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The rods of length $L_1$ and $L_2$ are made of materials whose coefficients of linear expansion are $\alpha _1$ and $\alpha _2$. If the difference between the two lengths is independent of temperatures
$\left( {\frac{{{L_1}}}{{{L_2}}}} \right) = \left( {\frac{{{\alpha _1}}}{{{\alpha _2}}}} \right)$
$\frac{{{L_1}}}{{{L_2}}} = \frac{{{\alpha _2}}}{{{\alpha _1}}}$
$L_1^2{\alpha _1} = L_2^2{\alpha _2}$
$\alpha _1^2{L_1} = \alpha _2^2{L_2}$
Solution
$\mathrm{L}_{1}^{1}-\mathrm{L}_{2}^{1}=$ constant $=\mathrm{L}_{1}-\mathrm{L}_{2}$
$\left(\mathrm{L}_{1}+\mathrm{L}_{1} \alpha \Delta \mathrm{T}\right)-\left(\mathrm{L}_{2}+\mathrm{L}_{2} \alpha_{2} \Delta \mathrm{T}\right)=\mathrm{L}_{1}-\mathrm{L}_{2}$
$\mathrm{L}_{1} \alpha_{1}=\mathrm{L}_{2} \alpha_{2}$
$\frac{\mathrm{L}_{1}}{\mathrm{L}_{2}}=\frac{\alpha_{2}}{\alpha_{1}}$
