Abstract

In the author’s electromagnetic field theory the theory about the mutual energy flow, there is a 90 degree phase difference between the electric field and the magnetic field of the electromagnetic wave. The author calls this electromagnetic wave reactive power wave. The author thinks that the far field of electromagnetic wave emitted by transmitting antenna is reactive power wave. This is different from Maxwell’s electromagnetic theory. If the radiation of the transmitting antenna is reactive power, the reader will ask, how does the antenna radiate electromagnetic waves? In fact, in the author’s electromagnetic field theory, the receiving antenna and the absorber of the environment also generate radiation. This radiation is a advanced wave. Therefore, electromagnetic wave has two kinds of current elements. One is the radiation source, including the transmitting antenna, the primary coil of the transformer, and the radiator charge. One is sink, including receiving antenna, secondary coil of transformer and absorber charge. The retarded wave generated by the source and the advanced wave generated by the sink together constitute the mutual energy flow. The mutual energy flow has all the properties of photons and can be regarded as photons themselves. It is mutual energy flow photons that constitute electromagnetic radiation. The author’s electromagnetic theory and Maxwell’s electromagnetic theory have different starting points and different calculation results. But the author tries to consider a transmitting antenna based on the author’s electromagnetic theory, and consider that the boundary of the universe or the sphere with infinite radius is filled with materials that absorb electromagnetic waves. These absorber materials can radiate advanced waves and thereby absorb electromagnetic waves. Therefore, there are two kinds of current elements in this system, one is the source at the coordinate origin, and the other is the current element of the absorber charge distributed on the spherical surface with infinite radius. With two current elements, the electromagnetic field can be calculated by the method of mutual energy theory proposed by the author, and the energy flow from the source to the absorber on the sphere can be calculated accordingly. In this paper, it is proved that the radiant energy obtained by the author’s method is consistent with the Poynting vector energy flow obtained by Maxwell’s method. The author believes that Maxwell’s electromagnetic theory must take into account the far field boundary condition, namely the Sliver Mueller condition. In fact, this condition has implied that the boundary of the universe is full of absorber charges. In principle, the author considers that there are N charges in an empty universe. In the author’s electromagnetic theory, the interaction can only come from the charge inside the sphere. If we consider the absorber charge distributed on the large spherical surface with infinite radius in the author’s electromagnetic theory, we should get the same or almost the same conclusion as Maxwell’s electromagnetic theory. This paper tries to prove this point.

1. Introduction

The author established the theory of electromagnetic mutual energy ^{7, 8, 9, 10}. This theory is affected by two aspects. One is the theory of mutual energy theorem in the author’s early work ^{6, 15, 16}. There are three reciprocity theorems close to the mutual energy theorem, Welch’s reciprocity theorem in 1960 ¹⁴ Rumsey’s reciprocity theorem in 1963 ¹³. De Hoop’s related reciprocity theorem in the end of 1987 ⁵. The other is the theory about the existence of advanced waves, including the absorber theory of Wheeler and Feynman ^{1, 2}, and Cramer’s transactional interpretation of quantum mechanics ^{3, 4}. The axioms of mutual energy theory of electromagnetic field consists of the law of conservation of energy, the theorem of mutual energy flow, and the principle of self energy. The electromagnetic field can be solved and the mutual energy flow can be calculated. In the author’s electromagnetic field theory, the self energy flow of the transmitting antenna is described by Poynting’s vector, which does not transfer electromagnetic energy. The author calls this energy flow self energy flow. The self energy flow is reactive power. Therefore, the electromagnetic wave corresponding to the self energy flow is the electromagnetic wave of reactive power. Although this electromagnetic wave carries energy, it does not transfer energy, because the energy of these electromagnetic waves reversely collapses back to the source. Energy is transferred only by mutual energy flow. The mutual energy flow is composed of the retarded wave from the source and the advanced wave from the sink. Mutual energy flow can transfer active power. For the author’s theory on mutual energy flow and photons, refer to ^{11, 12, 17, 18}.

