Question

show that the coefficient of area expansion (ΔA/A)/ΔT, of a rectangular sheet of the solid is twice its linear expansivity,∝1.

Answer 1

Answer:

please refer to the attached photo, hope you will be able to understand

Explanation:

You can mainly follow the 2nd method.

Hope you will be able to understand.

Answer 2

proved that, coefficient of area expansion = 2 × coefficient of linear expansion. i.e., 2α = β

let a rectangular sheet of length a and breadth b.

here, α is coefficient of linear expansion.

from linear expansion, after increasing temperature ∆T,

new length, a' = a(1 + α∆T)

new breadth, b' = b(1 + α∆T)

also β represents coefficient of area expansion.

now new area, A' = A(1 + β∆T)

⇒a(1 + α∆T)b(1 + α∆T) = ab(1 + β∆T) [ as area of rectangle, A = length × breadth = ab]

⇒(1 +α∆T)² = (1 + β∆T)

from Binomial expension,

if 1 >> x, ( 1 + x)ⁿ ≈ 1 + nx

here, 1 >> α∆T so, (1 + α∆T)² ≈ 1 + 2α∆T

so, 1 + 2α∆T = 1 + β∆T

⇒2α = β [hence proved]

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