Elastic potential energy density of a given stretch wire is proportional

Options

A. `(Stress)^2`
B. Strain
C. Stress
D. (Strain)^-1

Answer

To solve the question regarding the elastic potential energy density of a stretched wire, we will follow these steps:

### Step 1: Understand the Formula for Elastic Potential Energy Density
The elastic potential energy density (u) stored in a stretched wire can be expressed using the formula:
\[ u = \frac{1}{2} \text{stress} \times \text{strain} \]

### Step 2: Define Stress and Strain
- **Stress (σ)** is defined as the force (F) applied per unit area (A):
\[ \sigma = \frac{F}{A} \]

- **Strain (ε)** is defined as the change in length (ΔL) divided by the original length (L):
\[ \varepsilon = \frac{\Delta L}{L} \]

### Step 3: Analyze the Proportionality
From the formula for elastic potential energy density:
\[ u = \frac{1}{2} \sigma \varepsilon \]

We can see that:
- The elastic potential energy density (u) is directly proportional to both stress (σ) and strain (ε).

### Step 4: Identify the Proportional Relationships
Since \( u \) is proportional to both stress and strain, we can summarize:
- \( u \propto \sigma \) (stress)
- \( u \propto \varepsilon \) (strain)

### Step 5: Evaluate the Options
Given the question asks which quantities the elastic potential energy density is proportional to, we can conclude:
- It is directly proportional to stress (option 3) and strain (option 2).

### Final Conclusion
Thus, the elastic potential energy density of a stretched wire is proportional to both stress and strain.