Radiation — Stefan and Wien Laws

Every body at temperature $T$ emits electromagnetic radiation. The Stefan-Boltzmann law gives the total power; Wien's law gives the wavelength of peak emission.

Concept

Stefan-Boltzmann law (black body): $P = \sigma A T^4$ where $\sigma = 5.67\times 10^{-8}$ W/m²·K⁴.

For a body with emissivity $\varepsilon$ (gray body): $P = \varepsilon \sigma A T^4$.

Net radiation when a body at $T$ is in surroundings at $T_0$: $P_{net} = \varepsilon\sigma A(T^4 - T_0^4)$

Wien's displacement law: $\lambda_{max}\,T = b$, $b = 2.898\times 10^{-3}$ m·K.

Newton's law of cooling (small $\Delta T$): $dT/dt = -k(T-T_0)$; cooling is exponential to ambient.

Derivation

For small temperature excess $\theta = T - T_0$ with $\theta \ll T_0$: $T^4 - T_0^4 \approx 4T_0^3 \theta$

So $P_{net} \approx 4\varepsilon\sigma A T_0^3 \theta$. Energy balance: $mc\,d\theta/dt = -P_{net}$, giving $\frac{d\theta}{dt} = -\frac{4\varepsilon\sigma A T_0^3}{mc}\theta = -k\theta$

— this is Newton's law of cooling with $k = 4\varepsilon\sigma A T_0^3/(mc)$. Solution: $\theta(t) = \theta_0 e^{-kt}$.

JEE Worked Example

Problem: The sun ($T = 5800$ K, $R = 7\times 10^8$ m) radiates as a black body. Find peak wavelength and total power.

Solution: $\lambda_{max} = 2.898\times 10^{-3}/5800 \approx 500$ nm (green).

Power: $P = \sigma\cdot 4\pi R^2 \cdot T^4 = 5.67\times 10^{-8}\cdot 4\pi(7\times 10^8)^2\cdot(5800)^4 \approx 3.9\times 10^{26}$ W.

Traps

• $T$ must be in kelvin (never Celsius) in Stefan's law.
• Wien's $b$ refers to peak of spectral radiance vs. wavelength, not vs. frequency.
• Newton's law of cooling is valid only for small temperature differences.
• Emissivity $\varepsilon$ equals absorptivity (Kirchhoff's law) at thermal equilibrium.
• The $T^4$ scaling means doubling temperature gives $16\times$ power.

Key Takeaways

• $P = \varepsilon\sigma A T^4$; net $\propto T^4 - T_0^4$.
• $\lambda_{max} T = b$ (Wien).
• Newton's cooling: exponential decay of excess temperature.
• Hotter bodies emit more and peak at shorter wavelengths.