First slide

Thermal expansion

Question

Two different wires having length $L_{1}$ 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:

Moderate

A

$\frac{α_{1} + α_{2}}{2}$
B

$2 \sqrt{α_{1} α_{2}}$
C

$4 \frac{α_{1} α_{2}}{α_{1} + α_{2}} \frac{L_{2} L_{1}}{{(L_{2} + L_{1})}^{2}}$
D

$\frac{α_{1} L_{1} + α_{2} L_{2}}{L_{1} + L_{2}}$

Solution

Thermal expansion for effective wire
$\begin{array}{l} L_{e f f} α_{e f f} Δ θ = L_{1} α_{1} Δ θ + L_{2} α_{2} Δ θ \\ \Rightarrow α_{e f f} = \frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}} \end{array}$

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