AIIMS 2003 Physics Black Body Radiation MCQ Question

To understand the question regarding black body radiation curves at different temperatures $T_1$ and $T_2$ (where $T_2 > T_1$), we need to delve into the physics of black body radiation, as described by Planck's law.

Explanation of Black Body Radiation

A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The spectral radiance $B(\lambda, T)$ of a black body at temperature $T$ is given by Planck's law:

$$B(\lambda, T) = \frac{2 \pi hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda kT}} - 1}$$

Where:

• $B(\lambda, T)$ is the spectral radiance (intensity) at wavelength $\lambda$,
• $h$ is Planck's constant,
• $c$ is the speed of light,
• $k$ is Boltzmann's constant,
• $T$ is the absolute temperature of the black body.

Key Characteristics of Black Body Radiation Curves

1. Temperature Dependence: As the temperature increases, the peak of the black body radiation curve shifts to shorter wavelengths. This phenomenon is described by Wien's Displacement Law, which states:

$$\lambda_{\text{max}} T = b$$

where $b$ is Wien's displacement constant (approximately $2898 \, \mu m \, K$). Therefore, for $T_2 > T_1$, $\lambda_{\text{max}}(T_2) < \lambda_{\text{max}}(T_1)$.

1. Intensity of Radiation: The total intensity of radiation emitted by a black body also increases with temperature. According to the Stefan-Boltzmann Law, the total power emitted per unit area is given by:

$$j^* = \sigma T^4$$

where $\sigma$ is the Stefan-Boltzmann constant. This means that as the temperature increases, the area under the curve (total intensity) increases significantly.

Analysis of the Correct Answer (C)

In the correct plot (C), we would observe that:

• The curve corresponding to temperature $T_2$ is positioned to the left of the curve corresponding to temperature $T_1$, indicating that the peak wavelength for $T_2$ is shorter.
• The curve for $T_2$ is higher than that for $T_1$ at all wavelengths, illustrating that the intensity of radiation at higher temperatures is greater.

Clarification of Incorrect Options

If other options (A, B, D, etc.) depict:

1. Incorrect Peak Wavelengths: If they show the peak wavelength of $T_2$ being longer than $T_1$, they violate Wien's law.

2. Incorrect Intensity Levels: If they show the curve for $T_2$ below $T_1$, they contradict the Stefan-Boltzmann Law, which states that higher temperatures correspond to higher radiation intensity across all wavelengths.

Summary

The correct black body radiation curve for temperatures $T_1$ and $T_2$ (where $T_2 > T_1$) must show:

• A shift of the peak to shorter wavelengths for $T_2$.
• A greater intensity for $T_2$ across the spectrum.

Thus, option C accurately represents these characteristics, while other options fail to do so based on the fundamental principles of black body radiation.