The Transfer of Energy Between Two Surfaces by Radiation
The methods that we use to develop the mathematics that are used to describe energy transfer by radiation are established by concepts that are not part of our experiences that underlay all of the skills that we have acquired since we were born to guarantee our survival in the world in which we live. How many of those skills or parts of them are transferred to our progeny through our genes is now a serious scientific study (think spider webs).
The formulas that govern where a photon goes and how much energy it carries have arisen as a result of observed physical phenomena and an attempt to construct an expression (usually mathematical) that allows us to predict how energy is transmitted from one place to another. The idea that it was carried in bundles resulted in a Nobel Prize being awarded to Einstein. The properties of these bundles (quanta of energy called photons) require several different mathematical frameworks to explain how they travel from place to place. They are treated as particles, waves, entities having the property of mass as they are even affected by gravitational fields, and the apparent paths that they seem to take are governed by their surroundings. To develop the equations governing their behavior may requires such a bizarre mental pictures that no two theoretical physicists would tell you the same story if you asked him/her the simple question, “In twenty words or less, tell me what is a quantum of energy”. (Or maybe 50 or 100.)
As a result, simple expressions have been developed to allow us to calculate the results of the transmission of energy by photons. The most useful is the Stefan-Boltzmann equation describing the rate of energy transfer from one body to another which is written: Q = A ε σ (T14 – T24 ). The assumptions made concerning the emission of the radiation and its absorption, are given by the symbol ε and the geometries that are involved are encompassed by the symbol A. As an example, the spectrum emitted by a body at the temperature T1 will have a different absorptivity and emissivity at the second body if they are made of different materials or if there is a big difference in the temperature.
You will note that the there are two bodies involved, designated by the two subscripts on the temperatures, and we are reminded about the unidirectionality of the energy transfer by the equation. If T1 is higher than T2 then Q comes out positive for energy flowing from the first object to second. If the temperature of the second object is higher than the first, Q comes out negative and the concept of negative energy states is how the first antiparticle, the positron, was discovered when Paul Dirac wrote the quantum mechanical equation describing the electron. It is convenient that the equation is written in this form as it reminds us that the energy only flows in one direction, from hot to cold.
A proper quantum mechanical description of the process includes the nature of the environment in which this physical process takes place. Numerous mathematical pictures involving electromagnetic fields and the absorption of electromagnetic energy by materials have been written that have their own descriptions in the construction of the formulas that properly describe what takes place in nature. Some are easier to understand than others but as long as they give the right answers to what has been measured by careful observation of what takes place, no matter how bizarre they seem to be, one is as good as another but one of our criteria in the choice is that there is beauty in simplicity. The latest theory is the one developed by Claes Johnson and if you have some understanding of electromagnetic radiation and antenna theory it may be worthwhile to read his paper.
The validity of the Stefan-Boltzmann equation in the form in which it is written above is beyond dispute. Since it was first set down by pen to paper, physicists and engineers have never found it is contravened in an energy transfer by radiation. The nature of the other two symbols in the equation depends on where the energy transfer is taking place. The constant σ which includes several fundamental physical constants is the only quantity that is considered fixed.
As an aid to understanding energy flow between two surfaces let us look at the mechanical picture of energy flow between two surfaces of equal areas separated by a short distance, about which there has been considerable argument. Let us assume the plates are square and the distance between them is 1/1000 of the length of each side and they are in a container in which the pressure has been reduced until the mean free path between the collisions between the molecules is of the order of the separation between the plates. Let us make plate 1 have a higher temperature than plate 2. Now consider how the heat is transported. A molecule collides with plate one and is heated to the temperature of the plate. The temperature of the molecule is given by its kinetic energy. It leaves and travels to plate 2. It collides with plate 2 and transfers some kinetic energy to it. The transfer of heat by collision is easier for us to understand as these constitute the majority of the kinds of processes to which we have been exposed while we were growing up. The molecule leaves plate 2 cooler than it was when it arrived, with the kinetic energy appropriate to the temperature of that plate. It travels to plate 1 and collides but it can’t give energy to plate 1 which is at a higher temperature than plate 2 so it absorbs some from plate 1 making it a little cooler, and the molecule a little warmer, and carries it to plate 2 which now becomes a little hotter.
The picture of radiation transferring energy between two plates is basically the same except there is no photon carrying energy from plate 2 to plate one as the absorption of such a photon at plate 1 is forbidden by the rules of quantum mechanics that are required to describe the behavior of photons properly. One of the baffling things we find when we look at how photons travel is that sometimes they appear to go around corners without being reflected, and sometimes they don’t go where nothing seems to be blocking the route from where they were emitted to where they are absorbed. An easy little demonstration that you can do is to take a dark piece of paper about 10 cm in diameter and punch a round hole about 3 mm in diameter in the middle of it. Now hold it at about 20 cm away, close one eye and look at a fairly detailed object like the printing on a computer screen that is about 1 meter away. The photons prefer to go through the center of the hole which looks bright and sharp and not through the periphery. If you have made a hole with sharp edges, the periphery of the hole is even fuzzy and not sharp. It is easy to see the effect by looking at the print displayed on a computer screen but you can look at flowers in your garden..
Photons are not molecules although sometimes they behave like bullets but at the same time sometimes where they go is governed by the rules and equations we have developed to describe electromagnetic radiation. Have another look at Claes Johnson’s paper for a classical electromagnetic description of a quantum mechanical process. Unfortunately considerable training in the use of Maxwell’s equations and second order differential equations is necessary to completely understand it.
Many of the equations governing physical processes are described by mathematics that has been verified to be correct by observing natural phenomena. They allow us to predict will happen when we change the values of the free parameters, and see whether the results agree with the natural processes we observe. Unfortunately that is not the case for the so called man-made global warming phenomenon and no matter how well we think we understand the fairy tale, it has about as much validity as the arrival of Santa Claus and Rudolph on Dec. 25th. Another Rudolph whose last name was Clausius was the genius who developed the basis of the study of thermodynamics. His 2nd law was considered to be the most solid of all our physical laws by Albert Einstein, who was no slouch at developing mathematical equations to explain esoteric scientific phenomena we did not understand.
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