A metal surface is heated from 300 K to 600 K (doubling its absolute temperature). By what factor does its radiated power increase?
A2 — power is proportional to temperature
B4 — power is proportional to temperature squared
C8 — power is proportional to temperature cubed
D16 — power is proportional to the fourth power of absolute temperature
The Stefan-Boltzmann law gives P = εσAT⁴. Doubling T multiplies T⁴ by 2⁴ = 16. This T⁴ dependence makes radiation extraordinarily sensitive to temperature — far more so than conduction or convection, which scale roughly linearly with temperature difference. At high temperatures this steep dependence causes radiation to completely dominate the other heat transfer mechanisms.
Question 2 Multiple Choice
A matte black surface (ε ≈ 0.95) and a polished silver surface (ε ≈ 0.02) are both left in sunlight at the same temperature. Which surface heats up faster, and why?
AThe silver surface — it reflects most radiation so it stores more energy than it releases
BThey heat at the same rate — both surfaces receive the same incident solar radiation
CThe black surface — high emissivity means it absorbs more incident radiation (and by Kirchhoff's law, will also radiate more efficiently once hot)
DThe black surface only heats faster initially; once hot, its high emissivity causes it to cool faster than silver
By Kirchhoff's law, emissivity and absorptivity are equal at the same wavelength — a surface that emits well also absorbs well. The black surface (ε ≈ 0.95) absorbs about 95% of incident radiation; the silver surface (ε ≈ 0.02) reflects about 98% and absorbs only 2%. So the black surface heats up faster. Option D captures an important truth (it also radiates faster once hot), but the net effect is still that the black surface reaches a higher equilibrium temperature in sunlight — it absorbs so much more that it runs hotter despite also emitting more.
Question 3 True / False
A surface with high emissivity (close to 1) both absorbs incident radiation more efficiently AND emits more thermal radiation than a low-emissivity surface at the same temperature.
TTrue
FFalse
Answer: True
This follows directly from Kirchhoff's radiation law: for a surface in thermal equilibrium, absorptivity equals emissivity at each wavelength. A surface cannot be a good absorber without also being a good emitter, and vice versa. This symmetry is why matte black surfaces work well as both solar collectors (absorbing sunlight) and radiators (emitting heat), while polished surfaces reflect incident radiation and also emit very little.
Question 4 True / False
When two objects reach thermal equilibrium, they stop emitting thermal radiation because there is no longer any temperature difference to drive energy transfer.
TTrue
FFalse
Answer: False
Every object above absolute zero continuously emits thermal radiation, regardless of temperature equilibrium. At thermal equilibrium (T = T_env), each object emits exactly as much radiation as it absorbs from its surroundings, so P_net = εσA(T⁴ - T_env⁴) = 0. The radiation doesn't stop — it balances. This is consistent with the net exchange formula and with the broader thermodynamic principle that equilibrium is a dynamic balance, not a cessation of activity.
Question 5 Short Answer
Why does the T⁴ dependence of radiated power mean that radiation dominates heat transfer at high temperatures, even if conduction and convection are also present?
Think about your answer, then reveal below.
Model answer: Conduction and convection transfer rates scale roughly linearly with temperature difference ΔT. Radiation scales as T⁴ (or more precisely as T⁴ - T_env⁴, which for T >> T_env grows as T⁴). As temperature increases, the radiation term grows far faster than the linear terms. For example, tripling absolute temperature multiplies radiated power by 3⁴ = 81 but only triples conductive/convective transfer. At high enough temperatures, radiation's contribution becomes so large that the others are negligible in comparison.
This is why furnaces, stars, and incandescent filaments are dominated by radiation despite having surfaces that could also conduct and convect. At room temperature, conduction and convection often dominate because T⁴ is small relative to ΔT effects. But the crossover comes at surprisingly modest temperatures for engineering applications — understanding where each mode dominates is essential for thermal design.