The coefficients of linear expansion of brass and steel rods are $\alpha_1$ and $\alpha_2$, lengths of brass a

Question

The coefficients 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 $(l_2 - l_1)$ is maintained the same at all temperatures, which one of the following relations holds good?

Accepted Answer

For the difference in lengths $(l_2 - l_1)$ to remain constant at all temperatures, the change in length of both rods must be equal for any given change in temperature $\Delta T$.
Let the change in length of the brass rod be $\Delta l_1$ and that of the steel rod be $\Delta l_2$.
We know that $\Delta l = l \alpha \Delta T$.
So, $\Delta l_1 = l_1 \alpha_1 \Delta T$ and $\Delta l_2 = l_2 \alpha_2 \Delta T$.
Since $\Delta l_1 = \Delta l_2$, we have:
$l_1 \alpha_1 \Delta T = l_2 \alpha_2 \Delta T$
$\alpha_1 l_1 = \alpha_2 l_2$

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