A steel scale measures the length of a copper wire as 80.0 cm, when both are at 20^\circ C (the calibration temperature for scale). What would be the scale read for the length of the wire when both are at $40^\circ \mathrm{C}$ ? (Given α α steel $= 1.1 \times 10^{-6}$ per°C and α α copper $= 17 \times 10^{-6} \text{ per } ^\circ \mathrm{C}$ )
Text Solution
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With temperature rise (same 25°C for both), steel scale and copper wire both expand. Hence
length of copper wire w.r.t. steel scale or apparent length of copper wire after rise in temperature
$L_{\text{app}} = L'_{\text{cu}} - L'_{\text{steel}} = \left[L_0(1 + \alpha_{\text{Cu}} \Delta \theta) - L_0(1 + \alpha_{\text{s}} \Delta \theta)\right]$
⇒ ⇒ $L_{\text{app}} = L_0 (\alpha_{\text{Cu}} - \alpha_{\text{s}}) \Delta \theta$
$= 80(17 \times 10^{-6} - 11 \times 10^{-6}) \times 20$ = 80.0096 cm
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