Two metal wires -1 & 2 of lengths `L_(1)` and `L_(2)` and of same materical are fixed between two different sets of rigid supports at negligible tension. The cross-sectional area of wire-1 is greater than cross sectional area of wire-2 by `Delta A`. Wire-1 is cooled by `t^(@)C`. Wire-2 of cross-sectional area A is cooled `Delta t^(@)C` greater than wire-1. For same tension in both the wires, `Delta t` will be

Options

A. `(L_(1) t Delta A)/(L_(2) A)`
B. `(L_(2)t Delta A)/(L_(1) A)`
C. `(t Delta A)/(A)`
D. `(t A)/(Delta A)`

Answer

C

Explanation

`Y = (T)/(A alpha t)`
Wires are of same material,
`Y_(1) = Y_(2)` and `alpha_(1) = alpha_(2)`
`T_(1) = YA_(1) alpha t_(1)` and `T_(2) = YA_(2) alpha t_(2)`
So for `T_(1) = T_(2), A_(1) t_(1) = A_(2) t_(2)`
`(A_(1))/(A_(2)) = (t_(2))/(t_(1)) = (t + Delta t)/(t) = 1 + (Delta t)/(t)`
or `(A_(1) - A_(2))/(A_(2)) = (Delta t)/(t) = (Delta A)/(A)`
`Delta t = (t Delta A)/(A)`