Blackbody Radiation Spectrum (Spectral Emissive Power)

Dual Synchronized Panels: Exact match with Planck’s Law Spectral Emissive Power (Top) paired with Classical Deviations (Bottom)

Oven Temperature (T): 1650 K

Span:

PANEL 1: Spectral Emissive Power Spectrum \(E(\lambda)\) Peak & Infrared Tail

Exact match with Planck’s Law

Wien (1896): Drops in Infrared Tail

Rayleigh–Jeans: UV Catastrophe / Far-IR Equipartition

PTR Measured Points

Live Calculation Workbench (Verify with Your Calculator)

Inspection at λ = 12.0 µm | T = 1650 K

1. Planck Law & Quantum Parameter (x)

\(x = \frac{hc}{\lambda k_B T} = \frac{14388}{\lambda \cdot T}\)

x = 14388 / (12.0 × 1650) = 0.7267

Planck Reference Deviation:
Δ = 0.0% deviation (Exact Truth)

2. Wien's Law (1896 Empirical) Deviation

\(\Delta_{\text{Wien}} = -e^{-\frac{C_2}{\lambda T}} \times 100\%\)

e^(-0.7267) = 0.4835

Wien Deficit Result:
-48.4% deviation

3. Rayleigh–Jeans Law Deviation Calculation

\(\Delta_{\text{RJ}} = \left[\frac{e^x - 1}{x} - 1\right] \times 100\%\)

e^(0.7267) = 2.0682

(2.0682 - 1) / 0.7267 = 1.4700

Rayleigh–Jeans Excess Result:
+47.0% deviation

1. Planck Quantum Emissive Power Law (1900)

\[E(\lambda) = \frac{2\pi hc^2}{\lambda^5 \left(e^{\frac{hc}{\lambda k_B T}} - 1\right)}\]

Emissive power emitted into a hemisphere. Baseline reference standard across all wavelengths (\(\Delta = 0\%\)).

2. Wien's Empirical Distribution Law (1896)

\[E(\lambda) = \frac{C_1}{\lambda^5} e^{-\frac{C_2}{\lambda T}}\]

Historical empirical formula using constants \(C_1, C_2\) (formulated before quantum action \(h\) was introduced in 1900).

3. Rayleigh–Jeans Classical Emissive Power Law (1900)

\[E(\lambda) = \frac{2\pi c k_B T}{\lambda^4}\]

Classical equipartition law. Diverges in UV; converges to Planck's emissive power in Far-IR.

Historical Note: Wilhelm Wien formulated his law in 1896 using arbitrary empirical constants \(C_1\) and \(C_2\) prior to Planck's discovery of the quantum \(h\). The constant pre-factors cancel out completely when computing relative percentage deviations \(\Delta\).