Show that the coefficient of area expansion, $(\Delta A / A) / \Delta T,$ of a rectangular sheet of the solid is twice its Iinear expansivity, $\alpha_{1}$
Show that the coefficient of area expansion, $(\Delta A / A) / \Delta T,$ of a rectangular sheet of the solid is twice its Iinear expansivity, $\alpha_{1}$
Consider a rectangular sheet of the solid material of length $a$ and breadth $b$ . When the temperature increases by $\Delta T, a$ increases by $\Delta a=\alpha_{1} a\Delta T$ and $b$ increases by $\Delta b$ $=\alpha_{1} b \Delta T .$ the increase in area
$\Delta A =\Delta A_{1}+\Delta A_{2}+\Delta A_{3}$
$\Delta A =a \Delta b+b \Delta a+(\Delta a) \quad(\Delta b)$
$=a \alpha_{1} b \Delta T+b \alpha_{1} a \Delta T+\left(\alpha_{1}\right)^{2} a b(\Delta T)^{2}$
$ = {\alpha _1}ab\Delta T\left( {2 + {\alpha _1}\Delta T} \right) = {\alpha _1}A\Delta T(2 + {\alpha _1}\Delta T)$
since $\alpha_{1}=10^{-5} K ^{-1},$ from Table $11.1,$ the product $\alpha_{1} \Delta T$ for fractional temperature is small in comparision with 2 and may be neglected. Hence
$\left(\frac{\Delta A}{A}\right) \frac{1}{\Delta T} \simeq 2 \alpha_{l}$
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