The energy stored per unit volume of a strained wire is

Options

A. `(1)/(2)xx"(load)"xx"(extension)"`
B. `(1)/(2) (Y)/(("Strain")^(2))`
C. `(1)/(2)y ("strain")^(2)`
D. stress `xx` strain

Answer

To find the energy stored per unit volume of a strained wire, we can follow these steps:

### Step 1: Understand the Concept of Energy Density
The energy stored per unit volume in a material under strain is referred to as energy density. It is generally expressed in terms of stress and strain.

### Step 2: Recall the Formula for Energy Density
The formula for energy density (U) in a strained material is given by:
\[ U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]

### Step 3: Define Stress and Strain
- **Stress (σ)** is defined as the force (F) applied per unit area (A):
\[ \text{Stress} = \frac{F}{A} \]
- **Strain (ε)** is defined as the change in length (ΔL) divided by the original length (L):
\[ \text{Strain} = \frac{\Delta L}{L} \]

### Step 4: Substitute Stress and Strain in the Energy Density Formula
Using the definitions of stress and strain, we can express the energy density as:
\[ U = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{\Delta L}{L}\right) \]

### Step 5: Relate Young's Modulus to Stress and Strain
Young's modulus (E) relates stress and strain:
\[ E = \frac{\text{Stress}}{\text{Strain}} \implies \text{Stress} = E \times \text{Strain} \]

### Step 6: Substitute Young's Modulus into the Energy Density Formula
Substituting stress in terms of Young's modulus into the energy density formula gives:
\[ U = \frac{1}{2} \times (E \times \text{Strain}) \times \text{Strain} = \frac{1}{2} E \times \text{Strain}^2 \]

### Conclusion
Thus, the energy stored per unit volume of a strained wire is given by:
\[ U = \frac{1}{2} E \times \text{Strain}^2 \]

### Final Answer
The energy stored per unit volume of a strained wire is:
\[ \frac{1}{2} \times \text{Young's Modulus} \times \text{Strain}^2 \]

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