Question

For a blackbody, the wavelength at which the maximum monochromatic emissive power occurs is

• A. Independent of the absolute temperature of the blackbody
• B. Directly proportional to absolute temperature of the blackbody
• C. Inversely proportional to the absolute temperature of the blackbody
• D. Proportional to the fourth power of the absolute temperature of the blackbody

Hint:

Thermal Radiation Laws Summary: - Wien's Displacement Law: \(\lambda_{\max} \propto \frac{1}{T}\) (Governs the peak wavelength shift). - Stefan-Boltzmann Law: \(E_b = \sigma \cdot T^4\) (Governs the total power emitted across all wavelengths).

Answer 1

The Correct Option is C

Concept: All bodies above absolute zero emit thermal radiation across a spectrum of wavelengths. A blackbody is an ideal surface that absorbs all incident radiation and emits the maximum possible thermal radiation at any given temperature. The distribution of monochromatic emissive power across different wavelengths is described by Planck's distribution law. As the absolute temperature of a blackbody increases, the total emitted energy increases, and the peak of the emission spectrum shifts toward shorter wavelengths. This shift is quantified by Wien's Displacement Law.

Step 1: Formulating the mathematical statement of Wien's Displacement Law.
Wien's Displacement Law states that the wavelength corresponding to peak monochromatic emissive power (\(\lambda_{\max}\)) is inversely proportional to the absolute temperature (\(T\)) of the blackbody: \[ \lambda_{\max} \propto \frac{1}{T} \] This relationship can be written as an equation using Wien's displacement constant (\(b\)): \[ \lambda_{\max} \cdot T = b \] Where the constant value is experimentally and theoretically determined to be: \[ b \approx 2898\,\mu\text{m}\cdot\text{K} \quad \left(2.898 \times 10^{-3}\,\text{m}\cdot\text{K}\right) \]

Step 2: Interpreting the physical inverse relationship.
This inverse relationship means that as a object gets hotter, the peak of its radiation spectrum shifts toward shorter wavelengths (higher frequencies and higher energies):
• At moderate temperatures (\(\sim 1000\,\text{K}\)), the peak sits in the infrared region.
• At very high temperatures (\(\sim 6000\,\text{K}\), like the surface of the Sun), the peak shifts into the shorter-wavelength visible light spectrum. Therefore, the wavelength for maximum emissive power is inversely proportional to the absolute temperature, matching Option (C).