For a \(2.0-\mathrm{kK}\) blackbody, by what percentage is the Rayleigh-Jeans law in error at wavelengths of (a) \(1.0 \mathrm{mm},\) (b) \(10 \mu \mathrm{m},\) and (c) \(1.0 \mu \mathrm{m} ?\)

Blackbody radiation is a theoretical concept in physics that describes how an idealized physical body, called a blackbody, emits and absorbs electromagnetic radiation. According to this theory, the energy emitted by a blackbody is solely a function of its temperature, independent of its material composition or surface texture.

A blackbody in thermal equilibrium (that is, its temperature is constant) emits electromagnetic radiation called blackbody radiation. The radiation emitted encompasses a range of wavelengths and follows a specific distribution given by certain laws of physics, such as Planck's law. This distribution results in the characteristic 'blackbody spectrum', which has a peak intensity that shifts to shorter wavelengths as the temperature of the body increases—a phenomenon known as Wien's displacement law.

Understanding blackbody radiation is essential for various applications, including thermal imaging, astrophysics where it's used to study the temperature of stars, and the development of technologies such as incandescent light bulbs.

Planck's Law provides a mathematical description of the electromagnetic radiation emitted by a blackbody in thermal equilibrium at a definite temperature. This quantum theory-based formulation was introduced by Max Planck in 1900 and revolutionized the field of quantum mechanics by suggesting that energy is quantized, rather than continuous.

The law is expressed as: \( B_{\lambda}(T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda k_B T}}-1} \) where:\

• \( B_{\lambda}(T) \) is the spectral radiance of the blackbody
• \( h \) is Planck's constant
• \( c \) is the speed of light in vacuum
• \( \lambda \) is the wavelength of the emitted radiation
• \( k_B \) is Boltzmann's constant
• \( T \) is the absolute temperature of the body in Kelvins

Planck's Law significantly improved upon the earlier Rayleigh-Jeans Law at short wavelengths (ultraviolet catastrophe) where the latter predicted infinite energy radiance, which was not observed in reality.

Spectral radiance is a measure of the quantity of radiation that a surface emits or reflects in a given direction, per unit area, per unit wavelength, per unit solid angle. It is an important quantity in the study of electromagnetic phenomena, providing detailed information about how the intensity of radiation varies with wavelength. In other words, it gives a complete picture of the 'color' or spectral composition of the radiation.

Mathematically, if \([B_{\lambda}(T)] \) is the spectral radiance at a temperature \([T]\), then it will vary with wavelength \([\lambda]\) and temperature. It is essential when calculating the amount of energy a blackbody radiates, and it is at the heart of understanding both the Rayleigh-Jeans Law and Planck's Law. The spectral radiance not only helps in scientific research but also has practical applications in sensing technologies and developing light emission devices.

Percentage error calculation is a method used to quantify the accuracy of a measurement or value by comparing it against a known or accepted reference value. It is a crucial step in various scientific experiments and mathematical problems, as it helps determine how precise and reliable the results are.

The formula to calculate percentage error is:\

• \[% error = \frac{|Actual Value - Experimental Value|}{|Actual Value|} \times 100\% \]

This formula accentuates the relative size of the absolute difference between the actual value and the experimental value by turning it into a percentage of the actual value. The modulus is used to ensure the error is always positive, as the direction of deviation is not significant.

When applied to the results obtained from the Rayleigh-Jeans law and Planck's law, the percentage error calculation informs us how well the Rayleigh-Jeans law agrees with the more accurate Planck's law, particularly across different wavelengths as seen in the exercise for a blackbody at a temperature of 2000 Kelvin.