According to the author’s electromagnetic theory, if there is only one transmitting antenna, even if the antenna is connected to a power supply of a certain frequency, it will not radiate electromagnetic energy. Although this kind of electromagnetic wave carries energy to any point in space, the energy immediately reverses and collapses back to the transmitting antenna. Therefore, the antenna does not lose electromagnetic energy. How does the antenna radiate electromagnetic wave energy. The situation is different if the receiving antenna is used. The author thinks that the transmitting antenna transmits the retarded wave, the receiving antenna transmits the advanced wave, and the retarded wave and the advnaced wave can form a mutual energy flow. The mutual energy flow can be active power, and the energy is transferred from the transmitting antenna to the receiving antenna through the mutual energy flow. The retarded wave and the advanced wave themselves are reactive waves and do not transfer energy.

In this paper, we consider a case where the boundary of the universe or the sphere with infinite radius is filled with absorber charges, which are equivalent to receiving antennas. In this case, the mutual energy flow can be calculated. In this case, the calculated mutual energy flow should be consistent with the Poynting energy flow calculated for the transmitting antenna according to Maxwell’s theory. This paper proves this point. This also answers the question how the macroscopic electromagnetic wave is composed of photons.

The second chapter reviews Maxwell’s classical electromagnetic theory. The third chapter reviews the author’s electromagnetic theory. In the fourth chapter, it is proved that if the spherical surface with infinite radius is filled with absorber materials, the mutual energy flow can be formed by the source at the origin of the coordinate and the absorber on the spherical surface, namely the sink. The author wants to prove that this mutual energy flow is consistent with the radiation of the self energy flow calculated by Maxwell’s electromagnetic theory.

2. Review of Classical Electromagnetic Theory

Maxwell’s equation can be written as,

where means “is defined as”.

Poynting theorem can be deduced from Maxwell equation

Hence, there is,

If there is Poynting’s theorem of complex numbers in the frequency domain, there is

In the above AC current is considerred. Hence there is a factor in the fields and current intensity.

In fact, the contradiction of Poynting’s theorem is obvious. As we know, in the frequency domain

(1)

Consider a current element with a length of

We only consider the radiation field of this current element. Although the isolated current element does not exist, the current element always exists with the charges at both ends of the current element, forming a dipole antenna, we can still consider an ideal current element, which has zero charges at its two ends. Therefore, the electrostatic field of this current element is zero,

magnetic vector potential, which must be considered as a retarded potential,

Consider that the transmitting antenna is a dipole antenna, so is very small, which means that so there is where So there is so there is,

“~” simbole means proportional, when it is used only the phase and the direction of the vector is considerred.

The above equation is purely imaginary, so we have

“” means “taken the real part”. If we use the above formula to calculate the ideal dipole antenna, it will not radiate. But if we use (1) to calculate, the dipole antenna is radiating. Hence, we always have,

The above formula has actually revealed the problem of classical electromagnetic field theory. The Poynting theorem fails. However, due to the existence of the electric field in the vector potential, the electric field is divergent at the position of The exact value of the value on the left side of the above equation cannot be calculated anyway, so it is not very important what value it is. People can imagine that if the value is not divergent, it may not be a pure imaginary number. The left side of the above formula is divergent anyway, so people don’t care whether its real part is zero or not.

3. Review of the Author’s Mutual Energy Theory for Electromagnetic Fields

In Maxwell’s electromagnetic field theory, an antenna can constitute radiation. The radiation of the antenna has nothing to do with the absorption of electromagnetic waves. The mutual energy theory proposed by the author does not think so. Electromagnetic radiation must have two objects. One is the source and the other is the sink. The radiation source includes a transmitting antenna, a radiator charge, or a primary coil of a transformer. The sink includes a receiving antenna, an absorber charge, or a secondary coil of a transformer.

There are two current elements in Figure 1 below, is the radiation source, distributed inside . is a sink, distributed inside is the electromagnetic field of the source. is the electromagnetic field of the sink. is an arbitrary surface that separates two current elements. can be enclosing current cell , or surrounding current element . can also be an infinite plane, as shown by the green line in the Figure 1. The author’s electromagnetic theory must have at least two current elements. The electromagnetic fields of these current elements satisfy the following 3 axioms.

