Two different wires having lengths \(L _{1}\) and \(L _{2}\) and respective temperature coefficient of linear expansion \(\alpha_{1}\) and \(\alpha_{2},\) are joined end-to-end. Then the effective temperature coefficient of linear expansion is

At \(T +\Delta T \quad L _{ eq }= L _{1}+ L _{2}\)

where \(L _{1}= L _{1}\left(1+\alpha_{1} \Delta T \right)\)

\(L _{2}^{\prime}= L _{2}\left(1+\alpha_{2} \Delta T \right)\)

\(L _{ eq }^{\prime}=\left( L _{1}+ L _{2}\right)\left(1+\alpha_{ avg } \Delta T \right)\)

\(\Rightarrow\left(L_{1}+L_{2}\right)\left(1+\alpha_{avg } \Delta T \right)= L _{1}+ L _{2}+ L _{1} \alpha_{1} \Delta T + L _{2} \alpha_{2} \Delta T\)

\(\Rightarrow\left( L _{1}+ L _{2}\right) \alpha_{ avg }= L _{1} \alpha_{1}+ L _{2} \alpha_{2}\)

\(\Rightarrow \alpha_{ avg }=\frac{ L _{1} \alpha_{1}+ L _{2} \alpha_{2}}{ L _{1}+ L _{2}}\)