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Show that the coefficient of area expans...

Show that the coefficient of area expansions, `(DeltaA//A)//DeltaT`, of a rectangular sheet of the solid is twice its linear expansively, `alpha_(l).(alpha_(l)=10^(-5)K^(-1))`

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Suppose length of rectangle sheet of solid material is a and width is b.
When its temperature is increased by `DeltaT`, then increase in a, `Deltaa=alpha_(l)aDeltaT` and increase in b, `Deltab=alpha_(1)bDeltaT`.
Total increase in area of rectangular cube,
`DeltaA=DeltaA_(1)+DeltaA_(2)+DeltaA_(3)`
`=aDeltab+bDeltaa+(Deltaa)(Deltab)`
`=a(alpha_(l)dDeltaT)+b(alpha_(l)aDeltaT)+(alpha_(l)a DeltaT)(alpha_(l)bDeltaT)`
`=alpha_(l)abDeltaT+alpha_(l)abDeltaT+alpha_(l)^(2)ab(DeltaT^(2))`
`=alpha_(l)abDeltaT[2+alpha_(l)DeltaT]`
`=alpha_(l)ADeltaT" "[2+alpha_(l)DeltaT]" "[because" ab = A area"]`
By neglecting `a_(l)DeltaT` in comparison to 2,
`DeltaA=alpha_(l)ADeltaTxx2`
`(DeltaA)/(A DeltaT)=2alpha_(l)` hence, proved.
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Knowledge Check

  • A copper rod of 88cm and an aluminimum rod of unknown length have their increase in length independent of increase in temperature. The length of aluminium rod is: (alpha_(Cu) =1.7 xx 10^(-5) K^(-1) and alpha_(Al)= 2.2 xx 10^(-5) K^(-1))

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