1. Use the formula for elastic energy density

For a wire obeying Hooke’s law, the strain energy stored per unit volume is

$u = \frac{1}{2}(\text{stress})(\text{strain})$

2. Relate stress and strain using Young’s modulus

Young’s modulus is

$Y = \frac{\text{stress}}{\text{strain}}$

So,

$\text{strain} = \frac{\text{stress}}{Y} = \frac{S}{Y}$

3. Substitute into the energy density formula

$u = \frac{1}{2} \cdot S \cdot \frac{S}{Y}$

$u = \frac{S^2}{2Y}$

4. Match with the options

Option A is

$\frac{S^2}{2Y}$

So the correct answer is A.