Problem 81 - Master Resource Book in JEE Main Physics

Question

Solar radiation emitted by sun resembles that emitted by a black body at a temperature of \(6000 \mathrm{~K}\). Maximum intensity is emitted at a wavelength of about \(4800 \hat{A}\). If the sun were cooled down from \(6000 \mathrm{~K}\) to \(3000 \mathrm{~K}\), then the peak intensity would occur at a wavelength of (a) \(4800 \mathrm{~A}\) (b) \(9600 \mathrm{~A}\) (c) \(2400 \hat{A}\) (d) \(19200 \AA\)

Step-by-Step Solution

Answer

The peak intensity wavelength would occur at 9600 Å (option b).

Step 1: Identify the Concept

The problem involves the concept of black body radiation and Wien's Displacement Law, which states that the wavelength at which the intensity of radiation is maximum is inversely proportional to the temperature of the black body. Mathematically, it is given by \( \lambda_{max} T = b \), where \( b \) is Wien's constant.

Step 2: Apply Wien's Displacement Law

Given that for the sun at \(6000 \mathrm{~K}\), the maximum intensity is at \(4800 \AA\). Using Wien's Displacement Law: \( \lambda_{max,1} T_1 = \lambda_{max,2} T_2 \). Where \(T_1 = 6000\), \(T_2 = 3000\) and \( \lambda_{max,1} = 4800 \AA\).

Step 3: Calculate the New Wavelength

Solve for the new peak wavelength \( \lambda_{max,2} \) by substituting into Wien's Displacement Formula: \[ \lambda_{max,2} = \frac{\lambda_{max,1} T_1}{T_2} = \frac{4800 \times 6000}{3000} \].

Step 4: Simplify the Equation

Perform the arithmetic to find \( \lambda_{max,2} = \frac{4800 \times 6000}{3000} = 9600 \). Thus, the new peak intensity wavelength is \(9600 \AA\).

Key Concepts

Black Body Radiation

Black body radiation is a fundamental concept in physics that describes how an idealized object, called a black body, emits electromagnetic radiation. A black body is essentially a perfect emitter and absorber of radiation, meaning it doesn't reflect any light and absorbs all incoming radiation. This theoretical object provides us with a model to understand how real objects like stars, including our Sun, emit radiation.
The radiation emitted by a black body depends solely on its temperature. As the temperature of the black body increases, the intensity of emitted radiation also increases, and the radiation shifts towards shorter wavelengths. This is why hotter objects tend to have a bluish color, while cooler objects appear redder.
Key characteristics of black body radiation include:

• It produces a continuous spectrum of radiation.
• The radiation follows Planck's law, which describes the energy distribution across different wavelengths.
• It serves as an important reference for measuring the temperature and emission characteristics of stars and other celestial bodies.

Understanding black body radiation is crucial in fields like astronomy, meteorology, and climate science, where it helps explain not just the emission of stars but also Earth's own radiation patterns.

Solar Radiation

Solar radiation is the radiant energy emitted by the sun, primarily as a result of nuclear fusion within its core. This energy travels through space and reaches Earth, playing a critical role in providing light and heat to our planet, which supports life.
The sun's radiation resembles that of a black body because it emits a spectrum of electromagnetic radiation that is similar to a black body at a high temperature, approximately 6000 K. This means the sun emits radiation across a wide range of wavelengths from ultraviolet to infrared, with the peak being in the visible region of the spectrum, around 4800 Å (Angstroms).
Some important points about solar radiation:

• The sun's radiation intensity varies across different parts of its spectrum.
• It affects climate and weather patterns on Earth by influencing atmospheric processes.
• The energy from the sun is harnessed for solar power, which is a renewable energy source.

Understanding solar radiation is essential for various applications, from predicting weather and climate change to optimizing solar energy systems.

Peak Wavelength Calculation

The concept of peak wavelength calculation is closely tied to Wien's Displacement Law. This law is a key principle in the study of thermal radiation, specifically for objects like stars that can be approximated as black bodies.
Wien's Displacement Law states that the wavelength at which the emission of a black body is at its maximum (known as the peak wavelength) is inversely proportional to the temperature of the black body. Mathematically, this is represented as: \[ \lambda_{max} = \frac{b}{T} \]where:

• \( \lambda_{max} \) is the peak wavelength,
• \( T \) is the temperature in Kelvin,
• and \( b \) is Wien's constant, approximately \( 2.897 \times 10^{-3} \, m \cdot K \).

To find the new peak wavelength when the temperature changes, you can rearrange this formula. For example, if the sun's temperature decreases from 6000 K to 3000 K, the peak wavelength will shift to longer wavelengths. In our original problem, using the formula we find that decreasing the temperature halves it and doubles the peak wavelength from 4800 Å to 9600 Å.
Such calculations are crucial in determining the characteristics of celestial bodies and can also help in understanding how changing temperatures affect the color and energy distribution of stars and similar objects.