If 'S' is stress and 'Y' is young's modulus of material of a wire, the energy stored in the wire per unit volume is

Options

A. (a) `(S^2)/(2Y)`
B. (b) `2S^2Y`
C. (c) `(S)/(2Y)`
D. (d) `(2Y)/(S^2)`

Explanation

To find the energy stored in a wire per unit volume when given stress (S) and Young's modulus (Y), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The energy stored per unit volume (E) in a material under stress is given by the formula:
\[ E = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]

2. **Define Stress and Strain**:
- Stress (S) is defined as the force applied per unit area.
- Strain (β) is defined as the change in length divided by the original length.

3. **Relate Young's Modulus to Stress and Strain**: Young's modulus (Y) is defined as the ratio of stress to strain:
\[ Y = \frac{S}{\beta} \]
From this, we can express strain in terms of stress and Young's modulus:
\[ \beta = \frac{S}{Y} \]

4. **Substitute Strain in the Energy Formula**: Now, substitute the expression for strain into the energy formula:
\[ E = \frac{1}{2} \times S \times \left(\frac{S}{Y}\right) \]

5. **Simplify the Expression**: This simplifies to:
\[ E = \frac{1}{2} \times \frac{S^2}{Y} \]

6. **Final Result**: Therefore, the energy stored in the wire per unit volume is:
\[ E = \frac{S^2}{2Y} \]

### Final Answer:
The energy stored in the wire per unit volume is given by:
\[ E = \frac{S^2}{2Y} \]

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