For the above two current elements, a source and a sink, the following law of conservation of energy exists.

(2)

If in the frequency domain, the corresponding formula is,

(3)

The body current of the above formula can be changed to the line current,

For example, for an transformer, the subscript of primary coil is 1, and that of secondary coil is 2. The left side of the above formula is the power provided by the primary coil, and the right side is the power consumed by the load of the secondary coil circuit. We assume that the transformer is an ideal transformer. The above formula can be written as,

(4)

If considered,

(5)

(6)

“” means “to define something”. Substitute formulas (5, 6) into (4) to get

or

(7)

The two sides above are two mutual inductions. Neumann gave the mutual inductance formula in 1845,

(8)

(9)

Since the integral limits of the above two equations can be exchanged, there are

(10)

is a real number. Therefore, the formula (7) is automatically satisfied. It seems mediocre, but it is not. In fact, we have verified the law of conservation of energy (7).

Eq.(2) can be written as,

The above is the energy conservation law of two elements. It can be generalized as,

(11)

The above is energy conservation law of elements. The above laws are self-evident. The energy lost by current element will increase the energy of element and the total energy will remain unchanged.

Considering that there is a considerable distance between the primary coil and the secondary coil of the transformer, the primary coil is called the transmitting antenna and the secondary coil is called the receiving antenna. In this case, the electromagnetic field of the transmitting antenna should consider the retarded effect. Therefore, the formula (8) should be rewritten as,

In this case, the formula (7) is required to be satisfied,

Hence, there is,

The retarded potential can be defined by the internal integral of the above equation,

Define advanced potential,

The retarded potential obtained by converting the line current into the body current is,

Advanced potential is,

Changed from frequency domain to time domain, we have,

is speed of the light. The above two equations are retarded potential and advanced potential, which meet the vector potential wave equation,

(12)

in the above equation, , merge and write as . We know that the scale potential is

The corresponding retarded potential should be

The corresponding advanced potential is,

The wave equation satisfied by the scalar potential is,

(13)

The Lorenz guage condition can be verified by the formula (12,13),

(14)

(15)

Considering the definition of electromotive force,

where “” means “to define somesing”. Hence there is,

or

or

or

Same reason

(16)

Obtain,

Consider the mathematical formula,

Define

(17)

(18)

Among them

and

Consider,

Eq.(18) can be written as,

Eq.(17) can be written as

Consider,

Obtain,

So we have a quasi-static equation,

Same reason,

Magnetic quasi-static. In the magnetic quasi-static electromagnetic field, if can be negligible, there is,

Same reason,

The above are magnetic quasi-static electromagentic fields.

If retarded potential and advanced potential are used to replace non retarded potential, the difference is

Consider,

Obtain,

Consider,

There is,

From this we have corresponded the retarded potential to Maxwell’s equation,

Maxwell’s equation corresponding to advanced potential can be written as,

It is worth mentioning that the electromagnetic field obtained from the retarded potential advanced potential is not the same as that obtained from the magnetic quasi-static vector potential. they are two different things. This is why we use superscript and to indicate retardation and advance.

For Maxwell’s equation,

we have

Therefore, the Maxwell equation is as follows

(19)

(20)

Poynting’s theorem can be deduced from the above formula,

or

So there is Poynting’s theorem

(21)

Corresponding quasi-static equation

where

Poynting’s theorem corresponding to the quasi-static equation is,

If can be ignored, for example, when there is no capacitor in the system, this situation is called magnetic quasi-static field. The corresponding magnetic quasi-static field has Poynting’s theorem,

Consider current elements, ,.

