For an isotropic rectangular solid with initial dimensions and

PHYSICS · Multi-part · CBSE · Physics · Thermal Properties of Matter · Thermal Properties of Matter

Problem

For an isotropic rectangular solid with initial dimensions

l,\;b

and

t

, each linear dimension increases proportionally with temperature rise

\Delta T

. (i) Without assuming smallness, derive an exact expression for the percentage change in the area of the face

lb

when the temperature is raised by

\Delta T

. (ii) Show that for

\alpha\Delta T \ll 1

, the expression obtained in (i) reduces to

\beta\Delta T

where

\beta

is approximately

2\alpha

. (iii) Taking

\alpha = 4.0\times10^{-5}\,\text{K}^{-1}

and

\Delta T = 100\,\text{K}

, compute the percentage error introduced when the approximation

\beta = 2\alpha

is used instead of the exact relation.