Physics · Thermal Properties Of Matter · NEET
Take a square sheet of side L. Each side grows to L(1 + alpha delta T). New area = L squared (1 + alpha delta T) squared = L squared (1 + 2 alpha delta T + alpha squared delta T squared). Because alpha is very small (about 10^-5 per degree C), the alpha squared term is tiny and we drop it. So new area is about L squared (1 + 2 alpha delta T). This means delta A = 2 alpha A delta T, so beta = 2 alpha. The alpha squared piece is thrown away because it is negligible, not because the answer is alpha squared.
The change in area is delta A = beta A delta T = 2 alpha A delta T. Here A is the original area, delta T is the rise in temperature, and beta = 2 alpha is the coefficient of area expansion. The final area becomes A' = A(1 + 2 alpha delta T).
The hole gets bigger. Every part of the plate expands as if the hole were also made of metal. So the hole area increases by the same rule, delta A = 2 alpha A delta T, where A is the hole area. This is the same idea a blacksmith uses when heating an iron ring so it slips over a wheel.
The unit is per kelvin (K^-1) or equivalently per degree Celsius (degree C^-1). Since beta = delta A / (A delta T), the areas cancel and only the temperature is left in the denominator, so the unit depends only on temperature.
For an isotropic solid (expands equally in all directions), alpha : beta : gamma = 1 : 2 : 3. So beta = 2 alpha and gamma = 3 alpha, where gamma is the coefficient of volume (cubical) expansion. The numbers 1, 2, 3 come from length being 1 dimension, area 2 dimensions, and volume 3 dimensions.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Superficial expansion means the increase in the surface area of a solid when its temperature rises. It is measured by the coefficient of area expansion beta, defined by delta A = beta A delta T.
For a solid that expands equally in all directions, beta = 2 alpha, where alpha is the coefficient of linear expansion. Typical alpha values are around 10^-5 per degree C, so beta is around 2 x 10^-5 per degree C.
No. The formula delta A = 2 alpha A delta T works for any shape (square, circle, or a plate with a hole), because it only uses the original area A and the material's alpha.
It is nearly constant over the usual NEET temperature range (0 to 100 degree C), but strictly it can vary a little with temperature, just like alpha. For NEET problems you treat beta = 2 alpha as constant.
