Black-Body Radiation: An Explanation without Probability

Black-body Radiation –An Explanation without Probability



Devinder Kumar Dhiman



Devinder Kumar Dhiman

All rights reserved. Any kind of unauthorized reproduction, distribution, or exhibition of this copyrighted material in any medium or form is strictly prohibited.


Editing and Cover design

Chameleon Garcia


Shakespir Edition 2017




Table of Contents

A Word from the Author


Chapter I- Black-body Radiation Formula

Chapter II- Derivation of the Formula by Max Planck

Chapter III- Energy Distribution of Radiation – Independent of Probability



Other books by the Author




A Word from the Author

It was 21st December 2016, the Indian Journal of Science and Technology had published its 45th issue. My article, “Energy Distribution of Radiation Emitted by a Black-Body, Independent of Probability”, was in that issue.

I felt triumphant as my efforts in making a fundamental formula of Quantum Physics independent of probability had paid off. Publishing the article after passing a double blind review of the journal meant that the technique proposed by myself had been reviewed by at least two experts in the related field, and hence, had attained a credibility to spread it further. I wrote out messages on social media about this article. Congratulatory messages started pouring in and its circulation started increasing.

After a few days, I started getting feedback from the people who had downloaded the article and read it, informing me that it was challenging for them to comprehend the article. Hence, I decided to write this book, which is far more explanatory than the article, so every reader can recognize the importance of this article in Physics. Link to the article is [+ http://indjst.org/index.php/indjst/article/view/72642+]

Although it may not make any difference in the daily life of the reader whether the Energy Distribution of Radiation emitted by a Black-body is independent of probability or not, it should still prove interesting for the reader to know about its history and significance in physics.






In December 1900, Max Planck, a senior professor of thermodynamics at the University of Berlin, proposed a formula for energy distribution of black-body radiation using quantization of energy and probability. With this formula, he laid the foundation of quantum physics and deviated from the well-established classical physics. This event marked the birth of Modern physics, as well, and the use of probability by him encouraged other physicists to develop other formulae based on probability, which led to strange results. Quantum physics using such formulae is known to be absurd due to the use of probability, as Mr. Feynmann, a celebrated professor of Quantum physics, had famously quoted, “If you think you understand quantum mechanics, then you don’t.

The world over, physicists have tried to get rid of probability from physics but it has not been possible to do so. The use of probability is so deeply rooted in physics that it is impossible to imagine physics without it. My article, “Energy Distribution of Radiation Emitted by a Black-Body, Independent of Probability”, published in the Indian Journal of Science and Technology in its 45th issue in December 2016 may be a first step in the direction of making physics independent of probability. Therefore, in this book, we’ll discuss the importance of this article, and try to present it in a simple manner such that it becomes easy for the readers to understand it.





Chapter I

Black-body Radiation Formula

First of all, we should know what a black-body is, and what the importance is of energy distribution of radiation emitted from it. A black-body is any surface that absorbs maximum radiation falling on it. It is not necessarily black, but it is given this name because the color black absorbs maximum radiation among all visible colors. Black-body is not only a very good absorber of radiation, it is also a very good emitter. It is formed in the shape of a cubical box having a small hole in one of the sides. The box is insulated from any heat dissipation due to radiation or convection. The exit of radiation is allowed only through the hole as shown in Figure 1.

[* Figure 1- Black-body with an Orifice for Emission of Radiation *]

When a black-body is heated, radiations of different wavelengths are generated on the inner surfaces (outer surfaces are insulated). These radiations strike with one-another and get mixed up. A small amount of the mixture of all radiations finds its way out through the orifice. At a particular temperature, the wavelengths and the energy of the radiations are measured and a graphical curve is drawn. Then the temperature is raised, and the process is repeated, to get a set of graphical curves at different temperatures as shown in Figure 2. The horizontal axis of the graph represents the wavelength of the radiation, and vertical axis represent Energy emitted. Different curves are due to different temperatures

Figure 2 – Relation Between Energy radiated and Wavelength at Different Absolute Temperatures

Why do we require these curves? Let us take a look at the history of these curves.

When the electric incandescent bulb was invented in the nineteenth century, its filament was initially made of carbon, which evaporated slowly on heating, and deposited black soot on the inner surface of the bulb, rendering the bulb useless. The life of the bulb was only about ten to twelve hours at that time compared to thousands of hours of present day bulbs.

To increase the life of the bulb, it was necessary to find another material which could replace the carbon; one that would allow the bulb to exhibit maximum brightness with minimum electrical energy input.

In that era, Germany was the leader in this technology; so naturally, this responsibility came upon German physicists to investigate. Physicists, by that time, had already learnt that visible light is a part of electro-magnetic radiations. The electro-magnetic radiations having a larger wavelength than the color red are called infra-red radiations, while electro-magnetic radiations having a smaller wavelength than violet color of visible range are called ultra-violet radiation. They reasoned that if they study the behavior of electro-magnetic radiation emitted by various elements when heated to different temperatures, they shall be able to evaluate the behavior of light as well.

