Question

What is Stefan flow? Write the expression for Stefan's law and indicate what each variable represents.

Short answer

#Question# Explain the concept of Stefan flow and derive the expression for Stefan's law. Indicate what each variable in the expression represents. #Answer# Stefan flow, or thermal radiation, refers to the flow of electromagnetic radiation (mainly infrared light) emitted by an object due to its temperature. Stefan's law describes the power radiated per unit area by an object as a function of its temperature. It is derived from Planck's law of blackbody radiation by integrating the power radiated over all possible wavelengths: \[P = \sigma T^4\] In this expression, there are three variables: 1. \(P\) - Power radiated per unit area, representing the amount of energy emitted per unit area of the surface of the object (in W/m²). 2. \(\sigma\) - Stefan-Boltzmann constant, a constant of proportionality relating the power radiated per unit area to the object's temperature with a value of \(5.67 \times 10^{-8} W m^{-2} K^{-4}\). 3. \(T\) - Temperature of the object in Kelvin. The power radiated per unit area increases with the fourth power of the temperature, so the power radiated increases drastically when the object's temperature increases.

Step-by-step solution

Understand Stefan flow

Stefan flow, also known as thermal radiation, refers to the flow of electromagnetic radiation (mainly in the form of infrared light) emitted by an object due to its temperature. Any object with a temperature above absolute zero emits thermal radiation, and Stefan's law is used to describe the power radiated per unit area by an object as a function of its temperature.

Derive Stefan's law

Stefan's law is derived from the Planck's law of blackbody radiation. A blackbody is a hypothetical object that absorbs all incident radiation and perfectly emits radiation at every wavelength. The power radiated by a blackbody per unit area and per unit wavelength is given by Planck's law: \[B(\lambda, T) = \frac{2\pi hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}\] Where: - \(B(\lambda, T)\) is the spectral radiance (power per unit area and per unit wavelength) - \(\lambda\) is the wavelength of the radiation - \(T\) is the temperature of the blackbody - \(h\) is the Planck's constant - \(c\) is the speed of light - \(k_B\) is the Boltzmann constant Stefan's law is obtained by integrating the power radiated over all possible wavelengths: \[\text{Power per unit area} = \int_{0}^{\infty} B(\lambda, T)\, d\lambda\] This integration results in Stefan's law, which states: \[P = \sigma T^4\] Where: - \(P\) is the power radiated per unit area - \(\sigma\) is the Stefan-Boltzmann constant, and its value is approximately \(5.67 \times 10^{-8} W m^{-2} K^{-4}\) - \(T\) is the temperature of the object in Kelvin

Indicate the meaning of each variable in Stefan's law

In Stefan's law, there are three variables: 1. \(P\) - Power radiated per unit area: This represents the amount of energy emitted per unit area of the surface of the object. It has units of watts per square meter (W/m²). 2. \(\sigma\) - Stefan-Boltzmann constant: This is a constant of proportionality that relates the power radiated per unit area to the temperature of the object. Its value is approximately \(5.67 \times 10^{-8} W m^{-2} K^{-4}\). 3. \(T\) - Temperature: The temperature of the object in Kelvin. The power radiated per unit area increases with the fourth power of the temperature. So, when the temperature of an object increases, the power radiated per unit area increases drastically.