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

Answer

B

Explanation

Linear expansion of brass = `alpha_(1)`
Linear expansion of steel `= alpha_(2)`
Length of brass rod `= l_(1)`
Length of steel rod `= l_(2)`
On increasing the temperature of the rods by `Delta T`, new lengths would be `l._(1) = l_(2) (1 + alpha_(1) Delta T)` ..(i)
`l._(2) = l_(2) (1 + alpha_(2) Delta T)` ...(ii)
Subtracting eqn. (i) from eqn .(ii) we get
`l._(2) -l._(1) =(l_(2) - l_(1)) + (l_(2) alpha_(2) - l_(1) alpha_(1)) Delta T`
According to question, `l._(2) - l._(1) = l_(2) = l_(1)` (for all temperature)
`:. l_(2) alpha_(2) - l_(1) alpha_(2) = 0 or l_(1) alpha_(1) = l_(2) alpha_(2)`