To study the behavior of electro-magnetic radiations at different temperatures, they designed the black-body such that they could get a sample of all the radiations emitted by the material of the body at a particular temperature. Then, experiments were carried out with black-bodies of different materials, shapes, and at different temperatures. Data was recorded and graphical curves drawn as shown in figure 2.

The result of the experiments was surprising. They found that the energy given by the radiation neither depended on the shape of the black-body, nor the material. It depended only on the temperature of the black-body. Raising the temperature of the black-body gave out more energy of the radiation, and hence, it could be easily concluded that a higher temperature of the filament of the bulb was required in order to get brighter light. Material of the filament or the shape of the bulb was inconsequential. Therefore, a material with a high melting point, Tungsten, was chosen for the incandescent bulbs. Use of this material and evacuation of air from the bulb increased the life of the bulb.

Another important aspect of the experiments was the graphical curves themselves. Every graphical curve has a mathematical equation. Therefore, the physicists tried to give a mathematical equation to these curves also, but it was found to be a near impossible task as no mathematical equation was fitting these curves.

From figure 2, it can be observed that as we go from a starting point on the left toward the right side along the horizontal axis representing the wavelength, the energy emitted by the radiation increases sharply, becomes maximum, and then continuously decreases for a major part of the curve. Since the wavelength of the electro-magnetic radiation is inversely proportional to its frequency as per the formula λ=c/ν (where c is the speed of light), we can say that energy emitted by the radiation decreases as the frequency decreases for a major part of the curve, or, in other words, energy emitted by the radiation is proportional to the frequency of the radiation. In 1893, Wilhelm Wien used this relation to devise an empirical formula1 for the Energy emitted by radiation as:

B= 2hν^3e-hν/kT^/c3

where B is the amount of energy emitted at a frequency ν , T is the absolute temperature of the black-body, h is Planck constant, c is speed of light, and k is Boltzmann’s constant.

This formula provided accurate values of energy for short wavelengths, but it did not provide correct values at larger wavelengths of the radiation emitted, as can be seen by green curve in Figure 3. On the left hand side, it matched the black curve entirely, but on right hand side, it deviated from the black curve obtained from the experimental data.

In 1900, Lord Rayleigh derived a formula for the energy of radiation emitted from a black-body using classical electro-magnetic theory, and presented his formula2 as:


But this formula failed at short wavelengths of radiation, causing a condition known as ‘ultraviolet catastrophe’, where the energy contained in the radiation was divergent and reached infinity value for the radiation in the ultraviolet region, as shown in Figure-2 by a red line. The values obtained from the Rayleigh-Jean formula did not match with the black curve on the left side, whereas they matched exactly with the black curve on the right side which represented experimental data values.


[* Figure 3- Deviation of Wein’s and Rayleigh-Jean’s Curves from Experimentally Observed Values *].

Though Lord Rayleigh’s formula was not accurate, due to derivation of his formula from classical electro-magnetic theory, unlike Wein’s empirical formula, he provided a way forward for other physicists to improve on.

Lord Rayleigh used the Equi-partition theorem, which is the study of the average values of physical properties for very large groups of individual particles involved in random motion. The theorem3 states that the total energy contained in the assembly of a large number of individual particles exchanging energy among themselves through mutual collisions is shared equally (on the average) by all the particles. If the total energy available in a sample of a gas contained in a container is E, and number of molecules of the gas sharing that energy through mutual collision is N, then the average energy per molecule is E/N. Though the molecules may be differing in energy, some could be moving faster than others, yet the average energy will be E/N.

Secondly, Lord Rayleigh and Jeans observed that there is a similarity in the behavior of molecules of a gas and wavelengths of the electro-magnetic radiations which are emitted by heating any material. Therefore, they applied equi-partition theorem to the radiations generated in a black-body and derived the formula given in equation 2.

But this formula led to ‘Ultra-violet catastrophe’, as mentioned previously. The cause of this catastrophe was that, even if there were some similarities between molecules of a gas and waves of electromagnetic radiations, there was one major difference between the two. The number of gas molecules in a given container is always finite, whereas the number of possible waves generated by the black-body are infinite.

Max Planck solved this problem by assuming that the number of possible electromagnetic vibrations is not infinite, but there are a certain number of resonators which produce electromagnetic vibrations, and that these resonators have energy in certain multiples of a minimum amount of energy known as quantum of energy or quanta. This assumption was derived from Boltzmann’s concept that all matter is formed of indivisible atoms. Until then, Max Plank was a strong supporter of the theory of continuous matter where the matter could be divided into an infinite number of parts. He had accepted the idea of Boltzmann’s divisibility of matter and applied to energy as an ‘Act of desperation” to provide theoretical reasoning for his formula. He had used this concept as a temporary measure and expected to review this later and devise a different explanation, but could not do it. Later, this concept led to the development of Quantum Physics.

