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 establish the relationship between the lengths of the copper rod and the steel rod as they expand with temperature. Given that the difference in their lengths remains constant at 10 cm, we can set up the equations based on their coefficients of linear expansion.

### Step-by-Step Solution:

1. **Define Variables:**
Let:
- \( L_c \) = original length of the copper rod
- \( L_s \) = original length of the steel rod
- \( \alpha_c = 1.6 \times 10^{-5} \, \text{K}^{-1} \) (coefficient of linear expansion for copper)
- \( \alpha_s = 1.2 \times 10^{-5} \, \text{K}^{-1} \) (coefficient of linear expansion for steel)

2. **Establish Length Change Equations:**
The change in length due to temperature change \( \Delta T \) can be expressed as:
- Length of copper rod after temperature change:
\[ L_c' = L_c (1 + \alpha_c \Delta T) \]
- Length of steel rod after temperature change:
\[ L_s' = L_s (1 + \alpha_s \Delta T) \]

3. **Set Up the Length Difference Equation:**
According to the problem, the difference in lengths remains constant at 10 cm:
\[ L_c' - L_s' = 10 \, \text{cm} \]
Substituting the expressions for \( L_c' \) and \( L_s' \):
\[ L_c (1 + \alpha_c \Delta T) - L_s (1 + \alpha_s \Delta T) = 10 \]

4. **Rearranging the Equation:**
Expanding the equation gives:
\[ L_c + L_c \alpha_c \Delta T - L_s - L_s \alpha_s \Delta T = 10 \]
Rearranging this, we have:
\[ (L_c - L_s) + (L_c \alpha_c - L_s \alpha_s) \Delta T = 10 \]

5. **Recognizing the Constant Difference:**
Since the difference \( L_c - L_s \) must also equal 10 cm at all temperatures, we can set:
\[ L_c - L_s = 10 \]

6. **Substituting Back:**
Now, substituting \( L_s = L_c - 10 \) into the rearranged equation:
\[ (L_c - (L_c - 10)) + (L_c \alpha_c - (L_c - 10) \alpha_s) \Delta T = 10 \]
Simplifying gives:
\[ 10 + (L_c \alpha_c - (L_c \alpha_s - 10 \alpha_s)) \Delta T = 10 \]
This means:
\[ (L_c \alpha_c - L_c \alpha_s + 10 \alpha_s) \Delta T = 0 \]

7. **Conclusion:**
Since this must hold for all \( \Delta T \), we conclude that:
\[ L_c \alpha_c - L_c \alpha_s + 10 \alpha_s = 0 \]
Solving for \( L_c \):
\[ L_c (\alpha_c - \alpha_s) = -10 \alpha_s \]
\[ L_c = \frac{-10 \alpha_s}{\alpha_c - \alpha_s} \]
Substituting the values:
\[ L_c = \frac{-10 \times 1.2 \times 10^{-5}}{1.6 \times 10^{-5} - 1.2 \times 10^{-5}} = \frac{-12 \times 10^{-5}}{0.4 \times 10^{-5}} = -30 \]
Since lengths cannot be negative, we take the absolute value:
\[ L_c = 30 \, \text{cm} \]

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