Blackbody Emission From N N N Parallel & Infinite 2D Planes

Introduction

The concept of blackbody emission is a fundamental aspect of thermal radiation, describing the energy radiated by an object due to its temperature. The Stefan-Boltzmann law, which states that the total energy radiated per unit surface area of a blackbody across all wavelengths per unit time (also known as the power emitted per unit area), is given by the formula $\sigma T^4$, where $\sigma$ is the Stefan-Boltzmann constant and $T$ is the temperature of the blackbody. However, this law is primarily derived for objects with a finite surface area, such as spheres or cubes. In this article, we will explore the blackbody emission from $N$ parallel and infinite 2D planes, a scenario that has been puzzling many physicists.

Thought Experiment

A solid object will emit radiation with a flux surface of $\sigma T^4$, but what about a plane? Here is a thought experiment I came up with to help me make sense of this problem. Imagine a 2D plane, infinite in size, and at a temperature $T$. We can consider this plane as being composed of an infinite number of infinitesimally small areas, each of which is radiating energy according to the Stefan-Boltzmann law. However, since the plane is infinite, we need to consider the radiation emitted by each of these small areas and how they interact with each other.

Gauss's Law and the Electric Field

To understand the radiation emitted by the 2D plane, we can start by considering the electric field surrounding it. According to Gauss's law, the total electric flux through a closed surface is proportional to the charge enclosed within that surface. In the case of a 2D plane, we can imagine a Gaussian surface that is a cylinder with its axis perpendicular to the plane. The electric field lines will be perpendicular to the plane and will be confined within the cylinder. This means that the electric field will be zero outside the cylinder, and the flux through the cylinder will be proportional to the charge enclosed within it.

The Electric Field and Radiation

Now, let's consider the radiation emitted by the 2D plane. We can imagine a small area of the plane, which is radiating energy according to the Stefan-Boltzmann law. The radiation will be emitted in all directions, and the electric field will be perpendicular to the plane. However, since the plane is infinite, the radiation emitted by each small area will interact with the radiation emitted by other areas. This means that the total radiation emitted by the plane will be the sum of the radiation emitted by each small area, taking into account the interactions between them.

The Case of N Parallel 2D Planes

Now, let's consider the case of $N$ parallel 2D planes, each at a temperature $T$. We can imagine a Gaussian surface that is a cylinder with its axis perpendicular to the planes. The electric field lines will be perpendicular to the planes and will be confined within the cylinder. This means that the electric field will be zero outside the cylinder, and the flux through the cylinder will be proportional to the charge enclosed within it.

The Radiation Emitted by N 2D Planes

The radiation emitted by $N$ parallel 2D planes will be the sum of the radiation emitted by each plane, taking into account the interactions between them. Since each plane is radiating energy according to the Stefan-Boltzmann law, the total radiation emitted by the planes will be proportional to the sum of the temperatures of the planes. This means that the radiation emitted by $N$ parallel 2D planes will be given by the formula:

$$\sigma \sum_{i=1}^{N} T_i^4$$

where $\sigma$ is the Stefan-Boltzmann constant and $T_i$ is the temperature of the $i^{th}$ plane.

The Case of Infinite 2D Planes

Now, let's consider the case of infinite 2D planes, each at a temperature $T$. We can imagine a Gaussian surface that is a cylinder with its axis perpendicular to the planes. The electric field lines will be perpendicular to the planes and will be confined within the cylinder. This means that the electric field will be zero outside the cylinder, and the flux through the cylinder will be proportional to the charge enclosed within it.

The Radiation Emitted by Infinite 2D Planes

The radiation emitted by infinite 2D planes will be the sum of the radiation emitted by each plane, taking into account the interactions between them. Since each plane is radiating energy according to the Stefan-Boltzmann law, the total radiation emitted by the planes will be proportional to the sum of the temperatures of the planes. This means that the radiation emitted by infinite 2D planes will be given by the formula:

$$\sigma \sum_{i=1}^{\infty} T_i^4$$

where $\sigma$ is the Stefan-Boltzmann constant and $T_i$ is the temperature of the $i^{th}$ plane.

Conclusion

In conclusion, the blackbody emission from $N$ parallel and infinite 2D planes is a complex phenomenon that requires a deep understanding of the underlying physics. By considering the electric field and radiation emitted by each small area of the plane, we can derive the total radiation emitted by the plane. The case of $N$ parallel 2D planes is similar to the case of a single plane, with the total radiation emitted being proportional to the sum of the temperatures of the planes. The case of infinite 2D planes is more complex, with the total radiation emitted being proportional to the sum of the temperatures of the planes, but with an infinite number of terms.

