Thermal Radiation & Newton's Law of Cooling – Masterclass
Unlike conduction and convection which require a material medium, thermal radiation travels through vacuum at the speed of light (c = 3 × 108 m/s) in the form of electromagnetic waves (primarily in the infrared spectrum). Every body at a temperature above absolute zero (T > 0 K) continuously emits thermal radiation.
1. Basic Radiative Definitions & Kirchhoff's Law
- Emissive Power (e): Radiant energy emitted per second per unit surface area of a body.
- Absorptive Power (a): Ratio of radiant energy absorbed to total incident radiant energy (a ≤ 1).
- Perfect Blackbody: A body that absorbs 100% of all incident electromagnetic radiation of all wavelengths falling on it (a = 1). When heated, it emits maximum possible radiation at that temperature.
Kirchhoff's Law of Radiation
“At any temperature, the ratio of emissive power to absorptive power for any substance is constant and equal to the emissive power of a perfect blackbody at that temperature.”
Crucial Consequence: “Good absorbers are good emitters; bad absorbers are bad emitters.” A black coal piece heated to glowing shines brighter than surrounding polished metal. In solar spectra, Fraunhofer dark absorption lines coincide with bright emission lines of identical elements.
2. Stefan-Boltzmann Law of Radiation
The total radiant energy emitted per second per unit area of a perfect blackbody is directly proportional to the fourth power of its absolute temperature:
Real Body Emission (Emissivity e)
For a non-blackbody of emissivity e (0 < e < 1) and surface area A:
Net Radiant Power in Enclosure
If the body at temperature T is surrounded by an enclosure at temperature T0:
3. Wien's Displacement Law
A blackbody emits a continuous spectrum of electromagnetic radiation across all wavelengths. As temperature T rises, the total emission increases drastically (area under curve ∝ T4) and the wavelength of maximum spectral emissive power (λm) shifts towards shorter wavelengths.
λm × T = b ≈ 2.898 × 10−3 m K
Applications of Wien's Displacement Law
- Stellar Temperatures: Measuring λm of the Sun (λm ≈ 490 nm) gives the solar photosphere temperature: Tsun = (2.898 × 10−3) / (490 × 10−9) ≈ 5900 K.
- Colour Shift of Heated Iron: As an iron rod is heated: dull red (∼ 700 °C) → cherry red → bright orange → yellow → white-hot (> 1200 °C).
4. Newton's Law of Cooling
When a hot body cools in calm ambient surroundings at temperature T0, heat is lost by a combination of conduction, natural convection, and radiation. For small temperature differences (ΔT = T − T0 ≪ T0), Newton formulated his famous empirical law of cooling:
dQ / dt = − K' (T − T0) ⇒ − dT / dt = K (T − T0)
where K = K' / (m c) depends on the surface area, nature of the surface, and heat capacity of the body.
A. Exact Differential / Exponential Form
Integrating ∫ dT / (T − T0) = − K ∫ dt:
T(t) − T0 = (Tinitial − T0) e− K t
A plot of ln(T − T0) versus time t is a straight line with slope −K.
B. Approximate Average Method (for JEE/NEET)
For a body cooling from T1 to T2 in time t:
This linearised average approximation gives rapid, highly accurate numerical results when T1 − T2 is moderate.
Net heat loss: dQ/dt = e σ A (T4 − T04).
Let T = T0 + ΔT. Then T4 = T04(1 + ΔT/T0)4 ≈ T04(1 + 4 ΔT/T0) by binomial approximation.
⇒ T4 − T04 ≈ 4 T03 ΔT.
Therefore, dQ/dt ≈ (4 e σ A T03) (T − T0) ∝ (T − T0).
This proves that Newton's Law of Cooling is the first-order linear approximation of Stefan's radiation law for small temperature excess.
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