When an elastic material with Young's modulus Y is subjected to a stretching stress, the elastic energy stored per unit volume of the material is

Options

A. `Ys//2`
B. `s^(2)Y //2`
C. `s^(2)//2Y`
D. `s//2Y`

Explanation

To find the elastic energy stored per unit volume of an elastic material subjected to a stretching stress, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Young's Modulus (Y) is defined as the ratio of stress (s) to strain (ε).
- Stress (s) is defined as the force (F) applied per unit area (A).
- Strain (ε) is defined as the change in length (ΔL) divided by the original length (L).

2. **Relate Stress and Strain**:
- From the definition of Young's Modulus, we have:
\[ s = Y \cdot \epsilon \]
- Rearranging this gives us the strain in terms of stress:
\[ \epsilon = \frac{s}{Y} \]

3. **Elastic Energy Stored per Unit Volume**:
- The elastic energy (E) stored per unit volume in a material can be expressed as:
\[ E = \frac{1}{2} \cdot \text{stress} \cdot \text{strain} \]
- Substituting the expressions for stress and strain, we get:
\[ E = \frac{1}{2} \cdot s \cdot \epsilon \]

4. **Substitute Strain**:
- Now substitute the expression for strain (ε) into the equation:
\[ E = \frac{1}{2} \cdot s \cdot \left(\frac{s}{Y}\right) \]
- This simplifies to:
\[ E = \frac{1}{2} \cdot \frac{s^2}{Y} \]

5. **Final Expression**:
- Thus, the elastic energy stored per unit volume of the material is:
\[ E = \frac{s^2}{2Y} \]

### Conclusion:
The correct answer is \( \frac{s^2}{2Y} \).