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--- un:blackbody-radiation [2025/12/06 10:55] – asad
+++ un:blackbody-radiation [2025/12/06 10:58] (current) – asad
@@ Line 184: / Line 184: @@
 $$h\nu_{avg} \approx 2.70 kT$$
 This value provides a useful rule of thumb for characteristic photon energies. It is slightly lower than the energy at the frequency peak ($2.82 kT$) but provides a representative energy for interactions such as ionization or scattering within a thermal plasma.
-===== Insights =====
-  * The Planck function unifies the Rayleigh-Jeans (low frequency) and Wien (high frequency) regimes into a single quantum-statistical description dependent only on temperature.
-  * The specific intensity $I(\nu,T)$ is derived directly from the Bose-Einstein distribution, linking macroscopic radiation properties to microscopic quantum statistics.
-  * While the peak frequency scales as $T$, the peak wavelength scales as $1/T$, and the total power emitted scales rigorously as $T^4$ (Stefan-Boltzmann Law).
-  * The number density of photons in a blackbody field scales as $T^3$, which implies that the average energy per photon remains constant at $\approx 2.7kT$ regardless of the absolute temperature.
 ===== Inquiries =====
-  - Derive the Rayleigh-Jeans approximation starting from the Planck function using the Taylor expansion $e^x \approx 1+x$.
+  - Demonstrate how the Planck function reduces to the Rayleigh-Jeans approximation in the low-frequency limit ($h\nu \ll kT$) using the Taylor series expansion $e^x \approx 1+x$.
-  - Explain why the frequency of maximum emission $\nu_{peak}$ does not correspond to the wavelength of maximum emission $\lambda_{peak}$ using the relationship $c = \lambda \nu$.
+  - Explain why the spectral peak in frequency space ($\nu_{peak}$) corresponds to a different photon energy than the spectral peak in wavelength space ($\lambda_{peak}$), specifically referencing the relationship between the differentials $d\nu$ and $d\lambda$.
-  - Calculate the total energy density of the Cosmic Microwave Background today ($T=2.73$ K) and compare it to the energy density of starlight in the Galaxy ($\sim 10^{-13}$ J m$^{-3}$).
+  - Using the scaling relations provided, determine the factor by which the total luminosity of a spherical star increases if its temperature doubles while its radius remains constant.
-  - Discuss the physical significance of the radiation constant $a$ and its relationship to the Stefan-Boltzmann constant $\sigma$.
+  - Contrast the temperature dependence of the total photon energy density ($u \propto T^4$) with that of the total photon number density ($n \propto T^3$) and explain what this implies for the average energy per photon as temperature changes.
-  - Using the scaling relations, determine how the total number of photons in a box of blackbody radiation changes if the temperature is doubled.
+  - State the equation of state relating radiation pressure $P$ to total energy density $u$ for an isotropic blackbody field and identify the factor distinguishing it from the kinetic pressure of a non-relativistic gas.