Physics and Mathematics
Relation between Three Coefficients
1. Concept Overview
When a solid body is heated, it expands in length, area, and volume depending on its dimensions.
- The Coefficient of Linear Expansion (α) corresponds to change in length.
- The Coefficient of Areal (or Superficial) Expansion (β) corresponds to change in area.
- The Coefficient of Volumetric (or Cubical) Expansion (γ) corresponds to change in volume.
All three coefficients are interrelated, since expansion in one direction affects the others.
The mathematical relationships among them are:
[\beta = 2\alpha] [\quad \text{and}] [\quad] [\gamma = 3\alpha]
2. Explanation and Mathematical Derivation
(a) Linear Expansion
If a solid rod of initial length [L₀] is heated through a temperature rise [ΔT], its increase in length [ΔL] is given by:
[\Delta L = \alpha L₀ \Delta T]
Therefore, the new length is:
[L = L₀(1 + \alpha \Delta T)]
(b) Areal (Superficial) Expansion
Consider a rectangular sheet of the same material, having an initial length [L₀] and breadth [B₀].
After heating through [ΔT], they become:
[L = L₀(1 + \alpha , \Delta T)] [\quad \text{and}] [\quad] [B = B₀(1 + \alpha , \Delta T)]
Hence, the new area is:
[A] [= L \times B] [= L₀ B₀ (1 + \alpha \Delta T)^2]
Expanding and neglecting higher powers of [\alpha , \Delta T]:
[A] [= A₀(1 + 2\alpha \Delta T)]
Comparing with [A = A₀(1 + \beta , \Delta T)], we get:
[\beta = 2\alpha]
(c) Volumetric (Cubical) Expansion
For a cube of side [L₀], the initial volume is [V₀ = L₀^3].
After heating through [ΔT], the new side is:
[L = L₀(1 + \alpha \Delta T)]
Thus, the new volume is:
[V] = [L^3] [= L₀^3(1 + \alpha \Delta T)^3]
Expanding and neglecting higher powers of [\alpha , \Delta T]:
[V = V₀(1 + 3\alpha \Delta T)]
Comparing with [V = V₀(1 + \gamma \Delta T)], we get:
[\gamma = 3\alpha]
3. Dimensions and Units
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Coefficient of Linear Expansion | [\alpha] | K⁻¹ | [K⁻¹] |
| Coefficient of Areal Expansion | [\beta] | K⁻¹ | [K⁻¹] |
| Coefficient of Volumetric Expansion | [\gamma] | K⁻¹ | [K⁻¹] |
4. Key Features
- The three coefficients are not independent — they are mathematically related.
- These relations hold for small temperature changes, where expansion is approximately linear.
- For isotropic materials, expansion occurs uniformly in all directions.
- The relations are derived geometrically, using binomial approximation.
- The coefficients depend only on material type, temperature range, and state of matter.
- These relations are important in engineering, thermal design, and materials science.
5. Important Formulas to Remember
| Type of Expansion | Formula | Relation |
|---|---|---|
| Linear Expansion | [\Delta L = \alpha L₀ \Delta T] | — |
| Areal Expansion | [\Delta A = \beta A₀ \Delta T] | [\beta = 2\alpha] |
| Volumetric Expansion | [\Delta V = \gamma V₀ \Delta T] | [\gamma = 3\alpha] |
| Final Length | [L = L₀(1 + \alpha \Delta T)] | — |
| Final Area | [A = A₀(1 + \beta \Delta T)] | — |
| Final Volume | [V = V₀(1 + \gamma \Delta T)] | — |
6. Conceptual Questions with Solutions
1. Why are the coefficients of expansion related to one another?
Because expansion in one direction affects the overall area and volume of a body in proportion to its geometry.
2. What is the relation between β and α?
For small temperature changes, [\beta = 2\alpha].
3. What is the relation between γ and α?
For small temperature changes, [\gamma = 3\alpha].
4. Why is β not exactly equal to 2α in all cases?
Because for large temperature changes, higher powers of [\alpha \, \Delta T] cannot be neglected.
