The length of s steel rod exceeds that of a brass rod by 5 cm. If the difference in their lengths remains same at all temperature, then the length of brass rod will be: (`alpha` for iron and brass are `12xx10^(-6)//^(@)C` and `18xx10^(-6)//^(@)C`, respectively)

Options

A. 15 cm
B. 5 cm
C. 10 cm
D. 2 cm

Explanation

To solve the problem, we need to establish the relationship between the lengths of the steel rod and the brass rod, taking into account their coefficients of linear expansion.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( L_b \) be the original length of the brass rod.
- Let \( L_s \) be the original length of the steel rod.
- According to the problem, \( L_s = L_b + 5 \) cm.

2. **Linear Expansion Formula**:
- The length of a rod after expansion due to temperature change can be expressed as:
\[ L' = L + \alpha \Delta T \]
- Where \( L' \) is the new length, \( L \) is the original length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature.

3. **Lengths After Expansion**:
- For the brass rod:
\[ L_b' = L_b + \alpha_b \Delta T \]
- For the steel rod:
\[ L_s' = L_s + \alpha_s \Delta T \]

4. **Set Up the Equation**:
- The problem states that the difference in lengths remains the same at all temperatures:
\[ L_s' - L_b' = 5 \text{ cm} \]
- Substituting the expressions for \( L_s' \) and \( L_b' \):
\[ (L_s + \alpha_s \Delta T) - (L_b + \alpha_b \Delta T) = 5 \]

5. **Substituting \( L_s \)**:
- Replace \( L_s \) with \( L_b + 5 \):
\[ ((L_b + 5) + \alpha_s \Delta T) - (L_b + \alpha_b \Delta T) = 5 \]
- Simplifying this gives:
\[ 5 + \alpha_s \Delta T - \alpha_b \Delta T = 5 \]
- This simplifies to:
\[ \alpha_s \Delta T - \alpha_b \Delta T = 0 \]

6. **Factor Out \( \Delta T \)**:
- Factoring out \( \Delta T \):
\[ (\alpha_s - \alpha_b) \Delta T = 0 \]
- Since \( \Delta T \) cannot be zero, we conclude:
\[ \alpha_s = \alpha_b \]

7. **Using Coefficients**:
- Given \( \alpha_s = 12 \times 10^{-6} \) and \( \alpha_b = 18 \times 10^{-6} \), we can set up the equation:
\[ L_s = L_b + 5 \]
- Rearranging gives:
\[ L_b = \frac{5 \text{ cm}}{(\alpha_b - \alpha_s)/\alpha_s} \]

8. **Calculating Length of Brass Rod**:
- Since \( \alpha_b > \alpha_s \), we can find:
\[ L_b = 10 \text{ cm} \]

### Final Answer:
The length of the brass rod \( L_b \) is **10 cm**.