Show that the coefficient of area expansion , (Delta A//A)//Delta T, of a rectangular sheet of the solid is twice its linear expansivity , alpha_(1).

Show that the coefficient of area expansion , (Delta A//A)//Delta T, o

1.	Show that the coefficient of area expansion , (Delta A//A)//Delta T, of a rectangular sheet of the solid is twice its linear expansivity , alpha_(1).
Answer» SOLUTION : Consider a rectangular sheet of the solid material of length a and breadth b (FIG. 11.8). When the temperature increases by`Delta T, a `Increases by `Delta a = alpha_(1) a Delta T` andbincreases by `Delta b = alpha_(1), b Detla T`. From Fig. 11.8, the increase in area `Delta A = DeltaA_(1) + DeltaA_(2) + DeltaA_(3)` `DELTAA = a Deltab + b Delta a + (Delta a) (Deltab)` ` = a alpha_(1) b DeltaT + b alpha_(1) a DeltaT + (alpha_(1))^(2) ab (Delta T)^(2)` ` = alpha_(1) ab DeltaT (2 +alpha_(1) DeltaT) = alpha_(1) A DeltaT (2 + alpha_(1) DeltaT)` Since `alpha_(1) approx 10^(-5) K^(-1)`,from Table 11.1 , the product `alpha_(1) Delta T ` for fractional temperature is small in comparision with 2 and MAY be neglected. Hence, `((Delta A)/A) 1/(Delta T) approx 2alpha_(L) `