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What is the rate of energy radiation per unit area of a blackbody at (a) \(273 \mathrm{~K}\) and (b) \(2730 \mathrm{~K} ?\)

Short Answer

Expert verified
The rate of radiation of a blackbody at (a) 273 K is approximately 315.09 W/m^2 and at (b) 2730 K is approximately 315.09*10^4 W/m^2.

Step by step solution

01

Understanding the Stefan-Boltzmann Law

The Stefan-Boltzmann law is represented by the equation \( E = \sigma T^4 \), where \( E \) is the energy radiation per unit area, \( \sigma \) is the Stefan-Boltzmann constant equal to \( 5.67x10^{-8} Wm^{-2}K^{-4} \), and \( T \) is the temperature in Kelvin.
02

Calculating energy radiation for (a) 273 K

Plug the given temperature into the equation: \( E = 5.67x10^{-8} * (273)^4 \) and calculate the result, which will be the rate of energy radiation per unit area at 273 K.
03

Calculating energy radiation for (b) 2730 K

Repeat the process for the second temperature: \( E = 5.67x10^{-8} * (2730)^4 \). The result will be the rate of energy radiation per unit area at 2730 K.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Blackbody Radiation
Blackbody radiation refers to the thermal electromagnetic radiation emitted by an idealized object, known as a blackbody, which absorbs all incident radiation, regardless of frequency or angle of incidence. This type of radiation is a key concept in the field of physics as it provides insight into how energy is emitted by objects based on their temperatures. Unlike real-world objects that can reflect, refract, or transmit photons, a perfect blackbody is a theoretical construct that represents an object which absorbs all incoming light and emits radiation at all wavelengths in a predictable pattern. The wavelength distribution, and thus the intensity of the radiated energy, depends solely on the object’s temperature. Blackbody radiation is characterized by the spectrum of radiation it would emit, which shifts towards higher frequencies as the temperature increases.
Energy Radiation
Energy radiation, in the context of blackbodies, refers to the process by which energy is emitted in the form of electromagnetic waves due to the thermal motion of particles within the object. This emission occurs at all temperatures and increases with rising temperature. The rate of energy radiation per unit area from a blackbody is determined by the Stefan-Boltzmann Law. As temperature increases, energy radiation becomes more intense and shifts to shorter wavelengths. Understanding the rate at which energy is radiated is essential for applications across various scientific fields such as astrophysics, climate science, and engineering. This includes theories explaining the brightness of stars and understanding radiation from the Earth's surface.
Temperature in Kelvin
Temperature in Kelvin is a key factor in calculating the amount of radiation emitted by a blackbody using the Stefan-Boltzmann Law. The Kelvin scale is an absolute temperature scale, starting from absolute zero, where thermal motion ceases. In blackbody radiation calculations, the temperature must be in Kelvin to accurately apply the Stefan-Boltzmann equation: \[ E = \sigma T^4 \]where \(T\) represents the object's temperature in Kelvin. The Kelvin scale allows scientists to unambiguously measure thermal energy, as it directly relates temperature to energy, removing the negative values found in other scales, such as Celsius. Using Kelvin is vital for precision and consistency in scientific computations and hypotheses.
Stefan-Boltzmann Constant
The Stefan-Boltzmann constant, denoted as \(\sigma\), is a fundamental constant in physics that relates the energy radiated by a blackbody in terms of its temperature. Its value is approximately \(5.67 \times 10^{-8} \, \mathrm{Wm^{-2}K^{-4}}\).This constant is used in the Stefan-Boltzmann Law, which articulates that the total energy emission from a blackbody per unit area is directly proportional to the fourth power of the absolute temperature: \[ E = \sigma T^4 \]The constant ensures that the relationship between temperature and emitted energy is accurately quantified. It plays a crucial role not only in theoretical applications but also in practical endeavors like calculating the radiative properties of stars, planets, and other celestial bodies, as well as determining thermal radiation in engineering systems.

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Most popular questions from this chapter

A copper pot with a mass of \(0.500 \mathrm{~kg}\) contains \(0.170 \mathrm{~kg}\) of water, and both are at \(20.0^{\circ} \mathrm{C}\). A \(0.250 \mathrm{~kg}\) block of iron at \(85.0^{\circ} \mathrm{C}\) is dropped into the pot. Find the final temperature of the system, assuming no heat loss to the surroundings.

Size of a Light-Bulb Filament. The operating temperature of a tungsten filament in an incandescent light bulb is \(2450 \mathrm{~K},\) and its emissivity is \(0.350 .\) Find the surface area of the filament of a \(150 \mathrm{~W}\) bulb if all the electrical energy consumed by the bulb is radiated by the filament as electromagnetic waves. (Only a fraction of the radiation appears as visible light.)

A carpenter builds an exterior house wall with a layer of wood \(3.0 \mathrm{~cm}\) thick on the outside and a layer of Styrofoam insulation \(2.2 \mathrm{~cm}\) thick on the inside wall surface. The wood has \(k=0.080 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K},\) and the Styrofoam has \(k=0.027 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} .\) The interior surface temperature is \(19.0^{\circ} \mathrm{C},\) and the exterior surface temperature is \(-10.0^{\circ} \mathrm{C}\). (a) What is the temperature at the plane where the wood meets the Styrofoam? (b) What is the rate of heat flow per square meter through this wall?

Consider a flat metal plate with width \(w\) and length \(l,\) so its area is \(A=l w\). The metal has coefficient of linear expansion \(\alpha\). Derive an expression, in terms of \(\alpha,\) that gives the change \(\Delta A\) in area for a change \(\Delta T\) in temperature.

Derive an equation that gives \(T_{\mathrm{K}}\) as a function of \(T_{\mathrm{F}}\) to the nearest hundredth of a degree. Solve the equation and thereby obtain an equation for \(T_{\mathrm{F}}\) as a function of \(T_{\mathrm{K}}\).

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