Problem 72 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

At 273 K: \(315.6\) W/m\(^2\), at 2730 K: \(315,584,651\) W/m\(^2\).

Step 1: Identify the Formula

The total power radiated per unit area by a blackbody is given by the Stefan-Boltzmann law. The formula is \( E = \sigma T^4 \), where \( E \) is the energy radiated per unit area, \( \sigma \) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4)\), and \( T \) is the temperature in Kelvin.

Step 2: Calculate for 273 K

Using the formula \( E = \sigma T^4 \), plug in \( T = 273 \) K. So, \( E = 5.67 \times 10^{-8} \times (273)^4 \). Compute the result to find the energy radiated per unit area at 273 K.

Step 3: Calculate for 2730 K

Similarly, plug in \( T = 2730 \) K into the formula. Thus, \( E = 5.67 \times 10^{-8} \times (2730)^4 \). Compute the result to find the energy radiated per unit area at 2730 K.

Step 4: Final Calculation

Calculate \((273)^4 = 55568161\) and multiplying it by \(5.67 \times 10^{-8}\) gives \(E_{273} \approx 315.6\) W/m\(^2\). For \((2730)^4 = 5556809801\) and multiplying it by \(5.67 \times 10^{-8}\) gives \(E_{2730} \approx 315584651\) W/m\(^2\).

Key Concepts

Blackbody Radiation

Blackbody radiation is an essential concept in the field of physics. A blackbody is an idealized object that absorbs all radiation that falls on it, without reflecting any. This makes it a perfect emitter of radiation. The radiation emitted by a blackbody depends only on its temperature and is independent of its material or shape. This kind of radiation covers a spectrum of wavelengths, from infrared to visible light.
Understanding blackbody radiation is important because it serves as a model for real-world objects and stars that emit radiation due to their temperature. For example, the Sun behaves like a near-perfect blackbody. This property makes blackbody radiation a vital tool in studying the thermal characteristics of various astronomical and terrestrial bodies.

Energy Radiated

The energy radiated by a blackbody per unit area is a measure of how much energy is emitted by the object into its surroundings. This emitted energy plays a crucial role in understanding heat transfer and thermodynamics. According to the Stefan-Boltzmann law, the amount of energy radiated by a blackbody is proportional to the fourth power of its temperature. This implies that even a small increase in the temperature can lead to a significant increase in the radiation energy.

• Energy per unit area increases rapidly with temperature change.
• The concept helps in calculating the heat loss or gain by bodies in various scientific fields.

Such calculations are fundamental in designing systems that involve heat exchangers or thermal management solutions in engineering.

Temperature Dependence

Temperature plays a significant role in the amount of energy radiated by a blackbody. As shown in the Stefan-Boltzmann Law, the radiation energy is proportional to the fourth power of temperature, represented as \( E = \sigma T^4 \). This demonstrates that radiation energy grows exponentially with temperature.
Higher the temperature, greater is the energy radiated, making the temperature a critical factor in fields requiring thermal analysis, like climate science and materials engineering.

• Temperature increase leads to significant rise in radiated energy.
• Key to understanding the thermal balance in natural systems, like the Earth's climate.

The exponential nature of this relationship often surprises new learners, highlighting the importance of understanding how small changes in temperature can have large effects on radiation.

Stefan-Boltzmann Constant

The Stefan-Boltzmann constant is a fundamental constant in physics, denoted by \( \sigma \). Its value is \( 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4 \). It provides the proportionality factor in the Stefan-Boltzmann law, which relates the energy radiated by a blackbody to its temperature. This constant ensures that the equation gives results in units of energy per unit area per unit time.
The constant was derived from experimental data, and it plays a crucial role in blackbody radiation theory.

• Essential for calculating energy emission from bodies of different temperatures.
• Helps to model and predict characteristics of star radiation in astrophysics.

The Stefan-Boltzmann constant is utilized in understanding not only theoretical models but also practical applications like designing radiative heat transfer systems.