(22)

(23)

(24)

The following is the Poynting theorem of the N current elements,

(25)

The above formula can be disassembled into,

(26)

(27)

That is to say, the above two formulas are added together to get (25). We are trying to prove the energy conservation law Eq.(11), i.e.,

(28)

To prove that the above equation is zero, we need to prove that,

(29)

(30)

(31)

(32)

(33)

It is proved that (29) first transforms it into the frequency domain

Since the formula is purely imaginary in the frequency domain

“” means take the real part. This means that (29) is established. Proof (30), also in frequency domain

So there are (30). It is proved that (31) because this is a quasi-static condition, in the quasi-static condition, the electric field and magnetic field are attenuated with

When the radius of sphere tends to infinity,

This proves the formula (31). The same reason can be proved (33). Proof (32)

Among them

is the energy at the end of the process. is the energy at the beginning of the process, both of which can be taken as 0. This proves that (33)

When we prove the formula (29-33), we prove the law of conservation of energy (28) from Poynting’s theorem (25). That is to say, the law of conservation of energy (28) is proved to be true under the magnetic quasi-static condition.

Substituting the superposition principle (22-24) into Poynting’s theorem (21),

Divide Poynting’s theorem into,

(34)

(35)

To prove the law of conservation of energy,

(36)

It must be proved (29-33), in addition,

And

The proof of these two terms is the same as the formula (29,32). Therefore, it will not be repeated. Now reconsider the formula (31) and formula (33), because the radiated electromagnetic field must be considered now. The electric field and magnetic field both decay with .

Therefore, the original method can no longer prove the formula (33), but if considered and , one is retarded wave, the other is advanced wave. The retarded wave and advanced wave do not reach the spherical surface with infinite radius Γ at the same time. Therefore,

Therefore (33) is still valid. Finally, consider the formula (31), where

Is the Poynting vector of the i-th antenna. For any antenna,

Therefore, the formula (31) cannot be proved. In fact, we can’t prove the conservation law of energy (36). We still can prove Eq. (36) is a energy theorem, but now it is not the energy conservation law! Since there exist other energy flow, the self-energy is not zero.

The author thinks that the law of conservation of energy can not be proved, which shows that Maxwell’s equation has problems. The law of conservation of energy can be proved from the magnetic quasi-static equation, which shows that the magnetic quasi-static equation has passed this test. In the next section, we give the correction of Maxwell’s theory.

Do time integration on the formula (34,35)

(37)

(38)

The energy items disappear after time integration. Although we know that according to Maxwell’s equation,

(39)

But we think it is the fault of Maxwell’s equation, not the fault of the law of conservation of energy, so there should be,

(40)

We believe that the electromagnetic field calculated according to the magnetic quasi-static electromagnetic field equation or quasi-static equation is still electric field and magnetic field, but the electric field and magnetic field calculated according to Maxwell’s equation are no longer electric field and magnetic field. The true electric and magnetic fields should be determined by the formulas (40) and (37).

It can be proved that the formula (34) is equivalent to Maxwell’s equation, that is, it can be derived from Maxwell’s equation (34). Maxwell equation (19, 20) can also be derived from (34). The formula (37) has one more time integral than the formula (34), so it is properly relaxed by one step. The formula (34) is called mutual energy principle. The formula (37) is the relaxed mutual energy principle. Because of this relaxation process, We can add another equation (40) to the relaxed equation.

If , we have,

Or it can be written as,

Or

(41)

The formula (40) can be written as,

(42)

(43)

The above three formulas are the author’s electromagnetic field equation. The correct electromagnetic field can be obtained by replacing Maxwell’s equations with these three equations Eq. (41, 42,43).

According to Maxwell’s electromagnetic theory, Poynting’s theorem is,

Do time integration for Poynting theorem, and consider,

In the above formula

is the energy at the end of the electromagnetic process, which is . is the energy without the start of the electromagnetic process, which is . So we get,

We know from Maxwell’s classical electromagnetic theory that the radiation of any antenna is not zero. therefore

This shows that the author’s electromagnetic theory is different from Maxwell’s electromagnetic theory. It’s different. The far-field electric field and magnetic field of Maxwell electromagnetic theory antenna have the same phase. In the author’s theory (is defined by Eq.(41, 42,43)), there is a phase difference of 90 degrees between the electric field and the magnetic field.

Figure 2 shows two current elements. The surface in formula (41) can be taken arbitrarily, assuming that the surface contains only one current element there is,

See Figure 3. is the normal vector, which is from 1 to 2. The above formula can be written as following,

Assume that the surface contains only one current element , there is,

See Figure 4.

