A solid rectangular sheet has two different coefficients of linear expansion `alpha_1 `and `alpha_2` along its length and breadth respectively. The coefficient of surface expansion is (for `alpha_1 t ltlt 1, alpha_2 lt lt 1`)

Options

A. `(alpha_1 +alpha_2)/( 2)`
B. `2 (alpha_1 +alpha_2)`
C. `(4 alpha_1 alpha_2)/(alpha_1 +alpha_2)`
D. `alpha_1 +alpha_2`

Explanation

To solve the problem of finding the coefficient of surface expansion for a solid rectangular sheet with two different coefficients of linear expansion, α1 and α2, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a rectangular sheet with length \( L \) and breadth \( B \). The coefficients of linear expansion along the length and breadth are given as \( \alpha_1 \) and \( \alpha_2 \), respectively.

2. **Determine the New Dimensions**:
- When the temperature changes by \( \Delta t \), the new length \( L' \) can be expressed as:
\[ L' = L(1 + \alpha_1 \Delta t) \]
- Similarly, the new breadth \( B' \) can be expressed as:
\[ B' = B(1 + \alpha_2 \Delta t) \]

3. **Calculate the New Area**:
- The area \( A \) of the rectangular sheet is given by:
\[ A = L \times B \]
- The new area \( A' \) after the temperature change can be calculated as:
\[ A' = L' \times B' = (L(1 + \alpha_1 \Delta t))(B(1 + \alpha_2 \Delta t)) \]
- Expanding this expression:
\[ A' = LB(1 + \alpha_1 \Delta t)(1 + \alpha_2 \Delta t) \]
- Using the distributive property:
\[ A' = LB \left(1 + \alpha_1 \Delta t + \alpha_2 \Delta t + \alpha_1 \alpha_2 (\Delta t)^2\right) \]

4. **Neglect Higher Order Terms**:
- Since \( \alpha_1 \) and \( \alpha_2 \) are both much smaller than 1, the term \( \alpha_1 \alpha_2 (\Delta t)^2 \) becomes negligible. Therefore, we can simplify:
\[ A' \approx A \left(1 + (\alpha_1 + \alpha_2) \Delta t\right) \]

5. **Identify the Coefficient of Surface Expansion**:
- The coefficient of surface expansion \( \beta \) is defined as:
\[ \beta = \frac{A' - A}{A \Delta t} \]
- Substituting \( A' \) into this equation:
\[ \beta = \frac{A(1 + (\alpha_1 + \alpha_2) \Delta t) - A}{A \Delta t} \]
- Simplifying gives:
\[ \beta = \frac{A(\alpha_1 + \alpha_2) \Delta t}{A \Delta t} = \alpha_1 + \alpha_2 \]

### Final Answer:
The coefficient of surface expansion for the rectangular sheet is:
\[ \beta = \alpha_1 + \alpha_2 \]