Question

Total amount of energy emitted by a perfectly black body per unit area per unit time is directly proportional to (a) \(T\) (b) \(T^{2}\) (c) \(T^{4}\) (d) \(T^{3}\)

Short answer

The total amount of energy emitted by a perfectly black body per unit area per unit time is directly proportional to \(T^{4}\)

Step-by-step solution

Understanding the Concept

The Stefan-Boltzmann Law, which is fundamental in thermal physics, states that the total energy emitted per unit surface area of a black body in unit time, also referred to as the black body's emissive power, is directly proportional to the fourth power of the black body's absolute temperature. This is often written as \(E=\sigma T^{4}\) where \(E\) is the emissive power, \(\sigma\) is the Stefan-Boltzmann constant and \(T\) is the absolute temperature.

Matching the Concept with the Given Options

The options given in the exercise are different powers of the temperature \(T\): \(T\), \(T^{2}\), \(T^{3}\), and \(T^{4}\). As outlined in the previous step, according to Stefan-Boltzmann Law, the emissive power \(E\) is directly proportional to \(T^{4}\), not any other power of \(T\).

Key concepts

Black Body Radiation

Black body radiation is a foundational concept in the field of thermal physics. It involves the study of how objects absorb and emit radiation. A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. This means a black body is a perfect absorber and emitter of radiation.

A real-world object that approximates this behavior is termed a 'grey body' and has emissive properties guided by the same principles, although it does not absorb and emit all frequencies of radiation perfectly. Understanding black body radiation is crucial because it provides insights into the behavior of particles and quantum mechanics and helps us comprehend the radiation spectrum of objects.

For example, the radiation emitted by a star, though not a perfect black body, can closely be estimated using black body assumption and provides valuable information about the star's temperature, composition, and age.

Thermal Physics

Thermal physics is a sub-discipline of physics that deals with the relationships between heat, work, temperature, and energy. The laws of thermodynamics govern these relationships and are key to understanding how energy transfer occurs and how matter behaves at different temperatures. One of the principles of thermal physics is that objects emit energy in the form of heat and electromagnetic radiation.

The study of thermal physics is essential in various technological and scientific fields, including developing thermal insulation, engines, and even understanding the cosmological phenomena. It also plays a significant role in our everyday life, explaining how ovens heat food, how sunlight warms our skin, and why we use certain materials for warm clothing. It connects the macroscopic and microscopic worlds, explaining the behavior of atoms and molecules in terms of temperature-related phenomena.

Emissive Power

Emissive power, in the context of thermal physics, refers to the amount of thermal radiation energy that a body emits per unit area. It's an intrinsic property of materials and is heavily influenced by the temperature of the body and its surface characteristics. According to the Stefan-Boltzmann Law, the emissive power (E) of a black body is directly proportional to the fourth power of its absolute temperature (T), expressed through the formula \(E = \sigma T^{4}\).

The constant \(\sigma\) represents the Stefan-Boltzmann constant and is a measure of how efficiently a body can radiate energy. Materials with high emissive power are good emitters of energy as thermal radiation and play a critical role in the design of radiators, heaters, and thermal shielding, as they dictate how a material will react to changes in temperature, especially when it comes to heat transfer.