5. Why is γ not exactly equal to 3α?
For the same reason — higher order terms become significant when expansion is large.
6. What is the physical meaning of α = 10⁻⁵ K⁻¹?
It means the length increases by [10⁻⁵] times its original length for every 1 K rise in temperature.
7. Is α constant for all materials?
No, α depends on the material and the temperature range.
8. What happens if a cube expands isotropically?
Its linear, areal, and volumetric expansions maintain the ratios [1 : 2 : 3].
9. Does γ apply to liquids and gases?
Yes, because liquids and gases generally expand volumetrically.
10. How are these relations used in engineering?
They help in designing components like metal joints, bridges, and containers that experience temperature variations.
11. What mathematical principle is used in the derivation?
The **binomial approximation**, [(1 + x)^n ≈ 1 + nx], for small x.
12. If α = 2 × 10⁻⁵ K⁻¹, what is γ?
[\gamma] [= 3\alpha] [= 6 \times 10^{-5} \K^{-1}]
13. Why can we neglect higher powers of αΔT?
Because [\alpha \, \Delta T] is typically very small (≈10⁻³), making higher powers insignificant.
14. Does the relation hold for anisotropic materials?
No, it applies only to isotropic materials that expand equally in all directions.
15. Is the ratio β:γ always 2:3?
Yes, approximately, under small temperature changes.
7. FAQ / Common Misconceptions
1. Are α, β, and γ independent constants?
No, they are interrelated by geometry.
2. Is β always exactly twice α?
Only approximately, for small temperature ranges.
3. Is γ always exactly thrice α?
Only approximately, under linear expansion conditions.
4. Do these relations hold for liquids?
Only the volumetric relation applies to liquids.
5. Do gases follow these relations?
No, gas expansion follows the gas laws, not solid expansion relations.
6. Is α larger than β or γ?
No, α < β < γ for a given material.
7. Does temperature affect these coefficients?
Yes, slightly — they vary with temperature.
8. Are these coefficients dimensionless?
No, their unit is K⁻¹.
9. Can we use the same relations for anisotropic solids like crystals?
No, because such solids expand differently along different axes.
10. Is this relation derived experimentally?
No, it’s derived geometrically using the binomial expansion.
8. Practice Questions with Step-by-Step Solutions
Q1. If [\alpha = 2 \times 10^{-5} K^{-1}], find β and γ.
Solution:
[\beta = 2\alpha = 4 \times 10^{-5} K^{-1}]
[\gamma = 3\alpha = 6 \times 10^{-5} K^{-1}]
Q2. A square metal plate of area [1 m²] expands when heated. If [\alpha = 1.2 \times 10^{-5} K^{-1}] and [\Delta T = 50°C], find the percentage increase in area.
Solution:
[\beta = 2\alpha = 2.4 \times 10^{-5} K^{-1}]
[\dfrac{\Delta A}{A₀} = \beta \Delta T = 2.4 \times 10^{-5} \times 50 = 1.2 \times 10^{-3}]
Percentage increase = [0.12%]
Q3. A cube has a side of 10 cm. If [\alpha = 2 \times 10^{-5} K^{-1}] and [\Delta T = 100°C], find the increase in volume.
Solution:
[\gamma = 3\alpha = 6 \times 10^{-5} K^{-1}]
[\Delta V = \gamma V₀ \Delta T = 6 \times 10^{-5} \times (10)^3 \times 100 = 6 , cm³]
Q4. Show that for small temperature changes, [\gamma = 3\alpha].
Solution:
From [V = L³],
[\dfrac{\Delta V}{V₀} = 3\dfrac{\Delta L}{L₀}]
[\Rightarrow \gamma = 3\alpha]
Q5. If [\beta = 1.8 \times 10^{-5} K^{-1}], find [\alpha] and [\gamma].
Solution:
[\alpha = \dfrac{\beta}{2} = 0.9 \times 10^{-5} K^{-1}]
[\gamma = 3\alpha = 2.7 \times 10^{-5} K^{-1}]
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