Question

The coefficient of linear expansion of cyrstal in one direction is ${\alpha }_1$ and that in every direction perpendicular to it ${\alpha }_2$ . the coefficient of cubical expansion is :

• A. ${\alpha }_1+{\alpha }_2$
• B. $2{\alpha }_1+2{\alpha }_2$
• C. ${\alpha }_1+2{\alpha }_2$
• D. none of these

Answer 1

The correct option is C ${\alpha }_1+2{\alpha }_2$

$\begin{array}{c}V=V_0\left(1+\gamma \Delta \theta \right)L^3\\ =L_0\left(1+{\alpha }_1\Delta \theta \right)L_0^2{\left(1+{\alpha }_2\Delta \theta \right)}^2\\ =L_0^3\left(1+{\alpha }_1\Delta \theta \right){\left(1+{\alpha }_2\Delta \theta \right)}^2\\ Since,L_0^3=V_{0}and{L}^3=V\\ hence,1+\gamma \Delta \theta =\left(1+{\alpha }_1\Delta \theta \right){\left(1+{\alpha }_2\Delta \theta \right)}^2\\ \cong \left(1+{\alpha }_1\Delta \theta \right)\left(1+{\alpha }_2\Delta \theta \right)\cong \left(1+{\alpha }_1\Delta \theta +{\alpha }_2\Delta \theta \right)\\ g={\alpha }_1+2{\alpha }_2\end{array}$

Hence,

option $\left(C\right)$ is correct answer.