--- title: "Laws of thermal radiation" subject: "Heat Transfer" chapter: 4 key: "semester_5/heat_transfer:laws-of-thermal-radiation" source: https://www.bctengineer.com/q/semester_5/heat_transfer/laws-of-thermal-radiation site: BCT Engineering --- # Laws of thermal radiation Heat Transfer · Chapter 4: Radiation Heat Transfer - Asked in: year not given, 8 marks - Video: [Blackbody Radiation: the Laws of Stefan, Wien, and Planck!](https://www.youtube.com/watch?v=h5jOAw57OXM) — Sky Scholar - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/heat_transfer/laws-of-thermal-radiation - Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-4 ## Question State and explain Planck's law, Wien's displacement law, Stefan-Boltzmann law, Kirchhoff's law and Lambert's cosine law of thermal radiation. Show how Wien's law and the Stefan-Boltzmann law follow from Planck's distribution and explain the relation between temperature, frequency and wavelength of radiation. ## Short answer Planck's law: E_bλ = C1 λ^{-5}/(e^{C2/λT}-1). Wien's law: λ_max T = 2898 μm·K (peak shifts to shorter λ with T). Stefan-Boltzmann: E_b = σT^4 (integral of Planck). Kirchhoff: emissivity = absorptivity at equilibrium. Lambert: diffuse surface intensity constant, E = πI. Hotter bodies emit at higher frequency and more total energy. ## Answer Thermal radiation is electromagnetic radiation emitted by a body because of its temperature, in the range about $0.1$ to $100\ \mu$m (ultraviolet, visible and infrared). #### Planck's law It gives the spectral emissive power of a black body at wavelength $\lambda$ and absolute temperature $T$: $$ E_{b\lambda} = \frac{C_1\,\lambda^{-5}}{\exp\left(C_2/\lambda T\right)-1} $$ $C_1 = 3.742\times10^8\ \text{W}\,\mu\text{m}^4/\text{m}^2$, $C_2 = 14\,388\ \mu\text{m K}$. At every wavelength $E_{b\lambda}$ rises with $T$; each curve has a peak. #### Wien's displacement law Setting $dE_{b\lambda}/d\lambda = 0$ gives the wavelength at the peak. With $x = C_2/\lambda T$, the condition becomes $$ x\,e^{x} = 5\,(e^{x}-1)\;\Rightarrow\; x = 4.965 $$ so $$ \lambda_{max}T = \frac{C_2}{4.965} = 2898\ \mu\text{m K} $$ The peak moves to shorter wavelengths as $T$ rises (a heated iron goes from dull red to white). The sun (5800 K) peaks near $0.5\ \mu$m, a room-temperature body (300 K) near $9.7\ \mu$m. #### Stefan-Boltzmann law Integrating Planck's law over all wavelengths gives the total emissive power of a black body. Put $x = C_2/\lambda T$, so $\lambda = C_2/xT$: $$ E_b = \int_0^\infty E_{b\lambda}\,d\lambda = \frac{C_1T^4}{C_2^4}\int_0^\infty\frac{x^3}{e^x-1}\,dx = \frac{C_1T^4}{C_2^4}\cdot\frac{\pi^4}{15} = \sigma T^4 $$ $$ \sigma = \frac{\pi^4C_1}{15\,C_2^4} = 5.67\times10^{-8}\ \text{W/m}^2\text{K}^4 $$ For a real (grey) surface $E = \varepsilon\sigma T^4$. #### Kirchhoff's law At thermal equilibrium the emissivity of a surface equals its absorptivity, $\varepsilon_\lambda = \alpha_\lambda$. For a grey (diffuse) surface $\varepsilon = \alpha$ at all conditions. A good absorber is a good emitter. #### Lambert's cosine law The radiation intensity $I$ of a diffuse surface is the same in all directions, so emissive power in direction $\theta$ from the normal varies as $\cos\theta$, and total emissive power $E = \pi I$. #### Frequency, wavelength and temperature Wavelength and frequency are related by $\lambda = c/\nu$ ($c = 3\times10^8$ m/s in vacuum). A hotter body emits at higher frequency (shorter wavelength) for its peak, according to Wien's law, and also emits more energy at all frequencies according to the $T^4$ law. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Wien's law says peak wavelength: A. Increases as T rises B. Is independent of T C. Equals 5.67e-8/T D. Decreases as T rises, lambda_max T = 2898 um K Answer: D. Hotter bodies peak at shorter wavelengths. (Key point: Wien's law) 2. How does Stefan-Boltzmann follow from Planck's law? A. Integrate Planck over all wavelengths to get sigma T^4 B. Differentiate Planck to zero C. Set lambda = c/nu D. Divide by pi Answer: A. Integration over wavelengths gives E_b = sigma T^4; differentiating gives Wien's law. (Key point: Stefan-Boltzmann) 3. Kirchhoff's law states that at thermal equilibrium: A. Emissivity equals reflectivity B. Emissivity equals absorptivity C. Absorptivity is zero for black bodies D. Emissivity is 1 for all surfaces Answer: B. A good absorber is a good emitter. (Key point: Kirchhoff) 4. Lambert's cosine law states emissive power in direction theta varies as: A. sin theta B. 1/cos theta C. cos theta, with E = pi I D. cos^2 theta Answer: C. Intensity is the same in all directions for a diffuse surface. 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