Q. A steel scale measures the length of copper wire as $80.0cm,$ when both are at $20℃$ (the calibration temperature for scale). What would be the scale read for the length of the wire when both are at $40℃?$ (Given $\alpha_{steel} = 11 \times 10^{- 6}\ per℃\ and\ \alpha_{copper} = 17 \times 10^{- 6}per℃$)
A) $80.0096\ cm$
B) $80.0272\ cm$
C) $1\ cm$
D) $25.2\ cm$
Explanation
(a) With temperature rise (same $20℃$ 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
$L_{app} = L_{cu}^{'} - L_{steel}^{'} = \lbrack L_{0}\left( 1 + \alpha_{Cu}\mathrm{\Delta}\theta \right) - L_{0}(1 + \alpha_{s}\mathrm{\Delta}\theta)\rbrack$
$\Rightarrow L_{app} = L_{0}(\alpha_{Cu} - \alpha_{s})\mathrm{\Delta}\theta$
$= 80\left( 17 \times 10^{- 6} - 11 \times 10^{- 6} \right) \times 20 = 0.0096\ cm$
$\therefore$ Length of the wire read $= 80.0096\ cm$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.