Two different wires having lengths L 1 and L 2 and respective temperature coefficient of linear expansion α 1 and α 2 , are joined end-to-end. Then the effective temperature coefficient of linear expa

Question

Two different wires having lengths L 1 and L 2 and respective temperature coefficient of linear expansion α 1 and α 2 , are joined end-to-end. Then the effective temperature coefficient of linear expansion is :

TestHub · Accepted Answer

The wire are joined end-to-end, so they are in series combination, then total length of wire is given by
$L = L_{1} + L_{2} . . . (1)$

When the temperature is increase by $∆ θ$ , then its final length is given by using concept of linear expansion,
$L^{'} = L (1 + α ∆ θ) . . . (2)$

Using equation $(1)$ and $(2)$ for the series combination of wires, final length is given by

$L^{'} = {L_{1}}^{'} + {L_{2}}^{'} . . . (3)$

$\Rightarrow (L_{1} + L_{2}) (1 + α_{eff} ∆ θ) = L_{1} (1 + α_{1} ∆ θ) + L_{2} (1 + α_{2} ∆ θ)$

$\Rightarrow α_{eff} = \frac{L_{1} α_{1} + L_{2} α_{2}}{(L_{1} + L_{2})}$

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