the ratio of their strains (and hence extensions, since the lengths are the same) as:
\[ \frac{\text{Strain}_1}{\text{Strain}_2} = \frac{A_2}{A_1} \]

### Step 6: Substitute the area ratio
Substituting the area ratio we found:
\[ \frac{\text{Strain}_1}{\text{Strain}_2} = \frac{4}{1} \]

### Conclusion
Thus, the ratio of extension produced in the two wires is:
\[ \frac{\text{Extension}_1}{\text{Extension}_2} = \frac{1}{4} \]

### Final Answer
The ratio of extension produced in the wires by applying the same load is \( 1:4 \).

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