Problem 42 - University Physics with Modern Physics

Question

Determine \(\lambda_{\mathrm{m}}\) , the wavelength at the peak of the Planck distribution, and the corresponding frequency \(f,\) at these temperatures: (a) \(3.00 \mathrm{K} ;(\mathrm{b}) 300 \mathrm{K} ;(\mathrm{c}) 3000 \mathrm{K}\)

Step-by-Step Solution

Answer

(a) \(9.657 \times 10^{-4} \text{m}, 3.108 \times 10^{11} \text{Hz}\); (b) \(9.657 \times 10^{-6} \text{m}, 3.108 \times 10^{13} \text{Hz}\); (c) \(9.657 \times 10^{-7} \text{m}, 3.108 \times 10^{14} \text{Hz}\).

Step 1: Understand the Problem and Formula

We need to find the peak wavelength \( \lambda_{\mathrm{m}} \) for the Planck distribution given various temperatures. This requires the use of Wien's displacement law, which states \( \lambda_{\mathrm{m}} = \frac{b}{T} \), where \( b = 2.897 \times 10^{-3} \text{m K} \).

Step 2: Calculate for 3 K

Use Wien's law and substitute \( T = 3 \text{K} \). \( \lambda_{\mathrm{m}} = \frac{2.897 \times 10^{-3}}{3} = 9.657 \times 10^{-4} \text{m} \). To find the frequency, use \( c = f \lambda_{\mathrm{m}} \), \( f = \frac{c}{\lambda_{\mathrm{m}}} \), where \( c = 3 \times 10^8 \text{m/s} \). Hence, \( f = \frac{3 \times 10^8}{9.657 \times 10^{-4}} = 3.108 \times 10^{11} \text{Hz} \).

Step 3: Calculate for 300 K

Substitute \( T = 300 \text{K} \) into Wien's law: \( \lambda_{\mathrm{m}} = \frac{2.897 \times 10^{-3}}{300} = 9.657 \times 10^{-6} \text{m} \). Calculate the frequency: \( f = \frac{3 \times 10^8}{9.657 \times 10^{-6}} = 3.108 \times 10^{13} \text{Hz} \).

Step 4: Calculate for 3000 K

Use \( T = 3000 \text{K} \) in Wien's law: \( \lambda_{\mathrm{m}} = \frac{2.897 \times 10^{-3}}{3000} = 9.657 \times 10^{-7} \text{m} \). Find the frequency: \( f = \frac{3 \times 10^8}{9.657 \times 10^{-7}} = 3.108 \times 10^{14} \text{Hz} \).

Key Concepts

Planck Distribution

The Planck Distribution is a cornerstone concept in physics that describes the radiation emitted by a black body in thermal equilibrium. Imagine a perfect object that absorbs all radiation it comes in contact with, and the energy emitted by this object follows a very specific pattern known as the Planck Distribution. This pattern is determined by the temperature of the object. The distribution tells us how much energy is radiated at each wavelength.
The precise shape of this distribution is given by Planck's law of radiation, which is expressed as:\[I(\lambda, T) = \frac{8\pi hc}{\lambda^5} \cdot \frac{1}{e^{(hc/\lambda kT)} - 1}\]where:

• \(I(\lambda, T)\) is the spectral radiance
• \(h\) is Planck's constant
• \(c\) is the speed of light
• \(k\) is the Boltzmann constant
• \(T\) is the absolute temperature

The distribution not only tells us how energy is emitted over a range of wavelengths but also predicts quite accurately the position of the peak wavelength where most energy is emitted.

Peak Wavelength

When exploring the Planck Distribution, the peak wavelength is a key focus. It represents the point in the distribution where the intensity of radiation is at its maximum. The concept of peak wavelength is crucial because it helps in understanding the most energetic radiation being emitted from an object based on its temperature.
To determine this peak, Wien's Displacement Law is employed, and it is given by:\[\lambda_{\mathrm{m}} = \frac{b}{T}\]where:

• \(\lambda_{\mathrm{m}}\) is the wavelength at the peak
• \(b\) is Wien's displacement constant (approximately \(2.897 \times 10^{-3}\) m K)
• \(T\) is the absolute temperature of the black body

This law indicates an inverse relationship between the temperature and the peak wavelength: as the temperature increases, the peak wavelength decreases, meaning the object emits more energetic (shorter wavelength) radiation.

Thermal Radiation

Thermal Radiation is the emission of electromagnetic waves as a result of the temperature of a body. Every object, based on its temperature, emits radiation continually and this is what we observe as thermal radiation. It is a fundamental mechanism of energy transfer and becomes especially important in processes involving high temperatures.
Thermal radiation is not restricted to visible light; it spans across different parts of the electromagnetic spectrum, including infrared radiation for lower temperature bodies and even ultraviolet for extremely hot ones.
Factors governing thermal radiation include:

• The object's surface material and texture
• Its color and finish (dull or shiny)
• Most importantly, its temperature

With increasing temperature, the intensity and the color of the emitted radiation change, which is precisely explained through the Planck Distribution and the concept of peak wavelength using Wien’s Law.

Frequency Calculation

In the context of electromagnetic waves, frequency is an essential parameter. Once you know the wavelength, using the speed of light, you can easily calculate frequency. The relationship between frequency \(f\), wavelength \(\lambda\), and the speed of light \(c\) is described by the formula:\[f = \frac{c}{\lambda}\]This simple equation shows us how frequency and wavelength are inversely related. If you increase the wavelength, the frequency gets lower and vice versa.
Knowing the frequency at the peak wavelength is important in determining what kind of electromagnetic wave you're dealing with. For instance, a low frequency corresponds to radio waves, while a high frequency would be in the range of visible light or higher like ultraviolet or X-rays.

• \(c = 3 \times 10^8\) m/s, the speed of light in a vacuum is constant.
• \(\lambda\) is the wavelength determined from Wien’s Law or observed from the Planck Distribution.

This concept helps you tap into understanding the nature of radiation beyond the visible spectrum, providing a more comprehensive view of the electromagnetic spectrum.