The coefficient of linear expansion of brass and steel are ${\alpha _1}$ and ${\alpha _2}$. If we take a brass rod of length ${l_1}$ and steel rod of length ${l_2}$ at $0°C$, their difference in length $({l_2} - {l_1})$ will remain the same at a temperature if
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(d) ${L_2} = {l_2}(1 + {\alpha _2}\Delta \theta )$ and ${L_1} = {l_1}(1 + {\alpha _1}\Delta \theta )$
$ \Rightarrow ({L_2} - {L_1}) = ({l_2} - {l_1}) + \Delta \theta ({l_2}{\alpha _2} - {l_1}{\alpha _1})$
Now $({L_2} - {L_1}) = ({l_2} - {l_1})$ so, ${l_2}{\alpha _2} - {l_1}{\alpha _1} = 0$
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