Before the assumption of the above reasoning by Max Planck, he had already derived and publicized a new formula4 conjoining the formulae of Wein and Lord Rayleigh as:


This formula gave correct values, but it lacked a theoretical explanation. Then, Max Planck started his research to find a theoretical explanation because he thought he was obliged to give the explanation too, after giving the correct formula, and then he came up with the above mentioned reasoning. We shall now learn in detail in the next chapter how he derived the formula from this concept of quantization of energy and probability.





Chapter II

Derivation of the Formula by Max Planck

Max Planck imagined a black-body having perfectly reflecting mirrors on inner walls which could reflect the radiations falling on them without absorbing them. And, in place of molecules of gas, he imagined some kind of resonators which created electro-magnetic radiations (or waves). Next, he wished to calculate two properties:-

1). Energy of each resonator at particular temperature of the black-body and

2). Number of modes of the waves generated by resonators, which would give the number of resonators.

Then, it would be simple multiplication of energy of each resonator and number of modes of the waves generated by the resonators to get the total energy emitted by the black-body radiation as per equi-partition theorem.

The idea looked simple, but it was not because the number of resonators was very high and finding the energy of each resonator was also tricky. We shall further discuss how Max Planck achieved this task.

As already mentioned above, Planck had considered a cubical black-body with a small hole at one end from where all the radiation came out when the body was heated. The hot black-body was allowed to remain at that temperature for a fairly long time to reach equilibrium condition, so that any discrepancy in measurement of radiation coming out from the hole gets minimized. That allowed the formation of standing waves inside the cavity. First, we must understand what standing waves are.

When a string is attached at both ends and made to vibrate, it creates standing waves. The wavelength of the wave is either double the distance between both end, or an integral division of that distance as shown in figure 4.

Fig-4 Different Modes of Standing Waves


Here, four different types of waves are shown, these are called modes of the standing waves. Similarly, there can be more modes. For every mode, there is a different wavelength and hence a different frequency.

Planck imagined that the cavity inside the black-body contained a large number of resonators similar to the molecules of the gas, and emitted electromagnetic radiations of different modes depending upon the temperature of the black-body.

As the cavity is heated, the process of absorption of heat and emission of radiation continues until a point when heat supplied becomes equal to heat discarded through radiation and the temperature of the cavity becomes constant. This state is known as equilibrium state.

For every temperature of the cavity, there is one equilibrium state, and in this state, waves formed by the emitted radiation inside the cavity are standing waves, because standing waves have zero amplitude at the cavity walls; if the amplitude of radiation at the cavity walls is non-zero, then the radiation will dissipate energy and our condition of equilibrium is violated, so in that state only standing waves exist.

These waves are of different modes, and the number of modes depend upon the square of frequency of emitted radiation. The Formula for calculating the number of modes at a particular frequency is 8πν²/c³, where ν is the frequency and c is the speed of light.”


Derivation of the Formula for Number of Modes

Let us take a black-body, cubical in shape, and having each side of length L. For standing waves to exist in this body, the length of the body must be a multiple of half of the wavelength, only then can we have zero amplitudes at the walls, as nodes of the wave appear at every half of wavelength.

In the x-direction, the wavelengths of the waves which can be fitted into the box are those for which L= l x/2, where l is any integer number, and x any wavelength.

Similarly, if the wavelengths in y and z direction of the black-body are y and z, then the number of modes in y and z direction will be m and n respectively given by relations L= my/2, and L= nz/2

Therefore, the maximum possible modes in the cube will be l x m x n = 2L x 2L x 2L/ xyz

= 8L3/ xyz

This is the number of modes if the modes are counted individually in three directions and multiplied, but radiation waves are inter-related in all three directions, so only those waves will be permitted which can fit in the box shaped black-body. For that, we need to take into account the sinusoidal nature of the waves and the interrelation between three directions, and then modify the formula further. Without going into finer details of the effect of the sinusoidal nature of waves in calculating the number of modes, we write here the expression

N=8πL3/3λ^3^ (4)

where N= number of modes of the waves in the box, L is the side of the cube, λ is the wavelength.

By dividing this number by the volume of the cavity, we can obtain the density of the modes:

N/V = ρ~N~ = 8π/3λ^3^ (5)

Now, we differentiate this with respect to the wavelength to get the density of modes per unit wavelength,

dρ~N~ /dλ = -8π/λ 4

Converting this equation to the number of modes per unit frequency by using λ=c/ν, we get:

dρ~N~/dν = (dρ~N~/dλ) x (dλ/dν) = (-8πν^4^/c4) x (-c/ν^2^) = 8πν^2^/c3 (6)


Calculating the Energy of an Individual Resonator

After counting the number of modes per unit frequency, it was required to calculate the energy of individual resonators. To find the average energy of one resonator, Planck used Boltzmann’s method which uses probability technique.


