A wire of initial length L and radius r is stretched by a length l. Another wire of same material but with initial length 2L and radius 2r is stretched by a length 2l. The ratio of the stored elastic energy per unit volume in the first and second wire is?

Options

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

Explanation

To solve the problem, we need to find the ratio of the stored elastic energy per unit volume in two wires made of the same material but with different dimensions and elongations. Let's break it down step by step.

### Step 1: Understand the Formula for Elastic Energy per Unit Volume
The elastic energy stored per unit volume (E) in a material can be expressed using the formula:
\[ E = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]
Since we are looking for the energy per unit volume, we can set the volume to 1.

### Step 2: Relate Stress and Strain
Using Hooke's Law, we know that:
\[ \text{Stress} = Y \times \text{Strain} \]
where \( Y \) is the Young's modulus of the material. Therefore, we can rewrite the energy per unit volume as:
\[ E = \frac{1}{2} Y \times \text{Strain}^2 \]

### Step 3: Calculate Strain for Each Wire
- For the first wire:
- Initial length = \( L \)
- Stretched length = \( L + l \)
- Strain \( \epsilon_1 \) is given by:
\[ \epsilon_1 = \frac{l}{L} \]

- For the second wire:
- Initial length = \( 2L \)
- Stretched length = \( 2L + 2l \)
- Strain \( \epsilon_2 \) is given by:
\[ \epsilon_2 = \frac{2l}{2L} = \frac{l}{L} \]

### Step 4: Calculate the Ratio of Stored Elastic Energy
Now, we can express the energies for both wires:
- For the first wire:
\[ E_1 = \frac{1}{2} Y \left(\frac{l}{L}\right)^2 \]

- For the second wire:
\[ E_2 = \frac{1}{2} Y \left(\frac{l}{L}\right)^2 \]

### Step 5: Find the Ratio \( \frac{E_1}{E_2} \)
Now we can find the ratio of the stored elastic energy per unit volume in the first and second wire:
\[ \frac{E_1}{E_2} = \frac{\frac{1}{2} Y \left(\frac{l}{L}\right)^2}{\frac{1}{2} Y \left(\frac{l}{L}\right)^2} = 1 \]

### Conclusion
The ratio of the stored elastic energy per unit volume in the first and second wire is:
\[ \frac{E_1}{E_2} = 1:1 \]