Question

An isotropic solid has linear expansion (coefficient of $a_x,a_y$ and $a_z$ for three rectangular axes in a solid). The coefficient of cubical expansion is

• A. $a_x{a}_y{a}_z$
• B. $\frac{{\alpha }_x}{{\alpha }_y+{\alpha }_z}$
• C. $a_x+a_y+a_z$
• D. ${\alpha }_x^2+{\alpha }_y^2+{\alpha }_z^2$

Answer 1

The correct option is C $a_x+a_y+a_z$

Let the initial volume be $V=L_x{L}_y{L}_z$.

Thus volume after expansion$=V^{\prime }=L_x^{\prime }{L}_y^{\prime }{L}_z^{\prime }$

$=L_x(1+{\alpha }_x\Delta T)L_y(1+{\alpha }_y\Delta T)L_z(1+{\alpha }_z\Delta T)$
$=L_x{L}_y{L}_z(1+{\alpha }_x\Delta T+{\alpha }_y\Delta T+{\alpha }_z\Delta T)$

(Neglecting higher powers of ${\alpha }_x,{\alpha }_y,{\alpha }_z$)

$=V(1+({\alpha }_x+{\alpha }_y+{\alpha }_z\left)\Delta T\right)$

Thus the coefficient of cubical expansion is ${\alpha }_x+{\alpha }_y+{\alpha }_z$