Boltzmann’s Method of Finding the Energy of Individual Resonators:

According to Boltzmann’s method, if there are M resonators and P parts of Energy, then the number of ways P parts can be assigned to M resonators will give the entropy of one resonator. Formula for this is S= k logn[(M+P-1)!/ {(M-1)], where ‘S’ is the entropy of one resonator.

To explain this relation further, let’s use the following example. Imagine there are five chairs in a room and ten books. Then what is the average number of books that can be kept on the chairs? The answer is “Two.”

And the number of ways the books can be kept in these chairs will require a formula based on probability technique:

(n+m-1) x (m-1)!},

Using this formula, we can find the number of ways 10 books can be kept on 5 chairs as follows:

(10 + 5 -1)!/{10! X (5-1)!}

= 14!/ (10! X 4!)

= (14 × 13 × 12 × 11 × 10!)/ (10! x 4 × 3 × 2 × 1 )

= (14 × 13 × 12 × 11)/ 24

= 1001

This number does not look like it has any relation to the average number of the books that can be kept on chairs.

If we increase the number of books to 15 and keep the number of chairs the same, the average number of books on each chair becomes 3, and number of ways 15 books can be kept on 5 chairs is (15+5-1)!/ {15! X (5-1)!} = 3876

Similarly, if we increase the number of books to 20, we get the average number of books that can be placed on each chair as 4, whereas the number of ways the books can be kept on chairs increases to (20+5-1) X (5-1)!} = 10626

From this example, we can see that with an increase in the average number of books that can be kept on the chairs from 2, to 3, to 4, the number of ways increases from 1001, to 3876, to 10626. Though there does not appear any relation in these number Boltzmann found a relation in them using probability technique. According to him, a natural log of the number of ways something can be arranged is proportional to the average number. If the number of ways is W, then the average number is proportional to logW.

Boltzmann reasoned that if the number of ways the molecules of a gas can interact with each other increases, the order reduces. The measure of in-orderliness is known as entropy of a substance, therefore, the entropy increases. Using this relation of the number of ways the molecules of a gas can interact with each other and entropy, Boltzmann gave the formula for entropy of individual molecule of a gas as:

S= k logW,

where S is entropy, W is the number of ways molecules can interact with each other, and k is Boltzmann’s constant.

When we multiply entropy with the temperature of the substance, we get its internal energy. So, from the number of ways something can be arranged, we can get its average internal energy using Boltzmann’s statistical formula.


Energy of Individual Resonators:

In equilibrium state, at a particular temperature, when energy supplied is equal to the energy released, a balance is maintained between the set of resonators in the cavity and the radiation emitted. At this stage, Planck surmised that the average energy of one resonator can be calculated from the total energy by dividing with the number of resonators. He assumed M identical resonators inside the hypothetical cavity of a black-body, and assumed a total energy E distributed among them such that:

E=MU (7)

in which U represents the average energy of one resonator.

Similarly, the entropy of the cavity could be related as:

SM=MS (8)

in which SM represents the total entropy of the cavity and S represents the average entropy of one resonator.

Making a paradigm shift from the concept of continuous energy followed until then, Max Planck assumed that the energy exists in small indivisible parts, called ‘quantum’ akin to ‘atoms’ of matter. If the energy of a quantum is assumed to be e, and there are P number of quanta, then the total energy can be expressed as:E=Pe (9)

where P represents a large integer, while the value of e is yet uncertain. Planck laid the foundation of quantum physics with this step, which he called “An act of desperation”, because he was, until then, a strong supporter of classical physics advocating continuous energy. He had to take this step because he had no other choice. He felt obliged to provide an explanation for his formula.

Subsequently, Planck calculated the number of ways in which P energy elements, or quanta, could be distributed to M resonators. He used Boltzmann’s technique of statistics and expressed this number as W such that:

Because M and P were very large numbers, he ignored the minus ones in that formula and used Stirling’s approximation:

Thus the above Equation became:

He substituted this value in Boltzmann’s equation S= klogW

For M number of resonators, the entropy became:

SM = klog{(M+P)^(M+P)^/(MMPP)}

Which could be expanded to:

SM = k[(M+P)log(M+P) – MlogM – PlogP] (10)

Substituting the value P=MU/e (from equations 7 and 9) in above equation 10, he got:

SM = k[(M+MU/e)log(M+MU/e) – MlogM – (MU/e)log(MU/e)]

Taking M as common from the right hand side of the above equation, he proceeded further:

SM= kM[(1+U/e)logM(1+U/e) – logM – (U/e)log(MU/e)]

Expanding the above equation further, he got:

SM = kM[(1+U/e)logM + (1+U/e)log(1+U/e) – logM – (U/e)logM – (U/e)log(U/e)]

Cancelling out the equal positive and negative terms, he remained with the equation:

SM = kM[(1+U/e)log(1+U/e) – (U/e)log(U/e)] (11)

After dividing equation 11with M, he got:

SM/M= S = k[(1+U/e)log(1+U/e) – (U/e)log(U/e)] (12)

This equation gave him the entropy of individual resonator.


