Understanding Wein's displacement law is essential when studying blackbody radiation. This law describes the relationship between the temperature of a blackbody and the wavelength at which it emits the most radiation. Let's break down each statement to determine its correctness.

Wein's Displacement Law Explained

Wein's displacement law states that the wavelength corresponding to the maximum spectral emissive power of a blackbody is inversely proportional to its absolute temperature. Mathematically, it is expressed as:

$$\lambda_m T = b$$

Where:

• $$\lambda_m$$ is the wavelength at which the spectral emissive power is maximum.
• $$T$$ is the absolute temperature of the blackbody in Kelvin.
• $$b$$ is Wien's displacement constant, approximately $$2.898 \times 10^{-3} \text{ m}\cdot\text{K}$$.

This equation tells us that as the temperature (T) of a blackbody increases, the wavelength ($$\lambda_m$$) at which it emits the most radiation shifts to shorter wavelengths (e.g., from red light to blue light, or from infrared to visible light). Conversely, if the temperature decreases, the peak wavelength shifts to longer wavelengths.

Analyzing the Statements about Wein's Displacement Law

Let's examine each statement given in the question in detail:

Statement 1: Maximum spectral emissive power is displaced to longer wavelengths with increase in temperature.

• This statement contradicts Wein's displacement law. According to the law ($$\lambda_m T = b$$), if temperature $$T$$ increases, then $$\lambda_m$$ must decrease to keep the product constant.
• A decrease in $$\lambda_m$$ means the peak wavelength shifts to shorter wavelengths, not longer ones.
• Therefore, statement 1 is incorrect.

Statement 2: Maximum spectral emissive power increases with decrease in temperature.

• This statement is about the intensity or magnitude of the maximum spectral emissive power. According to Planck's law and the overall blackbody radiation curve, as the temperature of a blackbody decreases, the total energy radiated decreases, and the peak intensity (maximum spectral emissive power) also decreases.
• Imagine a blackbody radiation curve: as temperature drops, the entire curve lowers and flattens.
• Therefore, statement 2 is incorrect.

Statement 3: Maximum spectral emissive power is displaced to shorter wavelengths with increase in temperature.

• This statement directly aligns with Wein's displacement law ($$\lambda_m T = b$$). If temperature $$T$$ increases, then $$\lambda_m$$ must decrease.
• A decrease in $$\lambda_m$$ means the peak wavelength shifts to shorter wavelengths. This is why hotter objects glow blue-white (shorter wavelength), while cooler objects glow red (longer wavelength).
• Therefore, statement 3 is correct.

Statement 4: Maximum spectral emissive power decreases with decrease in temperature.

• This statement describes the behavior of the intensity of radiation. As the temperature of a blackbody decreases, the total energy emitted by it decreases, as described by the Stefan-Boltzmann law ($$E = \sigma T^4$$).
• Consequently, the peak spectral emissive power (the height of the blackbody radiation curve at its maximum) also decreases. The radiation becomes less intense overall.
• Therefore, statement 4 is correct.

Conclusion

Based on the analysis, statements 3 and 4 are correct. These statements accurately describe the behavior of blackbody radiation as temperature changes, according to Wein's displacement law and the general principles of thermal radiation.