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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 $40^\circ C$ $?$ (Given $\alpha_{steel} $ = $11 \times {10^{ - 6}}$per$°C$ and $\alpha_{copper}$ = $17 \times {10^{ - 6}}per\,^\circ C$)
$80.0096\,cm$
$80.0272\,cm$
$1\,cm$
$25.2\,cm$
Solution
(a) 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
${L_{app}} = L{'_{cu}} – \,L{'_{steel}} = [{L_0}(1 + {\alpha _{Cu}}\Delta \theta ) – {L_0}(1 + {\alpha _s}\Delta \theta )$
==> ${L_{app}} = {L_0}({\alpha _{Cu}} – {\alpha _s})\Delta \theta $
$ = 80(17 \times {10^{ – 6}} – 11 \times {10^{ – 6}}) \times 20 = 80.0096 cm$
