The length of the steel rod which would have the same difference in length with a copper rod of length 24cm at all temperatures.
`(alpha_("copper") = 18 xx 10^(-6) K^(-1) alpha_("steel") = 12 xx 10^(-6) k^(-1))` is -

Options

A. 20 cm
B. 18 cm
C. 24 cm
D. 36 cm

Explanation

To find the length of the steel rod that would have the same difference in length with a copper rod of length 24 cm at all temperatures, we can use the formula for linear thermal expansion:

\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]

where:
- \(\Delta L\) is the change in length,
- \(L_0\) is the initial length,
- \(\alpha\) is the coefficient of linear expansion,
- \(\Delta T\) is the change in temperature.

### Step-by-Step Solution:

1. **Identify the given values:**
- Length of the copper rod, \(L_{copper} = 24 \, \text{cm} = 0.24 \, \text{m}\)
- Coefficient of linear expansion for copper, \(\alpha_{copper} = 18 \times 10^{-6} \, \text{K}^{-1}\)
- Coefficient of linear expansion for steel, \(\alpha_{steel} = 12 \times 10^{-6} \, \text{K}^{-1}\)

2. **Write the equation for change in length for both materials:**
- For copper:
\[ \Delta L_{copper} = L_{copper} \cdot \alpha_{copper} \cdot \Delta T \]
- For steel (let the length of the steel rod be \(L_{steel}\)):
\[ \Delta L_{steel} = L_{steel} \cdot \alpha_{steel} \cdot \Delta T \]

3. **Set the change in lengths equal to each other:**
\[ \Delta L_{copper} = \Delta L_{steel} \]
Therefore,
\[ L_{copper} \cdot \alpha_{copper} \cdot \Delta T = L_{steel} \cdot \alpha_{steel} \cdot \Delta T \]

4. **Cancel \(\Delta T\) from both sides (assuming \(\Delta T \neq 0\)):**
\[ L_{copper} \cdot \alpha_{copper} = L_{steel} \cdot \alpha_{steel} \]

5. **Substitute the known values:**
\[ 0.24 \, \text{m} \cdot (18 \times 10^{-6} \, \text{K}^{-1}) = L_{steel} \cdot (12 \times 10^{-6} \, \text{K}^{-1}) \]

6. **Solve for \(L_{steel}\):**
\[ L_{steel} = \frac{0.24 \, \text{m} \cdot (18 \times 10^{-6})}{(12 \times 10^{-6})} \]
\[ L_{steel} = \frac{0.24 \cdot 18}{12} \, \text{m} \]
\[ L_{steel} = 0.36 \, \text{m} \]

7. **Convert to centimeters:**
\[ L_{steel} = 0.36 \, \text{m} = 36 \, \text{cm} \]

### Final Answer:
The length of the steel rod is **36 cm**.