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Heat Transfer: Conduction, Convection and Radiation

Heat moves in three ways. In conduction, heat passes from particle to particle while the particles stay in place; the rate through a slab is H = kA(T₁ − T₂)/L, where k is thermal conductivity. In convection, the fluid itself moves: hot fluid is lighter and rises, cool fluid sinks. In radiation, heat travels as electromagnetic waves and needs no medium. A blackbody absorbs all radiation and is the best emitter. Its peak wavelength falls as temperature rises (Wien: λmT = b), and its total emitted power grows as T⁴ (Stefan: P = σAT⁴). Newton's law of cooling says a body cools at a rate proportional to its temperature excess over the surroundings.

🎬 Step-by-step story

  1. One end of a metal rod sits in a flame. The hot particles shake harder and jostle their neighbours, passing the energy along. The wax pins fall one by one, near end first. This is conduction.
  2. Two equal rods, copper and glass, join a hot block to a cold block. Far more heat per second (orange dots) flows through copper. The rate is H = kA(T₁ − T₂)/L, and k is the thermal conductivity.
  3. Water at the bottom of a pot warms, becomes less dense and rises in the middle; cooler, denser water sinks at the sides. The moving water carries the heat. This is convection.
  4. Heat from the Sun crosses empty space where there is nothing to conduct or convect: this is radiation. The black ball absorbs more of it and gets hotter than the white one.
  5. A hot body glows. As its temperature rises, the colour moves from red to orange to white to bluish. The peak wavelength gets shorter (Wien's law), and the total power grows as T⁴ (Stefan's law).
  6. Your turn: slide the temperature. Double T and see λmax halve while the radiated power becomes 16 times bigger.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why do metals feel colder than wood at the same room temperature?

Metals are good conductors, so they pull heat away from your hand quickly. Wood conducts poorly, so your skin stays warm. Both are actually at the same temperature.

Why are cooking pots metal but handles wooden or plastic?

The pot must pass heat to the food quickly (high k). The handle must not pass heat to your hand (low k).

Why can't convection happen in solids?

Convection needs the material itself to flow. In a solid the particles are fixed in place, so only conduction (and radiation) can move heat.

How does sunlight reach us through empty space?

As electromagnetic waves. Waves do not need particles to travel, so radiation can cross the vacuum between the Sun and the Earth.

Why do stars have different colours?

By Wien's law the peak wavelength depends on temperature. Cooler stars peak in the red, hotter ones in the blue.

If a body radiates, why doesn't it cool down to 0 K?

It also absorbs radiation from its surroundings. The net loss is eσA(T⁴ − T₀⁴), which becomes zero when it reaches the surrounding temperature.

Three ways heat travels

Thermal conductivity

In steady state, heat flows through a bar of length L and cross-section A, with ends at T₁ and T₂ (T₁ > T₂), at the rate:

H = k A (T₁ − T₂) / L

H is in watts (J/s). k is the coefficient of thermal conductivity, unit W m⁻¹ K⁻¹. Values: silver ≈ 406, copper ≈ 385, aluminium ≈ 205, steel ≈ 50, glass ≈ 0.8, water ≈ 0.6, wood ≈ 0.12, air ≈ 0.024.

Thermal resistance R = L/(kA), so H = ΔT/R, just like I = V/R in electricity. Slabs one after another (in series) add: R = R₁ + R₂. Side by side (in parallel): 1/R = 1/R₁ + 1/R₂.

Uses: cooking pots have copper bottoms (spread heat fast); thermos flasks and double glass windows trap air (poor conductor); igloos keep Inuit warm because snow traps air.

Radiation and the blackbody

Every body above 0 K radiates energy. How much depends on its temperature and surface. Dark, rough surfaces absorb and emit well; shiny, white surfaces reflect and emit poorly. A good absorber is also a good emitter (Kirchhoff's idea).

A blackbody absorbs all radiation falling on it and reflects none. It is also the best possible emitter. A small hole in a hollow box with blackened walls acts almost like a perfect blackbody.

The radiation from a blackbody spreads over many wavelengths, with a peak at λm. As the body gets hotter, the whole curve rises and the peak shifts to shorter wavelengths. That is why an iron rod in a furnace glows dull red, then orange, then yellow-white.

Wien's displacement law

λm T = b, where b = 2.9 × 10⁻³ m K (Wien's constant). Hotter body ⇒ shorter peak wavelength.

The Sun's spectrum peaks near 500 nm, so its surface temperature is about 2.9 × 10⁻³ / 500 × 10⁻⁹ ≈ 5800 K. The human body (310 K) peaks near 9400 nm, in the infrared; thermal cameras see this.

Stefan–Boltzmann law

Total power radiated by a blackbody of area A at absolute temperature T:

P = σ A T⁴, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴

For a real body, P = e σ A T⁴, where the emissivity e is between 0 and 1 (e = 1 for a blackbody). If the body is at T in surroundings at T₀, the net power lost is P = e σ A (T⁴ − T₀⁴).

