Physics · Thermal Properties Of Matter · NEET
beta = 2 alpha, NOT alpha squared. The confusion comes from thinking area = length squared, so the coefficient must also be squared. But a coefficient is not the area itself. When length grows by a factor (1 + alpha dT), area grows by that factor squared = (1 + alpha dT)^2 = 1 + 2 alpha dT + (alpha dT)^2. The term (alpha dT)^2 is extremely tiny (alpha is about 10^-5), so we drop it and get 1 + 2 alpha dT. That leading number 2 is what beta equals: beta = 2 alpha.
Same reason as area. Volume grows as (1 + alpha dT)^3 = 1 + 3 alpha dT + 3(alpha dT)^2 + (alpha dT)^3. Only the first correction term 3 alpha dT matters because the higher power terms have (alpha dT)^2 and (alpha dT)^3 which are near zero. So the fractional change in volume is 3 alpha dT, meaning gamma = 3 alpha. The power 3 comes out as a multiplier, it does not stay as a cube.
It is 1 : 2 : 3. Since beta = 2 alpha and gamma = 3 alpha, dividing all three by alpha gives 1 : 2 : 3. Another handy form: gamma = beta + alpha, and beta = gamma - alpha. If a question gives you gamma = 5.1 x 10^-5 per degree C, then alpha = gamma/3 = 1.7 x 10^-5 per degree C.
Only for isotropic solids - solids that expand equally in all directions (like most metals, glass). For a crystal that expands differently along different axes (anisotropic), you get gamma = alpha_x + alpha_y + alpha_z and beta depends on the plane. For NEET, assume isotropic unless told otherwise, so use 1 : 2 : 3 directly.
Just multiply. beta = 2 x alpha and gamma = 3 x alpha. Example: aluminium has alpha = 23 x 10^-6 per degree C, so beta = 46 x 10^-6 per degree C and gamma = 69 x 10^-6 per degree C. Keep the same unit (per degree C or per K, they are equal in size for temperature differences).
Try the real previous-year questions from this chapter — each with the answer and a full solution.
beta = 2 alpha and gamma = 3 alpha, so alpha : beta : gamma = 1 : 2 : 3. Here alpha is linear, beta is area (superficial) and gamma is volume (cubical) expansion coefficient.
Area is length squared. When each length becomes (1 + alpha dT) times bigger, area becomes (1 + alpha dT)^2 = 1 + 2 alpha dT (dropping the tiny alpha^2 term). So the area expands by 2 alpha dT, giving beta = 2 alpha.
Volume is length cubed. Volume becomes (1 + alpha dT)^3 = 1 + 3 alpha dT after ignoring tiny higher-power terms. So volume expands by 3 alpha dT, giving gamma = 3 alpha.
Yes. Since beta = 2 alpha and gamma = 3 alpha, we have alpha + beta = alpha + 2 alpha = 3 alpha = gamma. This is a quick cross-check in problems.
For liquids we mainly use gamma (volume) since they have no fixed shape. The 1:2:3 relation is meant for isotropic solids. Gases are handled by the ideal gas equation, not by alpha.
