A copper rod and a steel rod maintain a difference in their lengths of 10 cm at all temperature . If their coefficients of expansion are `1.6xx10^(-5) K^(-1) and 1.2xx10^(-5) K^(-1)` , then length of the copper rod is

Options

A. 40 cm
B. 30 cm
C. 32 cm
D. 24 cm

Answer

To solve the problem, we need to find the length of the copper rod (L1) given that the difference in lengths between a copper rod and a steel rod is constant at 10 cm, and we know their coefficients of linear expansion.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( L_1 \) be the length of the copper rod.
- Let \( L_2 \) be the length of the steel rod.
- The difference in lengths is given as:
\[ L_2 - L_1 = 10 \, \text{cm} \]

2. **Coefficients of Linear Expansion**:
- The coefficient of linear expansion for copper (\( \alpha_1 \)) is:
\[ \alpha_1 = 1.6 \times 10^{-5} \, \text{K}^{-1} \]
- The coefficient of linear expansion for steel (\( \alpha_2 \)) is:
\[ \alpha_2 = 1.2 \times 10^{-5} \, \text{K}^{-1} \]

3. **Change in Length Formula**:
- The change in length due to temperature change can be expressed as:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]
- For the copper rod:
\[ L_1' = L_1 + L_1 \cdot \alpha_1 \cdot \Delta T \]
- For the steel rod:
\[ L_2' = L_2 + L_2 \cdot \alpha_2 \cdot \Delta T \]

4. **Expressing Lengths in Terms of Each Other**:
- Since \( L_2 = L_1 + 10 \):
\[ L_2' = (L_1 + 10) + (L_1 + 10) \cdot \alpha_2 \cdot \Delta T \]
- Substitute \( L_2' \) in the difference equation:
\[ L_2' - L_1' = 10 \]

5. **Setting Up the Equation**:
- Substitute the expressions for \( L_1' \) and \( L_2' \):
\[ [(L_1 + 10) + (L_1 + 10) \cdot \alpha_2 \cdot \Delta T] - [L_1 + L_1 \cdot \alpha_1 \cdot \Delta T] = 10 \]

6. **Simplifying the Equation**:
- Rearranging gives:
\[ (L_1 + 10) \cdot \alpha_2 \cdot \Delta T - L_1 \cdot \alpha_1 \cdot \Delta T = 0 \]
- Factor out \( \Delta T \):
\[ \Delta T \left[ (L_1 + 10) \cdot \alpha_2 - L_1 \cdot \alpha_1 \right] = 0 \]

7. **Solving for Lengths**:
- Since \( \Delta T \neq 0 \), we can set the expression inside the brackets to zero:
\[ (L_1 + 10) \cdot \alpha_2 = L_1 \cdot \alpha_1 \]
- Rearranging gives:
\[ L_1 \cdot \alpha_1 - L_1 \cdot \alpha_2 = 10 \cdot \alpha_2 \]
- Factoring out \( L_1 \):
\[ L_1 (\alpha_1 - \alpha_2) = 10 \cdot \alpha_2 \]
- Thus:
\[ L_1 = \frac{10 \cdot \alpha_2}{\alpha_1 - \alpha_2} \]

8. **Substituting Values**:
- Substitute the values of \( \alpha_1 \) and \( \alpha_2 \):
\[ L_1 = \frac{10 \cdot (1.2 \times 10^{-5})}{(1.6 \times 10^{-5}) - (1.2 \times 10^{-5})} \]
- Calculate:
\[ L_1 = \frac{10 \cdot (1.2 \times 10^{-5})}{0.4 \times 10^{-5}} = \frac{12 \times 10^{-5}}{0.4 \times 10^{-5}} = 30 \, \text{cm} \]

### Final Answer:
The length of the copper rod is \( L_1 = 30 \, \text{cm} \).