Then consider the law of conservation of energy formula (2), i.e.,

Obtain,

The above can be written as,

(44)

where

(45)

The symbol “” means “is defined as”.

Figure 5 shows the shape of the mutual energy flow. can be , It can also be any surface, the surface can separate the current element and current element . The formula (2) can be rewritten as,

(46)

2 is expanded to

(47)

The above equation (47) is the energy conservation law of current elements. Compared with the formula (2), this formula is self explanatory. Because we assume that there are charges in our space, and the motion of each charge constitutes a current element. Because already contains all the charges in the universe, the energy lost by some charges must be the energy obtained by others. The above formula expresses this meaning.

When the author saw the formula (2), he naturally thought that it was an energy theorem, so he called it the mutual energy theorem in 1987 in ref. ⁶. But when the author wrote the above equation (47), he understood that this formula is actually the law of conservation of energy (because it can be imagined that this contains all the charges in the universe, and there is no other charge in the universe). The above equation (47) is the law of conservation of energy, and the formula (2) is also the law of conservation of energy in case N=2.

As for proving that the above formula is the law of conservation of energy, it is not proved from Maxwell’s theory. If it can be proved from Maxwell’s equation, the above law of conservation of energy is not necessary to call as a law, but just a theorem. We can prove that the above equation is the law of conservation of energy from the magnetic quasi-static equation, but for the Maxwell equation containing the displacement current used as the radiation electromagnetic field, there are contradictory results. Therefore, Maxwell’s theory has made mistakes on this point. Therefore, the author believes that the above formula (47) is indeed the law of conservation of energy. According to this point, the author made a revision to Maxwell’s theory. The next section gives the revised electromagnetic theory.

However, equation (44) is not only the law of energy conservation, but also the law of localized energy conservation. The localization here is because the energy flows from the source to the sink can go step by step through the mutual energy flow.

Among the mutual energy flow,

is a mixed Poynting vector. It is also the energy flow density from the source to the sink.

it is the energy flow from the source to the sink.

is the energy that passes through the surface Γ. The author first called it mutual energy flow.

Is the work of the electric field of the source on the sink current element,

It is the work done by the source current element to overcome the electric field of the sink.

The above mutual energy flow or energy flow means that it can occur between the source and sink, and cannot occur between two sources or two sinks. Of course, sometimes the source and sink are inseparable. For example, when a metal sheet reflects electric field waves, it is both the source and sink. However, in this case, we divide it into a single source and a single sink. In this way, the source cannot transfer the energy to the source, and the sink cannot transfer the energy of the advanced wave to the sink. Therefore,

(48)

If and both are radiation sources or sinks. For example, the mutual energy flow between two transmitting antennas is zero. The mutual energy flow between the two receiving antennas is also zero.

The law that rediation does not overflow the universe can be written as following,

(49)

In the above formula, is the current element in region , assuming that

and

Substitute the above to Eq.(49)

(50)

The above means,

(51)

The above is the principle of self energy, all self energy flows do not overflow the universe. and

(52)

The above formula means that all mutual energy flows do not overflow the universe. In the above formula, a pair, , can be a pair of sources that emit retarded waves, a pair of sinks that emit advanced waves, or a source and a sink, so they are a retarded wave and a advanced wave. No matter what, radiation cannot overflow the universe.

The above two equations are the self energy principle, which tells us that the self energy flow is a reactive power wave. This wave radiates electromagnetic energy outward and inward at the same time. Moreover, the inward radiation is time reversal, which can be understood as reverse collapse. Therefore, this electromagnetic wave does not transfer energy. Note that this electromagnetic wave can carry electromagnetic energy, but the electromagnetic energy collapses back. So the energy transferred by the average effect is 0.

In frequency domain, the above formula is

Figure 6 shows the self energy current of current element. The self energy flow will collapse inward while radiating outward. Therefore, the average radiation in a period is zero. This is true for both sources and sinks. Therefore, self energy flow does not transfer energy.

