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

Options

A. `alpha_(1) + alpha_(2)`
B. `2alpha_(1) + alpha_(2)`
C. `alpha_(1) + 2alpha_(2)`
D. None of these

Answer

C

Explanation

`V = V_(0) (1+gammaDeltatheta)`
or `L^(3) = L_(0) (1 alpha_(1)Deltatheta) L_(0^(2))(1+alpha_(2)Deltatheta)^(2)`
`L_(0^(3)) (1+alpha_(1)Deltatheta) (1+alpha_(2)Deltatheta)^(2)`
Since `L_(0^(3)) = V_(0)`, hence
`1+ gammaDeltatheta = (1+alpha_(1)Deltatheta) (1+2alpha_(2)Deltatheta)^(2)`
`~= (1+alpha_(1)Deltatheta) (1+2alpha_(2)Deltatheta)`
`~= 1+ alpha_(1)Deltatheta + 2 alpha_(2)Deltatheta`
`:. gamma = alpha_(1) + 2alpha_(2)`