Kirchhoff's Law of Radiation
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Direct answer
Kirchhoff's law of thermal radiation states that at thermal equilibrium, a body's emissivity equals its absorptivity for every wavelength: eλ = aλ. A surface that soaks up a wavelength efficiently must also radiate that wavelength efficiently, so a perfectly black body — absorbing everything — is also the perfect emitter against which all real surfaces are graded by emissivity e between 0 and 1. The working consequence is the net Stefan–Boltzmann exchange, P_net = e σ A (T⁴ − T0⁴) with σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴: a body hotter than its surroundings radiates away more than it receives, and the fourth powers of absolute temperatures do the bookkeeping. For small excess temperatures the T⁴ difference linearises, which is precisely Newton's law of cooling — the two laws are one physics at different regimes.
What you must remember
- Kirchhoff's law: at equilibrium, absorptivity equals emissivity at each wavelength (aλ = eλ); dull black surfaces both absorb and emit strongly, polished silver surfaces do neither — which is why solar-cooker interiors are black and thermos walls are shiny.
- Emissivity grading: e = 1 for an ideal black body; roughly 0.05 for polished aluminium and silver, 0.6-0.9 for most oxidised and painted surfaces, near 0.95 for human skin and water.
- Stefan–Boltzmann with emissivity: emitted power P = eσAT⁴; net exchange with surroundings at T0 is eσA(T⁴ − T0⁴); σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, always with kelvin.
- Prevost's theory of exchanges: all bodies radiate at all temperatures above absolute zero — a body at room temperature "glows" in the infrared, exchanging radiation with everything it sees.
- Newton's law as a limit: T⁴ − T0⁴ = (T0 + ΔT)⁴ − T0⁴ ≈ 4T0³ΔT for small excess, so net power is proportional to ΔT — the linear cooling law falls straight out of Stefan's fourth powers.
- Selectivity consequence: glass transmits visible but absorbs infrared (high aλ there, hence high eλ there), which is the greenhouse trap — sunlight enters, re-radiated infrared cannot leave.
A body in a cold room
Compute the net loss of an engine block of emissivity 0.8 and area 0.5 m² at 327 degrees Celsius (600 K) in a 27-degree-Celsius (300 K) garage. P = 0.8 × 5.67 × 10⁻⁸ × 0.5 × (600⁴ − 300⁴) = 0.8 × 5.67 × 10⁻⁸ × 0.5 × 1.215 × 10¹¹ ≈ 2756 W. Nearly three kilowatts drain by radiation alone — and note how little the 300 K term subtracts: at the fourth power, the surroundings must be truly cold, or the body truly hot, for T0⁴ to matter at all. Drop the body to 310 K in 300 K surroundings and the same expression gives about 27 W — the order of a resting person's radiative loss per half square metre, the reason blankets work.
The linearisation is worth doing once by hand. With T = T0 + ΔT, T⁴ − T0⁴ ≈ 4T0³ΔT for small ΔT, so P_net ≈ 4eσAT0³ × ΔT — proportional to excess temperature. That coefficient is Newton's k wrapped around the Stefan constant, and knowing the derivation converts two chapters of memory into one line of algebra. JEE Advanced has asked for it as "show that Stefan's law reduces to Newton's law for small temperature differences".
Where students slip
The fourth-power subtraction is where arithmetic dies: both temperatures must be absolute (kelvin), and students who plug Celsius values get answers wrong by factors of ten or more — the options are spaced to catch it. Second, distinguish emitted power eσAT⁴ from net power eσA(T⁴ − T0⁴); questions asking "power radiated" in a room want the former or the latter strictly by wording, and careless reading flips answers. Kirchhoff-conceptual favourites: why a white-painted house stays cooler in summer (low absorptivity in visible, and by Kirchhoff low emissivity there too — but the infrared emissivity of paint is high either way, a subtlety Advanced questions have probed), and why a hole in a cavity is a better black body than any blackened surface (light entering bounces until absorbed, giving a ≈ 1 regardless of the wall material). Finally, the greenhouse effect's mechanism is selective absorption plus Kirchhoff-enabled re-emission at longer wavelengths, not "trapped hot air" alone — stating it correctly earns the assertion–reason mark.
Frequently asked questions
What does Kirchhoff's law of radiation state?
At thermal equilibrium a body's emissivity equals its absorptivity at every wavelength, so good absorbers are good emitters and no surface can excel at one while failing at the other.
What is the net radiated power of a real body?
P_net = eσA(T⁴ − T0⁴), with emissivity e, the Stefan constant 5.67 × 10⁻⁸ W m⁻² K⁻⁴, and both temperatures in kelvin.
How does Newton's law of cooling follow from Stefan's law?
Expanding (T0 + ΔT)⁴ − T0⁴ for small ΔT gives 4T0³ΔT, so net power becomes proportional to temperature excess — the exact statement of Newton's linear law.
Why do solar cookers use black interiors and thermos flasks shiny walls?
Black coatings have high absorptivity (and by Kirchhoff high emissivity) to capture sunlight, while shiny walls have low emissivity to refuse radiating the stored heat — each exploits the same law in opposite directions.
Why is a small hole in a cavity an ideal black body?
Radiation entering the hole reflects repeatedly inside until fully absorbed, so the hole's absorptivity is effectively unity regardless of the cavity wall material, making it the perfect emitter by Kirchhoff's law.