Heat Transfer — Conduction, Convection, Radiation

Heat always flows from a hotter body to a cooler one — never spontaneously in the reverse direction. But the way it flows depends on the medium and the circumstances. The three mechanisms of heat transfer — conduction, convection, and radiation — are fundamentally different in their physics, and each dominates in different situations.

Understanding these is essential not just for exam problems but for making sense of everyday phenomena: why a metal spoon in hot tea feels hotter than a wooden one, why rooms are heated from below (not above), and how the Sun warms the Earth despite the vacuum of space between them.

Key Terms & Definitions

Thermal conductivity (k): A material property measuring how well it conducts heat. Unit: W/(m·K). High $k$ → good conductor (metals); low $k$ → poor conductor/good insulator (wood, air, glass wool).

Temperature gradient: The rate of change of temperature with distance: $dT/dx$ . The driving force for conduction.

Newton’s Law of Cooling: The rate of heat loss of a body is proportional to the difference between its temperature and the ambient temperature (valid for small temperature differences).

Emissivity (ε): A dimensionless number (0 to 1) indicating how effectively a surface radiates energy compared to a perfect blackbody. $\varepsilon = 1$ for an ideal blackbody; $\varepsilon < 1$ for real surfaces. A shiny, polished surface has $\varepsilon \approx 0.02$ ; matte black surfaces have $\varepsilon \approx 0.95$ .

Stefan-Boltzmann constant (σ): $\sigma = 5.67 \times 10^{-8}\text{ W/m}^2\text{K}^4$

Wien’s Displacement Law: The wavelength at which a blackbody emits maximum radiation: $\lambda_{\max} T = 2.898 \times 10^{-3}$ m·K.

Conduction

Heat flows through a material from regions of higher temperature to lower temperature by direct atomic/molecular collisions or free electron transfer, without bulk movement of the material itself.

Mechanism: In metals, free electrons carry kinetic energy from hot regions to cold regions — this is why metals are far better conductors than non-metals. In non-metals, energy transfers through lattice vibrations (phonons).

\frac{dQ}{dt} = -kA\frac{dT}{dx}

Where:

$dQ/dt$ = rate of heat flow (W)
$k$ = thermal conductivity (W/m·K)
$A$ = cross-sectional area (m²)
$dT/dx$ = temperature gradient (K/m)
Negative sign: heat flows in direction of decreasing temperature

Thermal resistance: Analogous to electrical resistance. For a slab: $R_{th} = L/(kA)$ . For slabs in series: $R_{total} = R_1 + R_2 + ...$ . For slabs in parallel: $1/R_{total} = 1/R_1 + 1/R_2 + ...$

Practical applications:

Cooking utensils use metals (high $k$ ) for efficient heat distribution
Building insulation uses glass wool, foam (low $k$ ) to retard heat loss
Thermocol and still air are poor conductors — used in packaging
The “feel” of materials: metal at room temperature feels colder than wood because metal conducts heat away from your hand faster

Convection

Heat transfer by the movement of the heated fluid (liquid or gas) itself. The bulk motion of fluid carries thermal energy from one place to another.

Natural (free) convection: Driven by density differences due to temperature. Warm fluid becomes less dense, rises; cool fluid sinks. Circular convection currents form.

Forced convection: An external agent (fan, pump) drives fluid movement, speeding up heat transfer.

Mechanism: When a fluid is heated, thermal expansion decreases its density. Buoyant force lifts the lighter warm fluid while denser cool fluid sinks, creating circulation.

Newton’s Law of Cooling (convection context):

\frac{dQ}{dt} = hA(T - T_0)

Where $h$ = convection heat transfer coefficient, $A$ = surface area, $T$ = surface temperature, $T_0$ = fluid temperature.

Practical applications:

Household radiators heat air by convection — this is why heaters are placed near the floor (warm air rises, displacing cooler air above)
Ceiling fans circulate air, forcing convection and cooling people
Sea and land breezes (differential heating of sea and land creates convection)
Blood circulation distributes heat throughout the human body

For JEE numerical problems on Newton’s law of cooling: if the temperature difference is small, the rate of cooling is approximately $dT/dt \propto (T - T_0)$ . Integration gives $T(t) = T_0 + (T_i - T_0)e^{-kt}$ where $T_i$ is initial temperature. This exponential cooling law is tested directly.

Radiation

Heat transfer by electromagnetic waves (photons), requiring no medium. This is the only mechanism that works through a vacuum.

Mechanism: All objects with temperature > 0 K emit electromagnetic radiation due to the oscillation of charged particles. The intensity and spectrum depend on temperature.

Power radiated per unit area by a blackbody:

P = \varepsilon \sigma T^4

Net power exchanged between a body (temperature $T$ ) and surroundings (temperature $T_0$ ):

P_{net} = \varepsilon \sigma A (T^4 - T_0^4)

\lambda_{\max} \cdot T = b = 2.898 \times 10^{-3}\text{ m·K}

As temperature increases, the peak radiation wavelength shifts to shorter wavelengths (higher frequency).

Blackbody: An ideal absorber and emitter that absorbs all incident radiation (absorptivity = 1) and emits maximum radiation at a given temperature. Real objects approximate this with emissivity $\varepsilon < 1$ .

Kirchhoff’s Law: Good absorbers are also good emitters at the same temperature. $\varepsilon = \alpha$ (emissivity equals absorptivity). Shiny surfaces (low $\varepsilon$ ) both reflect more and emit less.

Practical applications:

Solar heating: Earth absorbs solar radiation (peak in visible spectrum). Greenhouse effect traps outgoing infrared.
Thermal cameras detect infrared emission from objects (used in night vision).
White/light-coloured clothing reflects solar radiation and keeps you cooler outdoors.
Dewar (thermos) flask: double-walled with vacuum (no conduction/convection) and silvered walls (minimise radiation).

