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_(o)(1+gamma Delta theta)`
`" " L^(3)=L_(o)(1+alpha_(1) Delta theta)L_(o)^(2)(1+alpha_(2) Delta alpha)^(2)`
`" "=L_(o)^(3)(1+alpha_(1) Delta theta)(1+ alpha_(2)Delta alpha)^(2)`
Since `" "L_(o)^(3)=V_(o) andL^(3)=V,`
Hence `" " 1+gamma Delta theta = (1+alpha_(1) Delta theta)(1+ alpha_(2) Delta theta)^(2)`
`" "~~(1+alpha_(1)Delta theta)(1+2alpha_(2)Delta theta)`
`" "~~(1+alpha_(1)Delta theta+2alpha_(2)Delta theta)`
`rArr" "gamma=alpha_(1)+2alpha_(2)`