Questions: Thermal Radiation and Stefan-Boltzmann Law

Question 1

A blackbody's temperature doubles from 300 K to 600 K. By what factor does its total radiated power increase?

• A. 2 — power scales linearly with temperature
• B. 4 — power scales with the square of temperature
• C. 8 — power scales with the cube of temperature
• D. 16 — power scales with the fourth power of temperature

Answer: (D)

The Stefan-Boltzmann law gives P = σAT⁴. Doubling temperature multiplies power by 2⁴ = 16. This steep T⁴ dependence is why radiation is negligible compared to conduction and convection near room temperature but completely dominates at furnace temperatures or stellar surfaces. Options A and B represent the incorrect intuitions that radiation behaves like conductive or convective heat transfer, which scale roughly linearly with temperature difference.

Question 2

A polished silver surface has an emissivity of ε ≈ 0.02. Compared to a blackbody at the same temperature, how does this surface behave?

• A. It emits only 2% as much radiation, but absorbs radiation just as readily as a blackbody
• B. It emits only 2% as much radiation and also absorbs only 2% as much incident radiation
• C. It emits full blackbody radiation but reflects most incident radiation away
• D. It emits 98% as much radiation because polished surfaces are near-perfect emitters

Answer: (B)

By Kirchhoff's law, emissivity equals absorptivity for a body in thermal equilibrium. A surface with ε = 0.02 both emits only 2% as much radiation as a blackbody and absorbs only 2% of incident radiation — the same factor governs both. This is why polished metal surfaces are used in thermos bottles: they suppress both emission and absorption, minimizing radiant heat exchange with surroundings. Option A is the common misconception that emissivity and absorptivity are independent properties.

Question 3

Unlike conduction and convection, thermal radiation can transfer heat through a perfect vacuum.

• T. True
• F. False

Answer: True

Thermal radiation is electromagnetic radiation — it requires no medium to propagate. This is why the Sun can heat the Earth across 150 million km of near-vacuum space, and why a glowing iron radiates heat even in a vacuum chamber. Conduction requires direct molecular contact; convection requires a fluid medium for bulk flow. Radiation is the only heat transfer mechanism that works in the absence of matter.

Question 4

Because the Stefan-Boltzmann law contains T⁴, radiation is the dominant heat transfer mechanism at most temperatures above absolute zero.

• T. True
• F. False

Answer: False

Although T⁴ grows faster than the linear temperature-difference dependence of conductive and convective heat transfer, at low temperatures (near room temperature) the absolute magnitude of radiated power is still small compared to conduction and convection in most practical situations. Radiation becomes *dominant* only at high temperatures — in furnaces, stellar surfaces, or space — where the T⁴ term pulls decisively ahead. The net radiation also depends on the T⁴ − T₀⁴ difference, which is small when T and T₀ are close.

Question 5

Why does doubling the temperature of a radiating body increase its emitted power by a factor of 16 rather than 2, and what practical consequence does this have for engineering design at high temperatures?

Model answer: The Stefan-Boltzmann law states that radiated power scales as T⁴. Doubling T gives (2T)⁴ = 16T⁴ — a 16× increase. This means radiation becomes overwhelmingly more important as temperatures rise: a furnace wall at 1200 K radiates 16 times more than the same wall at 600 K. Engineers designing high-temperature systems (furnaces, rocket nozzles, re-entry vehicles) must account for radiation as the dominant heat transfer mode, even if it was negligible at lower operating temperatures.

The T⁴ dependence comes from integrating the Planck spectrum over all wavelengths — the result of quantum statistical mechanics. Its practical consequence is that radiation 'races ahead' of conduction and convection as temperature rises, which is why emissivity control (low-ε coatings for insulation, high-ε coatings for efficient radiators) is central to high-temperature engineering design.