Show that Wien's displacement law results from Planck's radiation law. (Hint: substitute \(x=h c / \lambda k T\) and write Planck's law in the form \(I(x, T)=A x^{5} /\left(e^{x}-1\right)\) where \(A=2 \pi(k T)^{5} /\left(h^{4} c^{3}\right) .\) Now, for fixed \(T,\) find the position of the maximum in \(I(x, T)\) by solving for \(x\) in the equation \(d I(x, T) / d x=0 .)\),

At the heart of modern physics, especially when dealing with thermal radiation, lies Planck's radiation law. It's a quantum theoretical reformation of the law of blackbody radiation predicted by Planck in 1900. Planck's law describes the electromagnetic radiation emitted by a blackbody in thermal equilibrium at a definite temperature.

The law is expressed mathematically as the distribution of intensity of radiation as a function of wavelength or frequency for a given temperature. It implies that the spectrum of the radiation depends only on the body's temperature and peaks at a certain wavelength. This critical aspect becomes vital in understanding Wien's Displacement Law, which tells us the wavelength at which the emission of radiation is at its maximum for a blackbody at a given temperature.

Planck's law revolutionized the field of thermodynamics and brought about the quantum mechanics era, thereby providing a cornerstone for modern physics and chemistry. When engaging with this law, students can appreciate the blend of mathematics and physics that describes something as ubiquitous as thermal radiation.

Understanding blackbody radiation is a fundamental stepping stone in mastering concepts such as Planck's radiation law and Wien's displacement law. A blackbody is a theoretical or model body which absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That means it is a perfect emitter and absorber of radiation.

Blackbody radiation is the type of electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a blackbody held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depend only on the temperature of the body.

The study of blackbody radiation led to the development of Quantum mechanics after classical physics failed to explain the observed spectral energy distribution. Blackbodies are not just theoretical; in astronomy, stars are often approximated as blackbodies, and understanding their radiation helps scientists deduce key aspects like temperature and composition. In technology, blackbody models help make critical calculations in systems involving heat transfer.

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In essence, it describes how thermal energy is converted to and from other forms of energy and how it affects matter. Thermodynamics is governed by four fundamental laws, which explain phenomena such as why heat transfers from a hot object to a cold one.

The second law of thermodynamics introduces the concept of entropy, predicting the direction of thermal interactions and the inefficiency of energy conversions. This law has a direct relationship with Planck's radiation law and blackbody radiation, as it helps us understand the entropy changes as a blackbody emits radiation.

Furthermore, Wien's Displacement Law is a byproduct of thermodynamic principles applied to blackbody radiation, indicating that thermodynamics isn't just about heat engines and refrigerators but also fundamental to our understanding of the cosmic spectacles and everyday technologies. For students, mastering thermodynamics is essential as it underpins a vast array of natural and engineered processes.