A solid rectangular sheet has two different coefficients of linear expansion `alpha_(1) " and " alpha_(2)` along its length and breadth respectively. The coefficient of surface expansion is `("for " alpha_(1)t "<<"1, alpha_(2)t"<<"1)`

Options

A. `(alpha_(1)+alpha_(2))/2`
B. `2(alpha_(1)+alpha_(2))`
C. `(4alpha_(1)alpha_(2))/(alpha_(1)+alpha_(2))`
D. `alpha_(1)+alpha_(2)`

Answer

D

Explanation

Let, initial length and breadth of rectangular sheet are `a_(1) " and " b_(1)` respectively.
If the final length and breadth are `alpha_(2) " and " b_(2)` respectively in increase in temperature `t^(@)C`, then
`a_(2)=a_(1)(1+alpha_(1)) " and " b_(2)=b_(1)(1+alpha_(2)t)`
`therefore alpha_(2)b_(2)=a_(1)b_(1)(1+alpha_(1)t)(1+alpha_(2)t)`
or, `a_(1)b_(1)(1+betat)=a_(1)b_(1){1+(alpha_(1)+alpha_(1))t+alpha_(1)alpha_(2)t^(2)}`
or, `1+betat=1+(alpha_(1)+alpha_(2))t`
`" "`[neglecting the term `alpha_(1)alpha_(2)t^(2)`]
`therefore beta=alpha_(1)+alpha_(2)`
The option (D) is correct.