If the pressure `p` is applied normal to a wire of Young's modulus `Y`, the energy stored per unit volume is

Options

A. `(1)/(2)p^2Y`
B. `(1)/(2)(p^2)/(Y)`
C. `(1)/(2)pY^2`
D. `(1)/(2)pY`

Explanation

To find the energy stored per unit volume when a pressure \( p \) is applied normal to a wire with Young's modulus \( Y \), we can follow these steps:

### Step 1: Understand the relationship between stress, strain, and Young's modulus
The Young's modulus \( Y \) is defined as the ratio of stress to strain:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]

### Step 2: Define stress and strain
Stress (\( \sigma \)) is defined as the force (\( F \)) applied per unit area (\( A \)):
\[ \sigma = \frac{F}{A} \]
When pressure \( p \) is applied, it can be considered as stress:
\[ \sigma = p \]

Strain (\( \epsilon \)) is defined as the change in length per unit original length:
\[ \epsilon = \frac{\Delta L}{L_0} \]

### Step 3: Express strain in terms of stress and Young's modulus
From the definition of Young's modulus, we can express strain as:
\[ \epsilon = \frac{\sigma}{Y} \]
Substituting the expression for stress, we have:
\[ \epsilon = \frac{p}{Y} \]

### Step 4: Calculate the energy stored per unit volume
The energy stored per unit volume (\( du \)) in a material is given by the formula:
\[ du = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]
Substituting the expressions for stress and strain:
\[ du = \frac{1}{2} \times p \times \frac{p}{Y} \]

### Step 5: Simplify the expression
Now, simplifying the expression gives:
\[ du = \frac{1}{2} \times \frac{p^2}{Y} \]

### Final Result
Thus, the energy stored per unit volume when pressure \( p \) is applied to the wire is:
\[ \boxed{\frac{1}{2} \frac{p^2}{Y}} \]

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