References

• [1] Stefan-Boltzmann Law. In: Encyclopedia of Physics, 2nd edn. Springer, Berlin, Heidelberg (2005)
• [2] Gauss's Law. In: Classical Electrodynamics, 3rd edn. John Wiley & Sons, Inc., Hoboken, NJ (2007)
• [3] Blackbody Radiation. In: The Oxford Handbook of the History of Physics, Oxford University Press, Oxford (2013)
  Q&A: Blackbody Emission from N Parallel & Infinite 2D Planes ===========================================================

Q: What is blackbody emission?

A: Blackbody emission is the energy radiated by an object due to its temperature. The Stefan-Boltzmann law describes the energy radiated per unit surface area of a blackbody across all wavelengths per unit time.

Q: How does the Stefan-Boltzmann law apply to 2D planes?

A: The Stefan-Boltzmann law is primarily derived for objects with a finite surface area, such as spheres or cubes. However, for 2D planes, the law needs to be modified to account for the infinite surface area.

Q: What is the radiation emitted by a single 2D plane?

A: The radiation emitted by a single 2D plane is proportional to the temperature of the plane, given by the formula $\sigma T^4$, where $\sigma$ is the Stefan-Boltzmann constant and $T$ is the temperature of the plane.

Q: What is the radiation emitted by N parallel 2D planes?

A: The radiation emitted by $N$ parallel 2D planes is the sum of the radiation emitted by each plane, taking into account the interactions between them. The total radiation emitted is proportional to the sum of the temperatures of the planes, given by the formula $\sigma \sum_{i=1}^{N} T_i^4$.

Q: What is the radiation emitted by infinite 2D planes?

A: The radiation emitted by infinite 2D planes is the sum of the radiation emitted by each plane, taking into account the interactions between them. The total radiation emitted is proportional to the sum of the temperatures of the planes, given by the formula $\sigma \sum_{i=1}^{\infty} T_i^4$.

Q: How does the electric field affect the radiation emitted by 2D planes?

A: The electric field surrounding the 2D plane affects the radiation emitted by the plane. According to Gauss's law, the total electric flux through a closed surface is proportional to the charge enclosed within that surface. In the case of a 2D plane, the electric field lines will be perpendicular to the plane and will be confined within a Gaussian surface.

Q: What is the significance of Gauss's law in understanding blackbody emission?

A: Gauss's law is essential in understanding blackbody emission from 2D planes. It helps to determine the electric field surrounding the plane and how it affects the radiation emitted.

Q: Can you provide an example of how to calculate the radiation emitted by N parallel 2D planes?

A: Let's consider an example where we have 3 parallel 2D planes, each at a temperature of 300 K. Using the formula $\sigma \sum_{i=1}^{N} T_i^4$, we can calculate the total radiation emitted by the planes as follows:

$$\sigma (T_1^4 + T_2^4 + T_3^4) = \sigma (300^4 + 300^4 + 300^4) = 3 \sigma 300^4$$

: What are the implications of blackbody emission from 2D planes in real-world applications?

A: Blackbody emission from 2D planes has significant implications in various real-world applications, such as:

• Thermal imaging: Understanding blackbody emission from 2D planes is crucial in thermal imaging, where the radiation emitted by objects is used to create images of their temperature distribution.
• Heat transfer: Blackbody emission from 2D planes affects heat transfer between objects, which is essential in various engineering applications, such as heat exchangers and radiators.
• Materials science: The study of blackbody emission from 2D planes can provide insights into the properties of materials, such as their thermal conductivity and emissivity.

Q: What are the limitations of the current understanding of blackbody emission from 2D planes?

A: While significant progress has been made in understanding blackbody emission from 2D planes, there are still limitations to the current understanding. For example:

• Infinite surface area: The current understanding assumes an infinite surface area, which may not be realistic in many practical situations.
• Interactions between planes: The current understanding assumes that the interactions between planes are negligible, which may not be the case in real-world applications.
• Non-uniform temperature distribution: The current understanding assumes a uniform temperature distribution, which may not be the case in real-world applications.

Q: What are the future directions for research on blackbody emission from 2D planes?

A: Future research on blackbody emission from 2D planes should focus on:

• Experimental verification: Experimental verification of the current understanding of blackbody emission from 2D planes is essential to confirm its accuracy.
• Real-world applications: Investigating the implications of blackbody emission from 2D planes in real-world applications, such as thermal imaging and heat transfer.
• Non-uniform temperature distribution: Studying the effects of non-uniform temperature distribution on blackbody emission from 2D planes.
• Interactions between planes: Investigating the interactions between planes and their effects on blackbody emission.