Relation between Entropy and Frequency

Planck devised another formula between entropy and frequency using Wein’s displacement law. We shall not go into details of that, which is:

S = f(U/ν), (13)

where f is a function, S is entropy of individual resonator, U is the energy of individual resonator, and ν is frequency.

From Equation 12 and 13, we can deduce that e is proportional to ν, and hence substituting e for hν in equation 12, where h is a proportionality constant, we get the equation:

S=k[(1+U/hν)log(1+U/hν) – (U/hν)log(U/hν)] (14)

Using the d(uv)= udv+vdu rule of differentiation, this equation, when differentiated with respect to U, becomes:

dS = k[(1+U/hν)d{log(1+U/hν)} + {d(1+U/hν)}log(1+U/hν) – (U/hν)d{log(U/hν)} – {d(U/hν)}log(U/hν)]dU

After using the rule d(logx)= 1/x, and rearranging this equation, we get:

dS/dU = k[((1/hν){log(1+U/hν) – log(U/hν)}]

Which can be further rearranged as:

dS/dU = (k/hν)log(1+hν/U) (15)


Relation between Entropy, Internal Energy and Absolute Temperature

The formula showing the relation between entropy, internal energy, and absolute temperature is:

dS/dU = 1/T

Therefore, we can write the equation (15) as:

1/T = (k/hν)log(1+hν/U)

Which can be rewritten as:

e^hν/kT^ = 1+(hν/U) (16)

From equation (16), we can write:

U = hν/(e^hν/kT^ – 1) (17)

Which gives us the internal energy of an individual resonator.


The Energy of Radiation Emitted by a Black-body

Since the average energy of individual resonator is equal to the average energy per mode, the multiplication of the average energy of individual resonators, and density of the number of modes per unit frequency gives us the total energy of the radiation at that particular frequency.

(8πν^2^/c3) x (hν/(e^hν/kT^ – 1) = 8πhν^3^/c3(e^hν/kT^ -1) (18)

This formula derived by Planck using imaginary resonators and the number of possible ways these resonators could be assigned the modes of standing waves formed by the electromagnetic radiation, fit perfectly to the curve of energy emitted and frequency. But the use of probability in finding the average energy of an individual resonator was undesirable considering that physics is an exact science, not an occult science where probability can be used. Moreover, the resonators themselves were imaginary.

In the article that I have published, I have proved that there is no compulsion of imaginary resonators, number of modes of standing waves, and probability technique to derive this formula. In the next chapter we learn how we can do that.





Chapter III

Energy Distribution of Radiation – Independent of Probability

As discussed in previous chapters, for more than a century, energy distribution of radiation emitted by a black-body5 has always been derived by the traditional method of assumption of oscillators and standing waves inside the black-body. It is very difficult to make any alteration to this well-established method to make it independent of probability, therefore we follow an alternative approach.

The main aim of the experiments on a black-body was to find a way to increase the efficiency of incandescent bulbs. Therefore, instead of going through the black-body route, if we directly investigate the incandescent bulb itself, we should be able to derive the formula. Light emitted from a luminous bulb, is equivalent to black-body radiation, therefore, it should be able to give us the same formula without the assumption of oscillators and standing waves. This method has not been tried previously, so it is new to physics and requires a fresh outlook towards the subject.

Trading on the direct path, we consider an incandescent bulb as represented in Figure-5. When the bulb filament is heated by supplying electrical energy input, the bulb illuminates, and it has brilliance at a radial distance from the filament, at absolute temperature of the filament. As is evident from the common use of incandescent bulbs in daily life, the brilliance of an incandescent bulb is equal in all directions. Therefore, we shall represent the brilliance with a sphere around the bulb as shown around the first bulb in Figure-5, and we shall call it an energy-sphere.

[* Fig. 5- Brilliance of a Bulb at Different Energy Inputs *]

After that, when the bulb is supplied with additional electrical energy, the brilliance spreads up to a radial distance, as shown by the second bulb in Figure-5, where a second energy-sphere superimposes on the first energy-sphere. And with additional electrical energy supplied, values of brilliance at distances, and increase to, and respectively, as shown by the three bulbs in figure-5.

