Question

The coefficient of linear expansion of crystal in one direction is $\alpha_{1}$ and that in every direction perpendicular to it is $\alpha_{2}$. The coefficient of cubical expansion is

• A. $\alpha_{1}+\alpha_{2}$
• B. $2 \alpha_{1}+\alpha_{2}$
• C. $\alpha_{1}+2 \alpha_{2}$
• D. None of these

Answer 1

Answer: (C)

$V=V_{0}(1+\gamma \Delta \theta)$
$L^{3}=L_{0}\left(1+\alpha_{1} \Delta \theta\right) L_{0}^{2}\left(1+\alpha_{2} \Delta \theta\right)^{2}$
$=L_{0}^{3}\left(1+\alpha_{1} \Delta \theta\right)\left(1+\alpha_{2} \Delta \theta\right)^{2}$
Since $L_{0}^{3}=V_{0} \text { and } L^{3}=V$
Hence $1+\gamma \Delta \theta=\left(1+\alpha_{1} \Delta \theta\right)\left(1+\alpha_{2} \Delta \theta\right)^{2}$
$\cong\left(1+\alpha_{1} \Delta \theta\right)\left(1+2 \alpha_{2} \Delta \theta\right)$
$\cong\left(1+\alpha_{1} \Delta \theta+2 \alpha_{2} \Delta \theta\right)$
$\Rightarrow \gamma \alpha_{1}+2 \alpha_{2}$