Question

If λm denotes the wavelength at which the radioactive emission from a black body at a temperature T K is maximum, then

• A. λ_m is independent of T
• B. λ_m ∝ T
• C. λ_m ∝ T–1
• D. λ_m ∝ T–4

Answer 1

The Correct Option is C

To solve this problem, we need to understand the physical concept related to the wavelength at which a black body emits radiation most efficiently at a given temperature. The concept in question here pertains to Wien's Displacement Law.

**Wien's Displacement Law**

Wien's Displacement Law states that the wavelength \(\lambda_m\) corresponding to the maximum emission from a black body is inversely proportional to its temperature. Mathematically, this is expressed as:

\(\lambda_m \propto \frac{1}{T}\)

This indicates that as the temperature of the black body increases, the peak wavelength at which it emits radiation decreases. This law is derived from Planck's radiation law and is fundamental in understanding black body radiation.

**Step-by-Step Explanation**

1. Identify that the problem involves black body radiation.
2. Recall that the peak wavelength is governed by Wien's Displacement Law.
3. Understand that the relationship is such that \(\lambda_m \propto \frac{1}{T}\), meaning it inversely depends on the temperature.
4. Therefore, we conclude that the wavelength \(\lambda_m\) is proportional to T^{-1}.

**Conclusion**: The correct option is \(\lambda_m \propto T^{-1}\), which matches the statement: λ_m ∝ T–1.

**Rule Out Other Options**

• λ_m is independent of T: This contradicts Wien's Law as it implies no relationship with temperature.
• λ_m ∝ T: This is the opposite of actual behavior; higher temperatures lead to shorter wavelengths.
• λ_m ∝ T–4: This refers to Stefan-Boltzmann Law regarding total emitted energy, not the wavelength.