The Young's modulus of a wire Y. If the energy per unit volume is E, then the strain will be

Options

A. `sqrt((2E)/(Y))`
B. `sqrt(2EY)`
C. `EY`
D. `(E)/(Y)`

Answer

To find the strain in a wire given its Young's modulus (Y) and energy per unit volume (E), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between energy per unit volume, stress, and strain**:
The energy per unit volume (E) in a material can be expressed in terms of stress (σ) and strain (ε) using the formula:
\[ E = \frac{1}{2} \times \text{stress} \times \text{strain} \]

2. **Relate stress to Young's modulus and strain**:
Young's modulus (Y) is defined as the ratio of stress to strain:
\[ Y = \frac{\text{stress}}{\text{strain}} \implies \text{stress} = Y \times \text{strain} \]

3. **Substitute stress in the energy equation**:
Substitute the expression for stress from step 2 into the energy equation from step 1:
\[ E = \frac{1}{2} \times (Y \times \text{strain}) \times \text{strain} \]
This simplifies to:
\[ E = \frac{1}{2} Y \times \text{strain}^2 \]

4. **Rearrange to solve for strain**:
Rearranging the equation to isolate strain, we get:
\[ \text{strain}^2 = \frac{2E}{Y} \]
Taking the square root of both sides gives:
\[ \text{strain} = \sqrt{\frac{2E}{Y}} \]

5. **Final expression for strain**:
Thus, the strain in the wire is given by:
\[ \text{strain} = \sqrt{\frac{2E}{Y}} \]

### Conclusion:
The strain in the wire, given its Young's modulus \( Y \) and energy per unit volume \( E \), is:
\[ \text{strain} = \sqrt{\frac{2E}{Y}} \]