Coefficient of linear expansion of brass and steel rods are `alpha_(1)` and `alpha_(2)`. Length of brass and steel rods are `l_(1)` and `l_(2)` respectively. If `(l_(2) - l_(1))` is maintained same at all temperature, which one of the following relations holds good?

Options

A. `alpha_(1)l_(2)=alpha_(2)l_(2)`
B. `alpha_(1)l_(2)^(2)=alpha_(2)l_(1)^(2)`
C. `alpha_(1)^(2)l_(2)=alpha_(2)^(2)l_(1)`
D. `alpha_(1)l_(1)=alpha_(2)l_(2)`

Answer

D

Explanation

Let `Delta T` be the change in temperature.
For brass, `l'_(1) =l_(1)(1+alpha_(1)Delta T)`
For steel, `l'_(2) =l_(2)(1+alpha_(2)Delta T)`
`:. l'_(2)-l'_(1) =l_(2)(1+alpha_(2)Delta T) -l_(1)(1+alpha_(1)Delta T)`
`=(l_(2)-l_(1)) + (l_(2)alpha_(2)-l_(1)alpha_(1))Delta T`
Given, `l'_(2) -l'_(1) =l_(2) -l_(1)`, so
`l_(2)-l_(1)=(l_(2)-l_(1)) + (l_(2)alpha_(2)-l_(1)alpha_(1))Delta T`
or `l_(2)alpha_(2)-l_(1)alpha_(1)=0` or `l_(2)alpha_(2)-l_(1)alpha_(1)`