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PhysicsThermal Expansion

A solid sphere of radius $R$ has coefficient of linear expansion $\alpha$. When its temperature increases by $\Delta T$, the increase in its volume is:

R
Solution written and verified by Roshan, science educator with 5 years of experience teaching NEET and JEE aspirants. Last reviewed September 2026.
Options
1
$4\pi R^3 \alpha \Delta T$
2
$\frac{4}{3}\pi R^3 \alpha \Delta T$
3
$\frac{4}{3}\pi R^3 (3\alpha) \Delta T$
4
$4\pi R^2 \alpha \Delta T$
Correct Answer
$4\pi R^3 \alpha \Delta T$
Solution
1

$V = \frac{4}{3}\pi R^3$; $\gamma = 3\alpha$

2

$\Delta V = V\gamma\Delta T = \frac{4}{3}\pi R^3 \times 3\alpha\Delta T = 4\pi R^3\alpha\Delta T$

Answer: $4\pi R^3\alpha\Delta T$

$\Delta V = V \cdot 3\alpha \cdot \Delta T = \frac{4}{3}\pi R^3 \times 3\alpha\Delta T = 4\pi R^3\alpha\Delta T$
$\gamma = 3\alpha$; $\beta = 2\alpha$
Theory: Thermal Expansion
1. Thermal Expansion

Linear: $\Delta l = l\alpha\Delta T$; $l_T = l_0(1+\alpha\Delta T)$. Area: $\Delta A = 2\alpha A\Delta T$ ($\beta = 2\alpha$). Volume: $\Delta V = 3\alpha V\Delta T$ ($\gamma = 3\alpha$). Typical $\alpha$ values: metals $\sim 10^{-5}$ K$^{-1}$; glass $\sim 9\times10^{-6}$; invar (iron-nickel alloy) $\sim 10^{-6}$ (very low, used in precision instruments). Liquids: only volume expansion, $\gamma_{liquid} \approx 10^{-4}$ to $10^{-3}$ K$^{-1}$, generally larger than solids. Gases: $\gamma_{gas} \approx 1/273$ K$^{-1}$ at 0°C (from Charles\' law).

2. Newton's Law of Cooling

$\frac{dT}{dt} = -k(T - T_s)$ where $T_s$ = surrounding temperature. Solution: $T - T_s = (T_0-T_s)e^{-kt}$. Valid for small temperature difference. Stefan\'s law (general): $\frac{dQ}{dt} = \varepsilon\sigma A(T^4 - T_s^4)$. For small $\Delta T$: reduces to Newton\'s law. Practical application: estimating cooling time of objects, time of death estimation in forensics.

3. Calorimetry

Heat transfer: $Q = mc\Delta T$ (sensible heat); $Q = mL$ (latent heat, phase change). Specific heat of water: $c_w = 4200$ J kg$^{-1}$ K$^{-1}$ (highest of common substances). Principle of calorimetry: heat gained = heat lost (isolated system). Latent heat of fusion (ice): $L_f = 336$ kJ/kg. Latent heat of vaporisation (water): $L_v = 2260$ kJ/kg. $L_v \gg L_f$ (much more energy to convert liquid to gas than solid to liquid).

4. Heat Transfer Mechanisms

Conduction: $Q/t = kA\Delta T/l$ (Fourier\'s law). $k$ = thermal conductivity. Good conductors: metals (Cu, Ag, Al). Poor conductors (insulators): wood, glass, air. Convection: transfer by fluid motion. Natural: density difference drives flow (warm fluid rises). Forced: fan or pump. Radiation: $P = \varepsilon\sigma AT^4$ (Stefan-Boltzmann). $\varepsilon$ = emissivity (black body $\varepsilon=1$). Kirchhoff\'s law: good absorber = good emitter at same wavelength. Greenhouse effect: Earth absorbs solar radiation, emits longer IR; greenhouse gases (CO2, CH4) absorb outgoing IR → warming.

Frequently Asked Questions
1. What is the relation between linear and volume expansion? ⌄
$\gamma = 3\alpha$; $\beta$ (area expansion) = $2\alpha$. So: $\Delta l = l\alpha\Delta T$; $\Delta A = A\beta\Delta T = 2A\alpha\Delta T$; $\Delta V = V\gamma\Delta T = 3V\alpha\Delta T$.
2. How is volume expansion formula derived? ⌄
$V = \frac{4}{3}\pi R^3$. $\Delta V = \frac{dV}{dR}\Delta R = 4\pi R^2 \cdot (R\alpha\Delta T) = 4\pi R^3\alpha\Delta T$. Or: $\Delta V = V\gamma\Delta T = \frac{4}{3}\pi R^3 \times 3\alpha\Delta T = 4\pi R^3\alpha\Delta T$.
3. Does a hole expand or contract when heated? ⌄
A hole (cavity) in a solid EXPANDS when heated — the same as if the hole were filled with the same material. The material expands isotropically, so the hole grows too. Common misconception: hole shrinks because material expands inward. Correct: material expands outward AND inward, so hole gets bigger.
4. What is the anomalous expansion of water? ⌄
Water density is maximum at 4°C. Below 4°C: water expands on cooling (unusual). Above 4°C: normal expansion. This anomalous behaviour: ice floats on water (less dense than 4°C water), lakes freeze from top → marine life survives at bottom.
5. What is thermal stress? ⌄
If a rod is prevented from expanding: thermal stress = $Y\alpha\Delta T$ where $Y$ = Young\'s modulus. Thermal stress = $Y \times$ thermal strain = $Y \times \alpha\Delta T$. For large temperature changes: joints/gaps in railway tracks, bridges, expansion joints in concrete roads prevent damage from thermal stress.
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