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(D) Let the initial length be $L_{0}$ and initial breadth be $B_{0}$. The initial area is $S_{0} = L_{0} B_{0}$.When the temperature increases by $\De

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$\alpha_{1}+\alpha_{2}$

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(D) Let the initial length be $L_{0}$ and initial breadth be $B_{0}$. The initial area is $S_{0} = L_{0} B_{0}$.When the temperature increases by $\Delta t$, the new length $L_{t}$ and new breadth $B_{t}$ are given by:$L_{t} = L_{0}(1 + \alpha_{1} \Delta t)$$B_{t} = B_{0}(1 + \alpha_{2} \Delta t)$The new area $S_{t}$ is:$S_{t} = L_{t} 	imes B_{t} = L_{0}(1 + \alpha_{1} \Delta t) 	imes B_{0}(1 + \alpha_{2} \Delta t)$$S_{t} = L_{0} B_{0} (1 + \alpha_{1} \Delta t + \alpha_{2} \Delta t + \alpha_{1} \alpha_{2} (\Delta t)^{2})$Since $\alpha_{1} \Delta t \ll 1$ and $\alpha_{2} \Delta t \ll 1$, the term $\alpha_{1} \alpha_{2} (\Delta t)^{2}$ is negligible.$S_{t} \approx S_{0} (1 + (\alpha_{1} + \alpha_{2}) \Delta t)$Comparing this with the standard formula for surface expansion $S_{t} = S_{0} (1 + \beta \Delta t)$, where $\beta$ is the coefficient of surface expansion:$\beta = \alpha_{1} + \alpha_{2}$

$A$ 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 $\alpha_{1} \Delta t \ll 1, \alpha_{2} \Delta t \ll 1$):

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