The coefficient of linear expansion of brass and steel rods are $\alpha_1$ and $\alpha_2$. Lengths of brass an

Question

The coefficient of linear expansion of brass and steel rods are $\alpha_1$ and $\alpha_2$. Lengths of brass and steel rods are $L_1$ and $L_2$ respectively. If $(L_2-L_1)$ remains the same at all temperatures, which one of the following relations holds good?

Accepted Answer

Let the change in temperature be $\Delta T$.
The new lengths of the brass and steel rods will be:
$L_1' = L_1(1 + \alpha_1\Delta T)$
$L_2' = L_2(1 + \alpha_2\Delta T)$
Given that the difference in lengths $(L_2 - L_1)$ remains the same at all temperatures:
$L_2' - L_1' = L_2 - L_1$
$L_2(1 + \alpha_2\Delta T) - L_1(1 + \alpha_1\Delta T) = L_2 - L_1$
$L_2 + L_2\alpha_2\Delta T - L_1 - L_1\alpha_1\Delta T = L_2 - L_1$
$L_2\alpha_2\Delta T - L_1\alpha_1\Delta T = 0$
$\implies L_2\alpha_2 = L_1\alpha_1$
$\implies \alpha_1L_1 = \alpha_2L_2$

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