Question

What is the Stefan-Boltzmann law? Write down the formula and explain the variables.

Short answer

The Stefan-Boltzmann law states that energy emitted is proportional to temperature to the fourth power: \( E = \\sigma T^4 \).

Step-by-step solution

Understanding Stefan-Boltzmann Law

The Stefan-Boltzmann law is a principle in physics that defines how the total energy radiated per unit surface area of a black body is proportional to the fourth power of the black body's absolute temperature.

Mathematical Representation

The law is mathematically represented by the formula: \[ E = \sigma T^4 \]where \( E \) is the total energy radiated per unit area, \( T \) is the absolute temperature in kelvin, and \( \sigma \) is the Stefan-Boltzmann constant.

Explaining the Variables

- \( E \): Total energy radiated per unit area of the black body, usually measured in watts per square meter (W/m²).- \( T \): Absolute temperature of the black body in kelvin (K).- \( \sigma \): Stefan-Boltzmann constant, which is approximately \( 5.67 \times 10^{-8} \, \text{Wm}^{-2} \text{K}^{-4} \).

Key concepts

Black Body Radiation

Black body radiation refers to the type of electromagnetic radiation that a perfect black body can emit. A black body is an idealized physical object that perfectly absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. This means it does not reflect or transmit any light.
The concept of black body radiation is essential in understanding the Stefan-Boltzmann Law. The radiation emitted by black bodies has a characteristic distribution that depends only on the temperature of the body. As the temperature of a black body increases, it emits more energy across all frequencies, which peaks in intensity at a particular frequency determined by Wien's Displacement Law. This helps physicists predict how much energy an object will radiate based on its temperature.
Key points about black body radiation:

• Black bodies absorb all wavelengths of light.
• They emit energy in the form of electromagnetic radiation.
• The radiation depends solely on the body's temperature.

Absolute Temperature

Absolute temperature is a measure of temperature on a scale where zero is absolute zero, the point at which all molecular motion ceases. The most commonly used absolute temperature scale is the Kelvin scale, used in scientific research and engineering. In this scale, 0 Kelvin is equal to -273.15 degrees Celsius, which is the theoretical minimum temperature possible.
The Kelvin scale is crucial in applying the Stefan-Boltzmann Law since the formula uses temperature in Kelvin. This ensures accuracy because temperature in kelvin reflects true thermal states, unaffected by the properties of the material. The fourth power of the absolute temperature in the Stefan-Boltzmann Law illustrates how increases in temperature significantly impact the energy radiated by a body.
Key aspects of absolute temperature:

• Measured in Kelvin (K).
• Starts at absolute zero, where molecular motion stops.
• Essential for thermodynamic calculations and energy laws.

Stefan-Boltzmann Constant

The Stefan-Boltzmann constant is a fundamental constant in physics, represented by the Greek letter sigma (\( \sigma \)). It plays a vital role in the Stefan-Boltzmann Law, linking temperature to radiative energy emission, quantified in watts per square meter.
The value of the Stefan-Boltzmann constant is approximately \( 5.67 \times 10^{-8} \, \text{Wm}^{-2} \text{K}^{-4} \). It provides a measure of how much energy a body emits based on its temperature. The constant ensures that calculations of radiated energy are accurate and consistent with measurements made in experiments and nature.
Important facts about the Stefan-Boltzmann constant:

• Fundamental in linking temperature and energy radiation.
• The constant's value is derived from experimental measurements.
• Ensures uniformity in power calculations across different systems.

Energy Radiation

Energy radiation is the process by which energy is emitted from a system or body in the form of electromagnetic waves or particles. This concept is at the heart of the Stefan-Boltzmann Law, which calculates how much energy a black body emits based on its temperature.
In physics, energy radiation can occur in various forms, such as light, heat, X-rays, and radio waves. The energy radiated by a body depends on its temperature and surface characteristics. The Stefan-Boltzmann Law specifically measures this emission in watts per square meter, showing the correlation between temperature and the intensity of the radiative power.
Elements of energy radiation include:

• Emitted in the form of electromagnetic radiation.
• Dependent on both the surface area and temperature of the emitter.
• A key concept in thermodynamics and energy transfer processes.

Physics Formula

The Stefan-Boltzmann Law's formula is a mathematical representation of the relationship between a body's emitted energy and its temperature. The formula is expressed as:\[ E = \sigma T^4 \]where \( E \) is the total energy radiated per unit area, \( T \) is the absolute temperature in Kelvin, and \( \sigma \) is the Stefan-Boltzmann constant. This formula is simple yet powerful as it quantitatively describes how energy radiation scales with temperature.
By understanding each component of the equation, students and scientists alike can predict and measure the amount of energy radiated from a body, highlighting its relevance in fields like astrophysics, climatology, and engineering.
Components of the Stefan-Boltzmann formula:

• \( E \) represents radiated energy per unit area.
• \( T^4 \) signifies the pivotal role of temperature in energy emission.
• \( \sigma \) serves as a bridge between theoretical and experimental physics.