Following the same pattern, we can envisage that new energy-spheres get added over the previous energy-spheres, thus, the brilliance becomes maximum near the bulb filament, and decreases with radial distance from the bulb, in accord with the Inverse Square Law of Light6 which states that the brilliance of the bulb is inversely proportional to the square of the distance from the bulb.


Addition of Energy

For the bulb with brilliance spread up to a distance , the total energy supplied is, the absolute temperature of the filament is, the volume of the energy-sphere is, and highest frequency (the frequency in the radiation curve in Figure 3 at a point on the curve where it just touches the horizontal axis at the left most part of the curve. It is different from the frequency represented by the peak of the curve) of radiation emitted is ν~1~, as shown by the sphere on the left hand in Figure – 6.

[* Fig. 6- Conversion of Energy-sphere-1 to Energy-sphere-2 Due to the Addition of Electrical Energy E *]

Then, heat energy is provided electrically to increase the total energy of the bulb to; the absolute temperature of the filament will rise to, the volume of the energy-sphere will become , and the bulb will emit radiation of the highest frequency ν~2~, as shown by the sphere on the right-hand side in Figure-6.


Discarding of Additional Energy

When the bulb filament is in thermal equilibrium, the electrical energy supplied should be equal to the energy dissipated through heat and light radiation. Energy E added to Energy-sphere should, thus, get discarded completely through radiation after dissociating itself from as shown in Figure-7. The total energy of the bulb should then fall back to as it was before the addition of energy .

[* Fig. 7- Division of Energy-sphere-2 into Energy-sphere-E and Energy-sphere-1 *]


A few Points about Energy-spheres

1. It may be noted that the highest frequency of the radiation emitted by Energy-sphere-2 and Energy-sphere-E should be equal because the supply of energy enabled Energy-sphere-2 to emit radiation of frequency ν~2~.

2. Secondly, according to Rayleigh-Jean’s law, the total energy of radiation over a complete range of emission is 8πν3kT/c3, therefore, the energy of an energy-sphere should be proportional to the cube of the frequency of the radiation emitted.

3. The volume of the energy sphere increases from to, in the same proportion as the energy increase, therefore, the volume of the energy-sphere should also be proportional to the cube of the frequency.

4. The third point shows a similarity between energy contained and volume occupied by an energy-sphere, but there is a significant difference between these two values. The value of energy in the radiation curve in figure 3 starts reducing after reaching a peak value as the frequency increases, but the volume occupied by the energy sphere should keep increasing proportional to the cube of frequency. The reason being that the energy emitted by the black-body is dependent on the cube of frequency, and amount of radiation emitted. But the volume occupied by the energy sphere depends only on the cube of frequency.


Disintegration of Energy-sphere-E

For the reduction of Energy-sphere-2 back to Energy-sphere-1, the Energy-sphere-E should vanish completely, which is envisaged to happen by breaking of Energy-sphere-E into circumferential parts of energy e1, e2, e3 etc. as shown in Figure- 8, and escape in the form of radiation.

[* Fig. 8- Circumferential Partition of Energy-sphere-E *]


It is well known that mass and energy are equivalent as per Einstein’s famous equation7, therefore, the technique of mass disintegration of radioactive substances can also be applied here in calculating the value of energy of the disintegrated parts of Energy-sphere-E~.~

A radioactive substance disintegrates in such a way that it reduces to half of its original mass in a certain time period, which is termed as its half-life. If the reduction factor has any other value instead of half, then the number of steps for reduction change accordingly for arriving at same final value.

Let us assume the reduction factor for the total disintegration of Energy-sphere-E as α (instead of considering 2, as in half-life), and the number of steps as n. Then the ratio of energy values of Energy-sphere-2 and Energy-sphere-1, for complete disintegration of Energy-sphere-E into small packets of energy e1, e2, e3 etc. can be written as:

E2/E1 = α^n^, (19),

where α is an arbitrary number.

During disintegration, the first packet of energy released will be e1, which shall be accompanied by radiation of the highest frequency ν~2~. After releasing the packet of energy e1, the size of Energy-sphere-E will decrease, therefore, the second part e2, when released, will emit radiation of the highest frequency less than ν~2~ by an integral number.

Packets of energies e1, e2, e3 etc. may not necessarily result in a reduction of frequency one by one, they may do so by any other integral number ε (arbitrary number). So, the radiation of the highest frequency (ν~2~ – ε) should be emitted with the release of the second part of energy e2, and radiation of the highest frequency (ν~2~ – 2ε)) with the third part e3 and so on.

Following the above pattern, the highest frequency of radiation emitted with the release of energy eN in Nth step can be represented by the formula

{ν~2~ – (N-1)ε} (20)


The Number of Steps of Disintegration:

The highest frequency of the emitted radiation decreases by a fixed number ‘ε’ every time a packet of energy gets dissociated from Energy-sphere-E until it vanishes completely, therefore, the number of steps in the fragmentation of Energy-sphere-E can be written as n=ν~2~/ε.