Although the electromagnetic waves of the source and sink are both reactive power waves, they do not transfer energy. But energy can be transferred through mutual energy flow. The above two principles are the starting point or axiom system used by the author to solve electromagnetic field problems. The author’s electromagnetic theory is different from Maxwell’s electromagnetic theory. It is a corrected Maxwell’s theory.

4. Micro Maxwell’s Equation

The purpose of this paper is to approximate Maxwell’s electromagnetic theory from the author’s electromagnetic theory. The author believes that if sinks are uniformly distributed on a sphere with an infinite radius, the author’s theory can be equivalent to Maxwell’s electromagnetic theory. This idea was influenced by Wheeler Feynman absorber theory ^{1, 2}.

Maxwell’s electromagnetic theory holds that radiation of the current element is independent of the sink. That is, radiation is independent of absorption. However, in Maxwell’s radiation theory, the electromagnetic field is always required to meet the Silver-Mueller radation condition or Sommerfield radiation conditions. This condition actually includes absorption. The author’s electromagnetic theory does not require Silver-Mueller/Sommerfield condition. If we want to add this radiation condition, we must explicitly arrange an infinite number of absorbers or sinks on the boundary. The purpose of this paper is to attempt to derive Maxwell’s electromagnetic theory approximately from the author’s electromagnetic theory by adding sinks with uniform distribution on the boundary of the universe.

For the system shown in Figure 7, there is axiom 1 in the author’s theory: the law of conservation of energy,

(53)

Axiom 2. The following is the (mutual) energy flow law of electromagnetic field, and the word: “mutual” can be omitted. Because mutual energy flow is the only energy flow. The energy transfer of self energy flow has no contribution.

(54)

The mutual energy flow is defined as,

(55)

The surface is any surface which can separat two current sources , .

Axiom 3 Self energy flow principle,

(56)

This axiom shows that self energy does not transfer energy. The above Eq.(53-56) is the starting point of the author’s electromagnetic theory. It can be regarded as an axiom system.

Now the mutual energy flow theorem is the starting point of the author’s electromagnetic theory. Figure 8 shows the region inside of

The mutual energy flow theorem can be rewritten as,

Consider for

is the outer normal of the surface. We get,

(57)

For

is the outer normal of the surface. obtain

(58)

Consider the region within the sphere , where two small balls containing current elements are removed Charge . Because there is no current element in the region , is the region inside of the . . It can be obtained either by the formula (57) or (58),

If we assume that the normals of the inner surfaces and are outward, then there are,

Hence,

Considerred the formula(57,58), we obtain,

The above equation considers , within . is the region inside the surface . In this way, we can prove the relaxed mutual energy principle formula from the mutual energy flow theorem,

By the way, the relaxation in the above formula is because the author said that the principle of mutual energy refers to the following formula,

The relaxed mutual energy principle can be obtained by integrating the mutual energy principle with time. The relaxed mutual energy principle can be rewritten as,

By expanding 2 to we get,

Although a current element can be both an source and a sink, in this case, we regard the current element as an source and a sink. Suppose there are current elements, if we know

There are sources and sinks. According to the above Figure 9, we get the relaxed energy conservation law:

In the above formula, it is considered that the mutual energy flow between the two sources is zero, and the mutual energy flow between two sinks is zero, that is, the formula (48).

Next step, we assume that there is only one emitter, which is represented by the subscript 0, as shown in Figure 10, we get,

Or

Let

Obtain,

Considering is at the outside of the surface , see Figure 10, so there is,

(59)

In the above formula, we know that the electric field and magnetic field . The phase between and is different in the author's electromagnetic theory, and it is the same in Maxwell's electromagnetic theory. Let's assume that they are in phase temporarily. We need and and and is synchronized, so they are all in same phase, hence

The above means the advanced wave sends from the sinks are exactly same as the retarded wave sends from the source. Here we assume they are same. We get,

or

We have,

Here is the total electric field, and the total electric field is the sum of and .

The above factor seems to be reducible.

