Question

A block has a cavity inside, occupied by a photon gas. Briefly explain what the characteristics of this gas should have to do with the temperature of the block.

Short answer

The characteristics of the photon gas inside the block's cavity are directly proportional to the temperature of the block. This is due to the Stefan-Boltzmann law which states that the energy of radiation (or in this case, the energy of the photon gas) is proportional to the fourth power of the block's temperature. Thus, any change in temperature will affect the energy of the photon gas.

Step-by-step solution

Understanding Photon Gas

A photon gas is a theoretical gas made up of photons which could be considered as particles devoid of rest mass, in a constant state of motion at the speed of light. Due to their particle-wave duality, photons can also behave as waves, and have energy quantized, meaning it exists in discrete amounts. Photons also carry momentum and can exert pressure on objects.

Understanding the Link Between Radiation and Temperature

By virtue of the principles of thermodynamics, all bodies above absolute zero temperature radiate energy. In case of the block, the energy will be radiated through the photon gas in the cavity. The energy of radiation is proportional to the fourth power of the temperature of the body - a concept articulated in the Stefan-Boltzmann law which is expressed as \(E = \sigma T^4\), where \(E\) is the total emitted radiation, \(\sigma\) is the Stefan-Boltzmann constant, and \(T\) is temperature in kelvins. This means that increasing or decreasing the temperature of the block will influence the energy of the photon gas.

Correlating Photon Gas with Temperature of the Block

Given that a photon gas is essentially electromagnetic radiation, the temperature of the block and the energy possessed by the photon gas in the cavity are intrinsically linked. If the block's temperature increases, this means that more energy is being provided to the photon gas, thereby increasing its energy. Conversely, a decrease in temperature means a decrease in the energy of the photon gas. Thus, the characteristics of the photon gas (in terms of energy or spectral distribution) will be directly related to the temperature of the block.

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a cornerstone of thermodynamics and plays a crucial role in understanding how objects emit radiation. It states that the total energy radiated per unit surface area of a black body across all wavelengths is directly proportional to the fourth power of the black body's temperature. In simpler terms, if you double the temperature of an object (measured in Kelvin), it will radiate energy at sixteen times the rate. For students trying to relate this to the exercise, this means that the temperature of the block you are studying is essential for determining the energy output of the photon gas inside the cavity. If the block's temperature were to change, the energy of the photons, and therefore the radiation emitted, would change significantly, in line with the equation \(E = \sigma T^4\), where \(E\) represents the energy radiated per unit time, \(\sigma\) is the Stefan-Boltzmann constant, and \(T\) denotes the absolute temperature.

Particle-Wave Duality

Particle-wave duality is one of the most intriguing concepts in physics, asserting that all particles exhibit both wave and particle properties. This principle is fundamental to quantum mechanics and has profound implications when examining photon gases. Photons, which are quanta of light, can behave like particles, with definite amounts of energy and momentum, but also like waves, capable of displaying interference patterns. In the context of our exercise, this duality is important because it affects how the photon gas interacts with the walls of the cavity within the block. It also determines the energy distribution of the photons, since as waves they can occupy different modes within the cavity, and these modes are temperature-dependent. Understanding this duality helps students to grasp why changing the temperature of the block alters the photon gas characteristics in a specific manner.

Blackbody Radiation

Blackbody radiation refers to the theoretical emission spectrum of a black body—a perfect absorber and emitter of radiation—at a given temperature. This concept is closely related to the Stefan-Boltzmann Law and is key to understanding photon gas within a cavity. A black body at equilibrium emits electromagnetic radiation in a predictable distribution that depends solely on its temperature. So, in terms of the exercise, when addressing how the photon gas characteristics are related to the block's temperature, we infer that as the temperature shifts, so does the pattern of the blackbody radiation. This is demonstrated in the Planck's Law which accurately describes the spectral brightness of blackbody radiation as a function of wavelength for a given temperature. Students can thereby conclude that the blackbody radiation profile of the photon gas will alter with the temperature of the block, which in turn can be used to deduce the thermal characteristics of the cavity.

Thermodynamic Principles

Thermodynamics is the science of energy and its transformation, governed by fundamental laws. The principles that apply to photon gases and this exercise can be broken down into understanding how systems exchange energy (via work or heat) and reach thermal equilibrium. The first law of thermodynamics, often known as the law of energy conservation, implies that the energy cannot disappear; it can only be transferred or transformed. In our scenario, the photon gas within the block's cavity absorbs or releases energy as the temperature of the block changes. The second law of thermodynamics, which states that entropy within a closed system tends to increase, suggests that the energy distribution among the photons will tend towards equilibrium over time. This means the emission from the photon gas will ultimately stabilize, reflecting a new thermal equilibrium consistent with the block's temperature. For students, understanding that the photon gas adopts the temperature of its surroundings, aiming for equilibrium, is a key takeaway directly connected to the exercise.