A bulb filament will radiate more energy at higher temperature than at lower temperature. It implies that the size of the steps in the reduction of the energy is larger at high temperature than at low temperature. In other words, we can say ε is proportional to absolute temperature. Therefore, ε = μT2 where μ is another number related to arbitrary number ε, and T2 is absolute temperature.

From above two paragraphs, we can deduce that n=ν~2~/μT2 .


Energy Ratio:

Substitute the value of n in equation (19) to get:

E2/E1 = α^ν^2^/μT^2 (21)

After the release of packets of energy e1, e2, e3, e4 etc. from the Energy-sphere-E, the ratio of energies can be written as:

(E2 – e1) = α^(ν^2^/μT^2 – 1) ,

{E2 – (e1 + e2)} = α^(ν^2^/μT^2 – 2),

{E2 – (e1 + e2 +e3)} = α^(ν^2^/μT^2 – 3), and so on.

Thus, after the release of N packets of energy, the ratio will become:

E2 – Ʃei = α^(ν^2^/μT^2 – N) , where Ʃei denotes the summation of e1, e2, e3, e4 etc. for all values of i from 1 to N.

Substitute E2 = E1 + E in the above equation to get:

(E1 + E – Ʃei)/E1 = α^( ν^2^/μT^2 – N) (22)

After rearranging the terms, the equation changes to:

(E – Ʃei)/E1 = α^(ν^2^/μT^2 – N) – 1 (23)

The exponential of α in equation (23) can be rewritten as:

(ν~2~ – NμT2)/μT2

Since ε = μT2, we can rewrite the term as (ν~2~ – Nε)/μT2

After N times reduction of frequency in steps of ε from maximum frequency ν~2~ , we get the current frequency ν = (ν~2~ – Nε). Therefore, the exponential term will become equal to ν/μT2.

Substituting this value of exponential of α in equation (23), we get:

(E – Ʃei)/E1 = α^(ν/μT2)^ – 1 (24)

For all values of E1 ≠ 0 , we can inverse the equation (24) completely,

E1/(E – Ʃei) = 1/{α^(ν/μT2)^ – 1} (25)

This is the energy quotient of Energy-sphere-1 and the residual energy in Energy-sphere-E after N times the disintegration and dissipation of energy.


Energy Density:

We have learned so far that after N times the disintegration of Energy-sphere-E, the energy packet released will be able to emit highest frequency ν, and the volume of energy sphere is proportional to cube of maximum frequency of radiation released, therefore, the volume of the Energy-sphere before the release of the packet of energy eN should be V= γν^3^, where γ is a proportionality constant, and its energy should be E-Ʃei, where, i=N.

Therefore, the ratio of the residual energy and residual volume of Energy-sphere-E after N times disintegration can be written as:

ρ~e~ = (E-Ʃei)/γν^3^, (26)

where, ρ~e~ denotes energy density.

This energy density can be envisaged to behave akin to mass density, as mass and energy are equivalent. It is easier to break a part of a substance having less density than the one having more density, as the constituents of matter have less attraction towards each other than in the latter case. Therefore, if the energy density of Energy-sphere-E increases with its disintegration, it will become more difficult for Energy-sphere-E to disintegrate further, and as such, less radiation will escape from the bulb. Thus, the escape of energy that causes radiation, or the brilliance, should be inversely proportional to the energy density, i.e. B∞1/ρ~e~ .

It may also be noted that disintegrated parts of energy from Energy-sphere-E will scatter away from the bulb as emitted radiation due to repulsion from the energy of Energy-sphere-1 present at the bulb. Therefore, the radiated energy should be proportional to the energy of that sphere, i.e. B∞E1


Brilliance of the Bulb:

From the combination of the above two proportionalities, we have the relation B∞E1/ρ~e~ .

After introducing the proportionality constant β , the equation becomes:

B=βE1/ρ~e~ (27)

Substitute the value of ρ~e~ from equation (26) in equation (27) to get:

B = βE1/{(E-Ʃei)/γν^3^} (28)

After rearranging, it becomes:

B= βγν^3[^E~]1~/(E-Ʃe~i)~ (29)

Substituting the value of E1/(E – Ʃei) from equation (25) in equation (29) and changing the proportionality constant βγ to K, we get:

B= Kν^3^/{α^(ν/μT2)^ – 1} (30)

Since α is an arbitrary number (as mentioned in equation 19), we can replace it with e^η^, where η is another arbitrary number. Thus,

B= Kν^3^/{e^(ην/μT2)^ – 1} (31)

Comparing equation (31) with the formula in equation (18), we find that both equations become identical at T= T2, if we replace K with 8πh/c³, and η/μ with h/k.

