Spectral Distribution of Radiation Energy Graph

The spectral distribution of radiation energy graph is a visual representation of the intensity or energy density of electromagnetic radiation as a function of frequency or wavelength. This graph is crucial in understanding the characteristics of different types of radiation, including light emitted by various sources such as the sun, stars, and artificial light sources.

Blackbody Radiation

A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A blackbody also emits radiation in a characteristic, continuous spectrum. The spectral distribution of a blackbody's radiation depends only on its temperature, a relationship described by Planck's law.

Planck's Law

Planck's law describes the spectral density of electromagnetic radiation emitted by a blackbody in thermal equilibrium at a definite temperature. The law is mathematically expressed as:

$$ B(\nu, T) = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{\frac{h\nu}{k_BT}} - 1} $$

where:

• $B(\nu, T)$ is the spectral radiance (energy per unit area per unit time per unit solid angle per unit frequency).
• $\nu$ is the frequency of the electromagnetic radiation.
• $T$ is the absolute temperature of the blackbody.
• $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J s}$).
• $c$ is the speed of light in a vacuum ($3.00 \times 10^8 \text{ m/s}$).
• $k_B$ is the Boltzmann constant ($1.381 \times 10^{-23} \text{ J/K}$).

Wien's Displacement Law

Wien's displacement law states that the wavelength at which the emission of a blackbody spectrum is maximum, $\lambda_{\text{max}}$, is inversely proportional to the temperature:

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

where $b$ is Wien's displacement constant ($2.898 \times 10^{-3} \text{ m K}$).

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a blackbody across all wavelengths per unit time (also known as the blackbody radiant exitance or emissive power), $j^*$, is directly proportional to the fourth power of the blackbody's absolute temperature:

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

where $\sigma$ is the Stefan-Boltzmann constant ($5.670 \times 10^{-8} \text{ W/m}^2\text{K}^4$).

Spectral Distribution Graph

The spectral distribution graph typically has the wavelength or frequency on the x-axis and the spectral radiance or intensity on the y-axis. For a blackbody, the graph shows a continuous curve that peaks at a certain wavelength (or frequency) and then tapers off on either side. This peak shifts to shorter wavelengths (or higher frequencies) as the temperature increases, in accordance with Wien's displacement law.

Differences and Important Points

Feature	Description
Peak Wavelength/Frequency	The peak of the spectral distribution graph indicates the wavelength or frequency at which the radiation is most intense.
Temperature Dependence	The shape and peak position of the spectral distribution curve depend on the temperature of the emitting body.
Area Under the Curve	The total area under the spectral distribution curve represents the total power emitted per unit area, which is given by the Stefan-Boltzmann law.

Examples

Example 1: Sun's Radiation

The sun can be approximated as a blackbody with a surface temperature of about 5778 K. Using Wien's displacement law, we can estimate the peak wavelength of the sun's spectral distribution:

$$ \lambda_{\text{max}} = \frac{2.898 \times 10^{-3} \text{ m K}}{5778 \text{ K}} \approx 500 \text{ nm} $$

This peak wavelength is in the visible spectrum, which is why the sun appears bright to us.

Example 2: Incandescent Light Bulb

An incandescent light bulb has a filament that is heated to a high temperature, emitting light. If the filament temperature is approximately 3000 K, the peak wavelength of the emitted light would be:

$$ \lambda_{\text{max}} = \frac{2.898 \times 10^{-3} \text{ m K}}{3000 \text{ K}} \approx 965 \text{ nm} $$

This peak wavelength is in the infrared range, which means a significant portion of the energy is emitted as heat rather than visible light.

Understanding the spectral distribution of radiation energy graph is essential for various applications, including climate science, astrophysics, and the design of lighting and heating systems. It provides insight into the efficiency and characteristics of different light sources and helps in the study of objects based on their thermal radiation.