Solved Examples

Example 1 — CBSE Level: Conduction through a wall

Q: A glass window (thickness 4 mm, area 2 m², $k = 1\text{ W/m·K}$ ) has temperatures 25°C outside and 20°C inside. Find the heat flow rate.

Solution:

\frac{dQ}{dt} = kA\frac{\Delta T}{\Delta x} = 1 \times 2 \times \frac{25-20}{4 \times 10^{-3}} = \frac{10}{0.004} = 2500\text{ W}

2500 W (2.5 kW) of heat flows in — quite significant for just a temperature difference of 5°C!

Example 2 — JEE Main Level: Newton’s Law of Cooling

Q: A body at 60°C cools to 50°C in 5 minutes when room temperature is 25°C. How long will it take to cool from 50°C to 40°C?

Solution: By Newton’s law: rate of cooling ∝ (average excess temperature).

Average temperature for first interval: $(60+50)/2 = 55°C$ . Excess: $55-25 = 30°C$ . Rate: $10°C/5\text{ min} = 2°C/\text{min}$ .

Average temperature for second interval: $(50+40)/2 = 45°C$ . Excess: $45-25 = 20°C$ .

New rate: $2°C/min \times (20/30) = 4/3°C/min$ .

Time for 10°C cooling: $10 ÷ (4/3) = 7.5$ minutes.

Example 3 — JEE Advanced Level: Wien’s Law

Q: The Sun emits maximum intensity at $\lambda = 500\text{ nm}$ . A furnace at 600 K emits maximum intensity at what wavelength?

Solution:

For Sun: $\lambda_{sun} T_{sun} = 2.898 \times 10^{-3}$ → $T_{sun} = 2.898 \times 10^{-3}/500 \times 10^{-9} = 5796\text{ K} \approx 5800\text{ K}$ .

For furnace ( $T = 600\text{ K}$ ): $\lambda = 2.898 \times 10^{-3}/600 = 4.83 \times 10^{-6}\text{ m} = 4830\text{ nm}$ (infrared region).

Exam-Specific Tips

CBSE Class 10: Qualitative descriptions of three modes, examples, and comparisons are expected. No derivations.

CBSE Class 11: Newton’s Law of Cooling, Stefan-Boltzmann law, Wien’s displacement law. Derivation of thermal resistance analogy.

JEE Main: Numerical problems on all three modes. Newton’s Law of Cooling numerical (the “average temperature” shortcut). Wien’s law for temperature/wavelength calculations.

JEE Advanced: Combination problems (radiation from one surface to another, conduction with varying cross-section, coupled Newton’s cooling problems).

Common Mistakes to Avoid

Mistake 1: Saying “vacuum doesn’t allow any heat transfer.” Radiation passes through vacuum (how else does sunlight reach Earth?). Only conduction and convection require a material medium.

Mistake 2: In Newton’s law problems, using the instantaneous temperature difference instead of the average temperature difference for the interval. The correct approach for finite intervals is to use the average of initial and final temperatures, then subtract ambient.

Mistake 3: In Stefan-Boltzmann law, using Celsius instead of Kelvin. The law requires absolute temperature in Kelvin. $P \propto T^4$ — a factor of $T$ in Kelvin, not °C. Always convert: 27°C = 300 K before substituting.

Mistake 4: Confusing thermal conductivity of a material with the rate of heat transfer in a specific situation. Thermal conductivity $k$ is a material property. The actual heat flow rate also depends on the geometry ( $A$ , $L$ ) and temperature gradient.

Practice Questions

Q1. Why does a metal spoon feel colder than a plastic spoon at the same room temperature when you touch them?

Both are at the same temperature. Metal has much higher thermal conductivity than plastic. When you touch metal, it conducts heat away from your hand much faster — your hand loses heat rapidly, creating a sensation of coldness. Plastic conducts heat away slowly. The feeling is about the rate of heat loss, not the temperature.

Q2. A blackbody radiates at rate P at temperature 300 K. At what temperature will it radiate 16P?

$P \propto T^4$ . $16P/P = (T'/300)^4$ → $16 = (T'/300)^4$ → $T'/300 = 2$ → $T' = 600$ K.

Q3. Why is the bottom of a cooking pan made of copper while the handle is plastic or wood?

Copper has very high thermal conductivity — it distributes heat evenly across the base of the pan, preventing hot spots. The handle is plastic or wood (poor conductors) to insulate the user’s hand from the heat. This is a practical application of choosing materials based on thermal conductivity.

Q4. At what wavelength does a human body (37°C = 310 K) emit peak radiation?

$\lambda_{max} = b/T = 2.898 \times 10^{-3}/310 \approx 9.35 \times 10^{-6}$ m = 9.35 μm (mid-infrared range). This is why thermal cameras can detect human body heat.

FAQs

Why are houses painted white in hot climates?

White surfaces have high reflectivity (low absorptivity) in the visible spectrum, where sunlight has most of its intensity. They absorb less solar radiation, keeping the house cooler. Black surfaces absorb more solar radiation and become hotter. This is Wien’s law + Kirchhoff’s law in action.

Is convection possible in solids?

No. Convection requires the bulk movement of the medium. In solids, atoms are locked in a lattice and cannot move in bulk. Conduction is the only heat transfer mechanism within a solid (apart from radiation from its surface).

Why does a thermos flask keep both hot and cold drinks?

A thermos flask has three features: (1) vacuum between double walls (eliminates conduction and convection), (2) silvered inner walls (reflect radiation back, reducing radiative loss), (3) tight seal (eliminates convection from open top). These features resist heat flow in either direction — hot liquids stay hot and cold liquids stay cold for the same physical reason.