Replacement of K with 8πh/c³ is straightforward because K is a proportionality constant and can have a fixed value designated by 8πh/c³.

Replacement of η/μ by h/k is also possible because η/μ is the division of two arbitrary numbers which can have a fixed value h/k.

Thus the formula in equation (18) can be derived without the assumption of the oscillators and standing-waves inside the black-body, through an entirely distinctive concept of formation, disintegration and dissipation of energy-spheres, when energy is supplied to a bulb filament.






For derivation of the formula for energy distribution of radiation emitted by a black-body, we have pursued a new technique in which additional energy supplied to the luminous bulb to maintain the temperature results in emission of heat and light, which expands the existing energy-sphere present around the bulb, and from that, an energy-sphere having energy equal to supplied energy gets released and promptly breaks up circumferentially into parts of energy. These energy fragments are simultaneously pushed away as radiation by the existing energy-sphere around the bulb. This gives us a technique of the disintegration of the energy similar to radioactive disintegration of matter, which has been mathematically constructed step by step in order to present a derivation for the formula for the distribution of energy of a black-body- independent of probability.


Future Possibilities

Because the size of the bulb can be imagined as small as possible, the technique can be applied to the smallest level of source of radiation, without requirement of a large data of oscillators and standing waves. This implies that the technique can be applied to the formation of radiation by sub-atomic particles. Presently, the behavior of electrons in an atom is described using probability. Further research and development of this technique of the generation of radiation, if applied to electrons, may help in better understanding of their behavior instead of continuing to be dependent on probability. This technique may be a first step in the direction of removal of probability from physics and in tandem with the saying of the great scientist Albert Einstein, “I, at any rate, am convinced that He does not play dice^8^.”






1. Marr J M, Wilkin F P. A better presentation of Plank’s radiation law. Am. J. Ph. 2012 April; 80(5): p.2; doi:10.1119/1.3696974

2. Kramm G, Herbet F. Heuristic Derivation of Blackbody Radiation Laws using Principles of Dimensional Analysis. [Internet]. 2008 Mar 22, p.6; available from: arXiv:0801.2197v2

3. Gamow, George. Thirty Years that Shook Physics: The Story of Quantum Theory (Kindle Locations 210-212). Dover Publications. Kindle Edition

4. Kramm G, Herbert F. Heuristic Derivation of Blackbody Radiation Laws using Principles of Dimensional Analysis, [Internet]. 2008 Mar 22, p.11; available from: arXiv:0801.2197v2

5. Kumar M. Quantum.UK: Icon Books; 2008, Chapter1, The Reluctant Revolutionary; p.3-30

6. The Inverse Square Law of Light. [Internet]. NASA. available from: https://www.nasa.gov/pdf/583137main_Inverse_Square_Law_of_Light.pdf

7. Agarwal N S. New Quantum Theory. Indian Journal of Science & Technology. 2012 November; 5(11): p.2; doi:10.17485/ijst/2012/v5i11/30649

8. Natarajan V. What Einstein meant when he said God does not play dice. Resonance- Journal of Science Education. 2008 July; 13(7): p 655-651





Other books by the Author

First book by the author is named Lines of Space-Source of fundamental forces and constituent of all matter in the Universe. The book is about lines of space which are assumed to exist in the empty space and causing the forces gravitational, electrostatic, strong nuclear force and weak nuclear force. Contraction of the lines gives matter. The concept is explained with a fictional story of father educating his children. The concept is well supported by the mathematical equations and data collected from using these equations on lines of space. It is available at http://amzn.to/15Yg5JY

Second book is named 'Big bang and Lines of Space- A simple dialogue on two different concepts". This book explains the creation of the universe from Big-bang and the qualms related to the acceptance of Big-bang theory in relation to the concept of Lines of Space as explained in first book. This book is also written in a conversational style for the ease of understanding. It is available at http://www.amazon.com/dp/B00NYJ9Y56




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Black-Body Radiation: An Explanation without Probability

In the year 1900, the famous professor of physics in the University of Berlin, Max Planck, derived a formula for energy distribution of black-body radiation. He gave the explanation of his formula using quantization of energy, which was a great shift from the classical physics and led to the birth of Quantum physics.In this formula, he also made use of probability,which led to strange results and a requirement was always felt to prove the formula without probability. The author had derived this formula of physics without probability and published the results in the Indian Journal of Science and Technology. Since the article published in the journal is difficult to assimilate for the layman, the author has written this book explaining the derivation of the formula by traditional method as well as without the use of probability.

  • ISBN: 9781370913374
  • Author: Devinder Dhiman
  • Published: 2017-05-22 21:20:13
  • Words: 7161
Black-Body Radiation: An Explanation without Probability Black-Body Radiation: An Explanation without Probability