When solid is heated , its length changes according to the relation `l=l_0(1+alphaDeltaT)` , where l is the final length , `l_0` is the initial length , `DeltaT` is the change in temperature , and `alpha` is the coefficient of linear is called super - facial expansion. the area changes according to the relation `A=A_0(1+betaDeltaT)`, where A is the tinal area , `A_0` is the initial area, and `beta` is the coefficient of areal expansion.
The coefficient of linear expansion of brass and steel are `alpha_1 and alpha_2` If we take a brass rod of length `I_1 ` and a steel rod of length `I_2` at `0^@C` , their difference in length remains the same at any temperature if

Options

A. `alpha_1l_2=alpha_2l_2`
B. `alpha_1l_2^2=alpha_2l_1^2`
C. `alpha_1^2l_2=alpha_2^2l_1`
D. `alpha_1l_1 =alpha_2l_2`

Answer

To solve the problem, we need to analyze the change in lengths of the brass and steel rods when they are heated. We will use the formulas for linear expansion of solids.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- Let the initial length of the brass rod be \( l_1 \).
- Let the initial length of the steel rod be \( l_2 \).
- Both rods are at an initial temperature of \( 0^\circ C \).

2. **Apply the Linear Expansion Formula**:
- The final length of the brass rod after heating can be expressed as:
\[ l_{f, brass} = l_1 \left(1 + \alpha_1 \Delta T\right) \]
- The final length of the steel rod after heating can be expressed as:
\[ l_{f, steel} = l_2 \left(1 + \alpha_2 \Delta T\right) \]

3. **Determine the Difference in Lengths**:
- The difference in lengths after heating is given by:
\[ \text{Difference} = l_{f, brass} - l_{f, steel} \]
- Substituting the expressions from step 2:
\[ \text{Difference} = l_1 \left(1 + \alpha_1 \Delta T\right) - l_2 \left(1 + \alpha_2 \Delta T\right) \]

4. **Simplify the Expression**:
- Expanding the equation:
\[ \text{Difference} = l_1 + l_1 \alpha_1 \Delta T - l_2 - l_2 \alpha_2 \Delta T \]
- Rearranging gives:
\[ \text{Difference} = (l_1 - l_2) + (l_1 \alpha_1 - l_2 \alpha_2) \Delta T \]

5. **Condition for Constant Difference**:
- For the difference in lengths to remain constant as the temperature changes, the coefficient of \( \Delta T \) must equal zero:
\[ l_1 \alpha_1 - l_2 \alpha_2 = 0 \]
- This leads to the condition:
\[ l_1 \alpha_1 = l_2 \alpha_2 \]

### Conclusion:
The difference in length between the brass and steel rods remains constant at any temperature if the relationship \( l_1 \alpha_1 = l_2 \alpha_2 \) holds true.