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 \(\alpha_1 t \ll 1, \alpha_2 t \ll 1\))
\(\alpha_1+\alpha_2\)
For a temperature rise \(t\), the length becomes \(l(1+\alpha_1 t)\) and the breadth becomes \(b(1+\alpha_2 t)\).
New area: \(A' = lb(1+\alpha_1 t)(1+\alpha_2 t) = A\left[1 + (\alpha_1+\alpha_2)t + \alpha_1\alpha_2 t^2\right]\).
Since \(\alpha_1 t \ll 1\) and \(\alpha_2 t \ll 1\), the term \(\alpha_1\alpha_2 t^2\) is negligible, so \(A' \approx A\left[1 + (\alpha_1+\alpha_2)t\right]\).
Comparing with \(A' = A(1+\beta t)\), the coefficient of surface expansion is \(\beta = \alpha_1+\alpha_2\).
Hence, the answer is \(\alpha_1+\alpha_2\).
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