(60)

Consider that the phases of and in this formula are in phase, however, Poynting’s theorem

(61)

In, and are in phase. Formula (60 and 61) has same energy flow and same phase of the electromagnetic field and magnetic field, hence they have same electromagnetic field. That means we can derive (61) from (60). We know that Poynting’s theorem is equivalent to Maxwell’s equations,

(62)

Hence we have derived the Maxwell’s equations. It is worth mentioning that in the above formula, in fact, in the author’s electromagnetic theory and is not in phase.

Actually they have 90 degree phase difference. For example, the electric field can be

“∼” means proportional to, it do not care the value, but keep the phase and direction. In this case see Figure 11, the current density on the infinite surface can be compared with the electric field consistent, all

The magnetic field generated by this current can be,

Above and , and although their phases are different, and are in phase, and are in phase. So there are still,

So we can still get Poynting’s theorem (61). Then Maxwell equation is obtained. Of course, Maxwell’s equation is not really obtained here, but from the perspective of energy flow, it is the same as the energy flow of the solution obtained from Maxwell’s equations.

The derivation above assumes that the source is concentrated at a point. If the sources are not at a point, we can get the Maxwell’s equations (62) satisfied by all points by using the superposition principle. In the Figure 11 we only show one sink along the axis. But actually it can be at all direction.

The key is that the electromagnetic field of the sink and the electromagnetic field of the source can be synchronized. In synchronization, according to the author’s theory, the electric field of the source is not synchronized with that of the sink. The magnetic field of the radiation source and the magnetic field of the sink are also not synchronized. But they are ingeniously combined to make and is exactly synchronized and therefore can be added.

In addition, according to the author’s electromagnetic theory, the electromagnetic and magnetic fields also meet the boundary conditions near the source and sink. According to this condition, the magnetic field should have the same phase with the current element according to the Ampere circuital law, and the electric field should have a phase difference of 90 degrees with the current element according to the Faraday electromagnetic induction law. In fact, the theory of the author is the theory of photons. Because the retarded wave is the waveguide of the advanced wave, the advanced wave is the waveguide of the retarded wave. Therefore, the electromagnetic field in a photon is like the electromagnetic field inside a waveguide, which is actually a quasi plane wave. For this quasi plane wave, the phases of electric field and magnetic field are consistent from beginning to end.

This cannot be achieved according to Maxwell’s electromagnetic theory. According to Maxwell’s electromagnetic theory, far field electric field and magnetic field have the same phase. Near field electric field and magnetic field have different phases. Therefore, there is always a place where the phase of the electric and magnetic fields have to change. This electromagnetic field cannot constitute a quasi plane wave, so it is impossible to describe photons.

5. Discussion

Wheeler Feynman put forward the absorber theory ^{1, 2}. According to this theory, half of the electromagnetic field is generated by the sources, and the other half is generated by the absorbers. It can be assumed that the absorbers are uniformly distributed on the spherical surface with infinite radius. Wheeler Feynman’s theory was developed by Cramer and put forward the transactional interpretation. Papers on transactional interpretation have received a large number of citations, with more than 1000 citations. It can be seen that this theory has broad support. The author also agrees with this theory. However, Wheeler Feynman Cramer’s theory is still a qualitative theory, belonging to the interpretation of quantum mechanics. The author adds a quantitative theory to this theory. This quantitative theory is electromagnetic field mutual energy theory. Mutual energy theory starts from a new set of axioms, axiom 1, the conservation of energy. Axiom 2, mutual energy flow theorem. Axiom 3, self energy principle. The mutual energy flow theorem is equivalent to a relaxed Maxwell’s equations. Because of this relaxation, the relationship between electric field and magnetic field can be properly adjusted. Then we can add a new constraint condition of the self energy, which is called the self energy principle. The principle of self energy tells us that the self energy flow of electromagnetic wave radiated by radiation source must be reactive power. Therefore, it does not radiate electromagnetic wave energy. The energy is not transferred by the self energy flow, that is, the energy flow corresponding to Poynting vector. According to the author’s electromagnetic theory, energy is transferred by the mutual energy flow between the source and sink.

