Thermal expansion — linear, area, volumetric

1.7.7 · Physics › Thermodynamics

Linear expansion ko scratch se derive karna

Toh $α$ ko proportionality constant ke roop mein define karo:

$\frac{d L}{L} = α d T$

Yeh step kyun? $d L / L$ fractional change hai — yahi physically meaningful quantity hai (ek fraction same rehta hai chahe rod mm mein ho ya miles mein). Constant $α$ ke liye integrate karo:

$\int_{L_{0}}^{L} \frac{d L}{L} = \int_{0}^{Δ T} α d T \Rightarrow ln \frac{L}{L _{0}} = α Δ T$

$L = L_{0} e^{α Δ T}$

Yeh step kyun? Exact form hai. Lekin $α \sim 1 0^{- 5} K^{- 1}$ bahut chota hai, toh $α Δ T ≪ 1$ . Use karo $e^{x} \approx 1 + x$ :

Area expansion — derive karo $β = 2 α$

Ek square plate lo, side $L_{0}$ , area $A_{0} = L_{0}^{2}$ .

$A = L^{2} = [L_{0} (1 + α Δ T)]^{2} = L_{0}^{2} (1 + α Δ T)^{2}$

Yeh step kyun? Dono sides same linear factor se expand hui (isotropic material). Square expand karo:

$A = A_{0} (1 + 2 α Δ T + α^{2} Δ T^{2})$

Yeh step kyun? Term $α^{2} Δ T^{2} \sim (1 0^{- 5})^{2} \sim 1 0^{- 10}$ negligible hai. Isko drop karo:

Volume expansion — derive karo $γ = 3 α$

Side $L_{0}$ ka cube, volume $V_{0} = L_{0}^{3}$ :

$V = L_{0}^{3} (1 + α Δ T)^{3} = V_{0} (1 + 3 α Δ T + 3 α^{2} Δ T^{2} + α^{3} Δ T^{3})$

Yeh step kyun? Teen directions → linear factor ka cube. Sirf first-order term rakho:

Figure — Thermal expansion — linear, area, volumetric

"Hole bhi expand hota hai" wala idea

Forecast-then-Verify

Recall Pehle forecast karo, phir check karo

Q: Ek pendulum clock ki brass rod garmiyon mein expand karti hai. Kya woh fast chalegi ya slow? Forecast: ... padhne se pehle commit karo. Verify: Lambi rod → lamba period $T = 2 π L / g$ → kam ticks → clock slow chalti hai. ✓ Expansion real hai aur measurable hai!

Common mistakes

Active recall

Linear expansion coefficient

α

define karo (units ke saath).

Fractional change in length per unit temperature rise,

d L / L = α d T

; units

K^{- 1}

β = 2 α

kyun hota hai?

Area

= L^{2}

hai, har side

(1 + α Δ T)

se grow karti hai, square karne pe

(1 + 2 α Δ T + \dots)

milta hai;

α^{2}

term drop karo.

γ = 3 α

kyun hota hai?

Volume

= L^{3}

hai,

(1 + α Δ T)

ka cube leading

1 + 3 α Δ T

deta hai.

Ratio

α : β : γ

1 : 2 : 3

Kya ek heated plate mein hole bada hota hai ya chota?

Bada — woh exactly waise expand karta hai jaise same metal se bhara ho.

Δ T

ke liye kya farak padta hai agar

^{\circ} C

use karo ya

K

Nahi, temperature ka difference dono mein same hota hai.

Linear expansion ka exact (un-approximated) form?

L = L_{0} e^{α Δ T}

Expansion possible kyun hoti hai (atomic reason)?

Asymmetric interatomic potential — average atom spacing bahar ki taraf drift karta hai jaise vibration energy badhti hai.

Ek pendulum clock ki rod garmi mein lambi ho jaati hai — fast chalegi ya slow?

Slow, kyunki period

T = 2 π L / g

badhta hai.

Recall Feynman: 12-saal ke bachhe ko samjhao

Socho ek bhare hue kamre mein sablog zyada jor se naachne lagte hain. Unhe zyada elbow room chahiye, toh woh thoda faile jaate hain — logo ki poori jagah thodi zyada ho jaati hai, jabki koi naya banda nahi aaya. Ek garam metal mein atoms zyada jor se "dance" karte hain aur unhe zyada room chahiye, toh metal thoda bada ho jaata hai. Agar tum ek flat sheet garam karo toh woh do directions mein badi hoti hai, aur ek solid block teen mein — toh blocks sabse zyada phulte hain, sheets uske baad, aur ek patli rod sabse kam.

Connections

Kinetic theory of gases — same atomic-jiggling idea; gases bahut zyada expand karte hain.
Interatomic potential energy curve — asymmetry root cause hai.
Stress and strain — block expansion + thermal stress jab constrained ho.
Anomalous expansion of water — famous exception ( $γ < 0$ from 0–4°C).
Pendulum and simple harmonic motion — clock-error example.
Gas laws — jahan $Δ T$ zaroor absolute hona chahiye, yahan ki tarah nahi.