Two wire of same radius and lengh the are subjected to the same load. One wire is of steel and the other is of copper. If the Young's modulus of steel is twice that of copper, the ratio the energy stored per unit volume in steel to that of copper wire is

Options

A. `1:2`
B. `2:1`
C. `1:4`
D. `4:1`

Explanation

To solve the problem, we need to find the ratio of the energy stored per unit volume in steel to that in copper when both wires are subjected to the same load.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two wires (one made of steel and the other made of copper) that have the same radius and length and are subjected to the same load. The Young's modulus of steel (Y_s) is given to be twice that of copper (Y_c).

2. **Young's Modulus Relation**:
The Young's modulus (Y) is defined as the ratio of stress (σ) to strain (ε):
\[ Y = \frac{\sigma}{\epsilon} \]
Rearranging gives us:
\[ \epsilon = \frac{\sigma}{Y} \]

3. **Energy Stored per Unit Volume**:
The energy (U) stored per unit volume in a material is given by the formula:
\[ U = \frac{1}{2} Y \epsilon^2 \]
Substituting ε from the Young's modulus relation, we get:
\[ U = \frac{1}{2} Y \left(\frac{\sigma}{Y}\right)^2 = \frac{\sigma^2}{2Y} \]

4. **Calculating Energy for Steel and Copper**:
Let’s denote:
- For steel: \( U_s = \frac{\sigma^2}{2Y_s} \)
- For copper: \( U_c = \frac{\sigma^2}{2Y_c} \)

5. **Finding the Ratio**:
Now, we need the ratio of energy stored per unit volume in steel to that in copper:
\[ \frac{U_s}{U_c} = \frac{\frac{\sigma^2}{2Y_s}}{\frac{\sigma^2}{2Y_c}} = \frac{Y_c}{Y_s} \]

6. **Substituting Young's Modulus Values**:
Given that \( Y_s = 2Y_c \):
\[ \frac{U_s}{U_c} = \frac{Y_c}{2Y_c} = \frac{1}{2} \]

### Final Answer:
The ratio of the energy stored per unit volume in steel to that in copper is:
\[ \frac{U_s}{U_c} = \frac{1}{2} \]