Question

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

Short answer

Rate is 315.6 W/m² at 273 K; 3.156 x 10⁶ W/m² at 2730 K.

Step-by-step solution

Understanding the Exercise

We need to determine the rate at which a blackbody radiates energy per unit area at two different temperatures: 273 K and 2730 K. This requires using the Stefan-Boltzmann law for blackbody radiation.

Introduction to the Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the energy radiated per unit area of a blackbody per unit time is proportional to the fourth power of the absolute temperature. The equation is given by:\[ E = \sigma T^4 \]where \( E \) is the energy emitted per unit area, \( \sigma \) is the Stefan-Boltzmann constant \( (5.67 \times 10^{-8} \, \text{W/m}^2\cdot \text{K}^4) \), and \( T \) is the absolute temperature in Kelvins.

Calculating Radiation at 273 K

Using the Stefan-Boltzmann law, we can substitute \( T = 273 \) K into the formula:\[ E_{273} = 5.67 \times 10^{-8} \times (273)^4 \]Performing the calculation gives:\[ E_{273} \approx 315.6 \, \text{W/m}^2 \]

Calculating Radiation at 2730 K

Now using the same law but with \( T = 2730 \) K:\[ E_{2730} = 5.67 \times 10^{-8} \times (2730)^4 \]Perform the calculation to find:\[ E_{2730} \approx 3.156 \times 10^6 \, \text{W/m}^2 \]

Conclusion

The rate of energy radiation per unit area of a blackbody at 273 K is approximately 315.6 W/m², and at 2730 K is approximately 3.156 x 10⁶ W/m².

Key concepts

Blackbody Radiation

Blackbody radiation is a fundamental concept in physics, referring to the type of electromagnetic radiation emitted by a perfect blackbody. A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It does not reflect or transmit any light. This makes it a perfect emitter of radiation when it is at thermal equilibrium, emitting a characteristic spectrum of light that depends solely on its temperature.

When something behaves like a blackbody, the energy it emits is uniformly distributed across different wavelengths. This property is key in many fields, including astrophysics and climate science, as it helps in understanding how stars emit light and even in gauging Earth's energy balance. The emitted radiation covers a range of wavelengths, and as the temperature changes, these wavelengths shift. This process follows the principles of the Stefan-Boltzmann law, which connects the temperature of a body to the radiation it emits.

Energy Emission

Energy emission refers to the release of energy, often in the form of electromagnetic radiation, from a source. In the context of blackbody radiation, energy emission is the radiation emitted by the blackbody per unit area. It depends on the temperature of the blackbody and is described using the Stefan-Boltzmann law.

The amount of energy emitted per unit area (denoted as \( E \)) is directly related to the fourth power of the absolute temperature \( (T) \) of the body. The Stefan-Boltzmann constant \( (\sigma) \) acts as the constant of proportionality, making the equation:

• \( E = \sigma T^4 \)
• Where \( \sigma \) is approximately \( 5.67 \times 10^{-8} \text{ W/m}^2 \cdot \text{K}^4 \)
• \( T \) is the absolute temperature in Kelvin

This equation helps compute how much energy a blackbody emits at any given temperature, as seen in calculating energy at 273 K and 2730 K in the provided exercise.

Temperature Dependence

The Stefan-Boltzmann law highlights the temperature dependence of energy emission from a blackbody. Temperature dependence means that the amount and characteristics of radiation emitted by the blackbody change as its temperature changes.

This relationship is specifically a fourth power dependence, as quantified by the equation \( E = \sigma T^4 \). It implies that even small changes in a blackbody's temperature can lead to large changes in the energy it emits. For instance, doubling the temperature increases the energy emission by a factor of sixteen \((2^4)\).

Due to this strong dependence, this law becomes crucial in predicting and understanding phenomena in various systems, from everyday objects to celestial bodies. It helps scientists explain why hotter objects (like the Sun) emit more energy than cooler ones (like the Earth).

Absolute Temperature

Absolute temperature is a measurement based on the thermodynamic scale, which starts at absolute zero—the point where particles have minimal thermal activity. Kelvin (K) is the unit for measuring absolute temperature.

The concept of absolute temperature is vital when dealing with thermal radiation calculations, such as those involving the Stefan-Boltzmann law. Since energy emission of a blackbody (as per the Stefan-Boltzmann equation) depends on \( T^4 \), it is crucial to have accurate temperature readings.

Absolute zero, zero Kelvin, is theoretically the lowest limit of temperature where no energy is emitted, as all particle movement ceases. By using Kelvin, we ensure that all temperature calculations are relative to this baseline, avoiding negative values and ensuring a standard reference for physical laws. This consistency is what allows formulas like the Stefan-Boltzmann law to be universally applicable across different systems and conditions.