Problem 65 - Master Resource Book in JEE Main Physics

Question

A black body radiates at two temperatures \(T_{1}\) and \(T_{2}\), such that \(T_{1}

Step-by-Step Solution

Answer

(a) less at \( T_1 \)

Step 1: Understanding Black Body Radiation

A black body is an idealized physical object that absorbs all incident electromagnetic radiation and re-emits it at corresponding frequencies determined by its temperature. One key aspect of this phenomenon is described by Wien's Displacement Law.

Step 2: Introducing Wien's Displacement Law

Wien's Displacement Law states that the frequency (\( u_{max} \)) at which the emission from a black body is maximized is directly proportional to its temperature \( T \). Mathematically, this is expressed as \( u_{max} \propto T \).

Step 3: Applying Wien’s Law to the Given Temperatures

Given two temperatures, \( T_1 \) and \( T_2 \) with \( T_1 < T_2 \), according to Wien's Displacement Law, the frequency corresponding to maximum intensity (\( u_{max} \)) increases with temperature. Therefore, \( u_{max, T_2} > u_{max, T_1} \).

Step 4: Evaluating the Multiple Choice Options

Based on Wien’s Displacement Law and the relation that the frequency of maximum intensity increases with temperature, the correct choice among the options given is (a) less at \( T_1 \), because \( u_{max, T_1} < u_{max, T_2} \).

Key Concepts

Wien's Displacement Law

Wien's Displacement Law is a fundamental concept in the study of black body radiation, often found in physics. It helps us understand how the wavelength or frequency of the light emitted by a black body shifts with temperature. According to Wien's Law, the frequency (\(u_{max}\)) at which the intensity of radiation is maximum is directly proportional to the temperature (\(T\)) of the black body. This means as temperature increases, the peak frequency – at which the body emits most strongly – also increases. This concept is depicted mathematically as:

• \( u_{max} \propto T \)

This implies that a hotter object will glow more brightly and at shorter wavelengths compared to a cooler one. By using Wien’s Law, scientists and engineers can determine the surface temperature of distant stars and other celestial bodies by just analyzing the frequency of the light they emit. This law provides an incredible tool for understanding the universe and the behavior of light in different conditions.

Electromagnetic Radiation

Electromagnetic radiation is a type of energy commonly known to encompass a range of different kinds of waves. It includes visible light, radio waves, X-rays, and more. These waves do not require a medium and can travel through the vacuum of space, which differentiates them from other waves like sound.

The spectrum of electromagnetic radiation is vast and is characterized by its frequency and wavelength. Imagine a team of waves moving at the speed of light; the main characteristic that sets different types of electromagnetic radiation apart is their frequency or wavelength.

• Radio waves have long wavelengths and low frequencies.
• Gamma rays have short wavelengths and high frequencies.

The spectrum is continuous and shows a progression from low frequency, long wavelength radiation to high frequency, short wavelength radiation. Understanding this spectrum is crucial because different frequencies and wavelengths have unique properties and interactions with matter. For instance, visible light interacts differently with objects compared to X-rays. Thus, knowledge of electromagnetic radiation and its spectrum helps in many practical applications like communication technologies, medical imaging, and even astronomical observations.

Temperature and Frequency Relation

The relationship between temperature and frequency in black body radiation is an essential piece of understanding how objects emit light. In essence, as the temperature of an object rises, the frequency at which it emits radiation most strongly also increases.

Wien's Displacement Law solidifies this relation by stating that higher temperatures push the peak of the emitted light towards the higher frequency, or equivalently shorter wavelength, end of the electromagnetic spectrum. This has profound implications not only in scientific theories but also in practical applications.

• Stars: Hotter stars emit light that peaks at shorter wavelengths, often appearing blue or white.
• Incandescent bulbs: These objects emit light in the visible spectrum but are much less efficient due to them peaking in the infrared.

This temperature and frequency relationship is a cornerstone for technologies and scientific fields that require precise measurement and control over light emissions, such as astronomy and thermography. It gives insight into the thermal properties of objects through non-invasive means, often being used to infer properties like star temperatures and even the thermal efficiency of devices on Earth. This relation is critical for advancing our understanding of both practical and theoretical physics.