Because of T⁴, doubling the temperature makes the radiated power 16 times. The greenhouse effect: the Earth re-radiates in the infrared; gases like CO₂ and water vapour absorb some of it and send it back, keeping the Earth warm.

Newton's law of cooling

For small temperature differences, the rate of cooling of a body is proportional to how much hotter it is than its surroundings:

−dT/dt = K (T − Ts)

So hot tea cools fast at first and slowly later. A useful average form: (T₁ − T₂)/t = K[(T₁ + T₂)/2 − Ts]. The graph of ln(T − Ts) against time is a straight line with negative slope.

Try it: two spoons and an ink drop

1) Stand a steel spoon and a wooden spoon in hot tea. After a minute, feel the handles: steel is warm (good conductor), wood is cool. 2) Drop one drop of ink at the bottom edge of a glass of warm water held over a candle-warm spot (ask an adult): the ink rises in the middle and sinks at the sides. 3) In the 3D, step 6, put T = 3000 K, note λmax, then set 6000 K. λmax should halve.

Key formulas and definitions

Worked examples

1. A copper rod 0.5 m long, area 1 × 10⁻⁴ m², has ends at 100 °C and 0 °C. Find the heat flow rate. (k = 385 W/m K)

Step 1: H = kAΔT/L. Step 2: = 385 × 10⁻⁴ × 100 / 0.5. Answer: 7.7 W.

2. A glass window pane is 1 m × 1 m, 4 mm thick. Inside 25 °C, outside 5 °C. Find heat lost per second. (k = 0.8 W/m K)

Step 1: ΔT = 20 K, L = 0.004 m. Step 2: H = 0.8 × 1 × 20 / 0.004. Answer: 4000 W. That is why double glazing (with air between) is used in cold places.

3. The Sun's radiation peaks at 480 nm. Estimate its surface temperature. (b = 2.9 × 10⁻³ m K)

Step 1: T = b/λm. Step 2: = 2.9 × 10⁻³ / 480 × 10⁻⁹. Answer: T ≈ 6040 K.

4. A blackbody at 1000 K emits power P. What does it emit at 2000 K?

Step 1: P ∝ T⁴. Step 2: ratio = (2000/1000)⁴ = 16. Answer: 16P.

5. Find the power radiated by a blackbody sphere of area 0.01 m² at 1000 K. (σ = 5.67 × 10⁻⁸)

Step 1: P = σAT⁴. Step 2: T⁴ = 10¹². Step 3: P = 5.67 × 10⁻⁸ × 0.01 × 10¹². Answer: 5670 W.

6. Two slabs of equal area are joined in series: copper (k = 385, L = 1 cm) and steel (k = 50, L = 1 cm). The outer faces are at 100 °C and 0 °C. Find the junction temperature.

Step 1: Same H through both: k₁(100 − T)/L = k₂(T − 0)/L. Step 2: 385(100 − T) = 50T. Step 3: 38500 = 435T. Answer: T ≈ 88.5 °C. Most of the temperature drop is across the poor conductor (steel).

7. A cup of tea cools from 80 °C to 60 °C in 5 minutes in a room at 20 °C. How long will it take to cool from 60 °C to 40 °C? (use the average form of Newton's law)

Step 1: First interval: (80 − 60)/5 = K(70 − 20) ⇒ 4 = 50K ⇒ K = 0.08 per min. Step 2: Second: (60 − 40)/t = 0.08 × (50 − 20) = 2.4. Step 3: t = 20/2.4. Answer: t ≈ 8.3 min. It cools more slowly as it nears room temperature.

Common mistakes

Practice quiz

1. Heat from the Sun reaches the Earth by:
2. The SI unit of thermal conductivity is:
3. Wien's law says:
4. If absolute temperature is doubled, radiated power becomes:
5. Sea breeze during the day is an example of:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is thermal conductivity?

It is the property that tells how well a material conducts heat. H = kA(T₁ − T₂)/L; the unit of k is W m⁻¹ K⁻¹. Metals have high k, air and wood low k.

What is Wien's displacement law?

The wavelength at which a blackbody emits most strongly is inversely proportional to its absolute temperature: λmT = 2.9 × 10⁻³ m K.

What is the Stefan–Boltzmann law?

The total power radiated by a blackbody is proportional to the fourth power of its absolute temperature: P = σAT⁴, with σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.

Where this is taught

PolandSzkoła podstawowa, klasa VIIThermal phenomena
Ukraine8 класThermal phenomena
CBSE (India)Class 11Properties of Bulk Matter
USA (Common Core, NGSS, AP)Grade 8MS-PS3 Energy
Germany (Bavaria)Jahrgangsstufe 9Heat and thermodynamics
FrancePremièrePhysics-chemistry: Energy
FranceTerminalePhysics-chemistry: Energy

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