Therefore, if there is no absorber in the environment, there is no earth, moon, stars and dust in space. The sun does not radiate. The author’s theory provides a method for calculating the electromagnetic energy flow from the source to the sink. But there is a special case that needs special treatment, that is, if we consider the situation that the absorber is distributed on the spherical surface with infinite radius. In this paper, based on the author’s electromagnetic theory, it is assumed that the absorber is uniformly distributed on the large spherical surface, so that Maxwell’s electromagnetic theory can be derived approximately. The approximation here refers to the same energy flow radiated in both cases from the perspective of energy flow.

Although Maxwell’s theory does not emphasize that the boundary is full of absorber materials. It is also not emphasized that the absorber material should also produce advanced waves, but Maxwell’s theory requires that the cosmic boundary meet the Sliver-Mueller or Sommerfeld radiation conditions. This radiation condition actually means that the boundary must be filled with absorber materials.

According to the author’s theory, the radiation energy flow of macro electromagnetic wave is consistent with the sum of many mutual energy flows. The author has been using mutual energy flow to interpret photons. Now the light source radiates a large number of photons to the absorber material at the boundary of the universe, and these photons and composite effects are the same as macroscopic electromagnetic waves. This further tells us that electromagnetic waves are actually composed of large amount of photons. Large amount of photons can form mutual energy flow. This mutual energy flow is close to the radiation energy flow of macroscopic electromagnetic wave calculated by Maxwell’s theory.

On the other hand, we can also compare two quantities, one is the wave Poynting vector, and the other is the mutual energy flow. We assume that

Here are the electric and magnetic fields in Maxwell's theory, are electromagnetic fields in the author's electromagnetic theory.

It takes into account

That is, we get,

Poynting vector and mutual energy flow correspond to the same energy flow. The left side of the above equation is the energy flow of Poynting’s theorem of Maxwell’s theory, and the right side is the mutual energy flow of the author. The two are equivalent. Of course, there is another case: Maxwell’s electric field

In this case

In this case, the left side is Poynting energy flow, and the right side is half of the mutual energy flow. In the above cases, it may be necessary to add a normalization factor to the mutual energy flow.

Hence the two theory offers the energy flows perhaps have a normalization factor upto 2. Otherwise the two theories Maxwell’s classical electromagnetic field theory and the author’s mutual energy theory offers same energy flow of radiation.

Of course, the electric field and magnetic field given by the two theories are different. But the electric and magnetic fields corresponding to electromagnetic waves are not measurable quantities. The real measurable quantity is the radiation power of the transmitting antenna and the receiving power of the receiving antenna. And the pattern of the transmitting antenna. For these measurable quantities, the two theories get the same value after suitable normailization.

6. Conclusion

The author reviewed his own theory of electromagnetic mutual energy. According to this theory, the radiation energy flow of the electromagnetic field can only occur between the transmitting antenna and the receiving antenna. A single transmitting antenna cannot radiate electromagnetic energy. But consider that a sphere with an infinite radius is full of materials that absorb electromagnetic waves, and this absorber can also act as a receiving antenna. Considering such an absorber, we can get almost the same results as Maxwell’s electromagnetic theory by using the author’s electromagnetic theory. Thus Maxwell’s electromagnetic theory can be regarded as the approximate result of the author’s electromagnetic theory. The two theories reach a certain unity. The author provides a quantitative theory corresponding to the qualitative theory of Wheeler Feynman and Cramer.

On the other hand, the electromagnetic theory proposed by the author has obvious advantages compared with Maxwell’s electromagnetic theory. First, it provides a better understanding of electromagnetic fields. The theory that energy is only transferred by mutual energy flow rather than self energy flow is established. The viewpoint that photons are mutual energy flow is clarified. Furthermore, the fact that a large number of mutual energy flows can form a macroscopic mutual energy flow shows that macroscopic electromagnetic waves can be composed of photons. So as to unify the waves and particles. The problem of wave particle duality is well solved. This can be extended to other particles in quantum mechanics, such as electrons. It can be said that electrons are also mutual energy flows. This mutual energy flow is composed of the retarded wave and advanced wave satisfying Dirac equation or Schrodinger equation.

References