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3B. Entity Field Electromagnetism (Part B)

## Contents

### 3.8. The Origin of Magnetic Phenomenon

All experiments prove that there is no magnetic monopole to form the source of a magnetic force. The so-called “magnetic field” is the result of a moving electron or proton. In term of entity field, a magnetic force is originated from a moving entity field. To get a clear picture of this new way of understanding magnetic phenomenon, let’s start with some simple experiments.

 Figure (3.8.1c) An electron moves in free space

Figure (3.8.1a) is a stationary electron in a magnetic force field FB. There is no magnetic force on it. In fact, it is nothing but a normal electron. This means that a magnetic force does not interact with entity electric field.

Then, we can imagine that this electron is moving perpendicularly in the magnetic force field FB with speed v, as shown in figure (3.8.1b). As the result of this motion, there is a magnetic force on the electron. This means that the moving electron produces a magnetic force F to interact with the external FB. By using right hand rule we can find out that the force on the moving electron is pointing out from the screen, or can be symbolized as ʘ.

Comparing with a rest electron, a moving electron not only gets a kinetic energy for its particle, but also gets a magnetic energy for its entity field to form the source of the magnetic force. Both energies are transformed from an accelerating energy.

As we know that the source of an electric force is the entity electric field of an electron. Similarly, the source of a magnetic force can only be the entity field of a moving electron. There is no other possible source involved. Thus, a magnetic force is the result of an entity field with magnetic property. We can simply call it an entity magnetic field.

As the electron moves on, it passes the area of magnetic force and gets into a free space, as shown in figure (3.8.1c). Since the magnetic energy is stored inside the moving electron as an entity magnetic field, the disappearance of the external magnetic force field FB only withdraws the interaction and does not affect the existence of its entity magnetic field. So, this moving electron should carry the same entity magnetic field and has the same magnetic force field around it.

A question here is that if a moving electron carries a magnetic force field, what is its direction? As we can see that it doesn’t have direction yet since there is no interaction occurred. The direction of a magnetic force field will be defined according to its interaction with other magnetic force. Like an electric force field, a magnetic force field has no absolute physical direction.

The electron then moves into another external magnetic force field as shown in figure (3.8.1d). But this time the external magnetic force FB is parallel to the electron’s moving direction. Although the electron still carries a magnetic force field caused by its entity magnetic field, it does not interact with the external magnetic force FB. Again, we cannot decide its direction.

Finally, the electron is decelerated to its original state – stationary state. The speed of its particle is dropped to zero and the entity magnetic field is emitted as quantified photons carrying electromagnetic wave, which we will discuss later. There is no more magnetic force field around it.

We understand now that an entity magnetic field is, in fact, an extra entity field with magnetic property gained by moving entity electric field of an electron. Thus, both entity magnetic field and entity electric field are unified as entity field. From the viewpoint of entity field, an electric field is an entity field with electric property and a magnetic field is an entity field with magnetic property.

Since an entity magnetic field is only related to a moving electron, its image is also related to a moving electron. This is shown in figure (3.8.2).

It is quite clear that an entity magnetic field co-exists with its moving electron and emits out as a photon carrying electromagnetic wave once the speed of its electron is reduced. In some situations such as in figure (3.8.1c) and figure (3.8.1d), entity magnetic field of a moving electron cannot be detected.

A moving proton also carries an entity magnetic field as the source of its magnetic force field. The direction of this force field is opposite to the direction of which of an electron. A photon is emitted if the proton is decelerated.

A moving atom also carries entity magnetic fields due to the contributions of its electrons and protons. Normally, no magnetic force can be detected from out side of a moving atom for the magnetic forces are interacting within the atom. It emits photons if its electron, proton or the whole atom reduces speed.

Every moving object carries entity magnetic fields. Normally, magnetic forces are interacting within the object so that there is no magnetic force field can be detected. If the moving object is stopped, entity magnetic fields are emitted as photons in the form of electromagnetic wave.

In the case of magnetic force fields of moving electrons are aligned in one direction, strong magnetic force can be detected and the object is called magnet.

If a moving object has rest mass m0, increased mass m and kinetic energy EK, its magnetic energy EM is decided by special relativity

mc2=m0c2+EK+EM                 (3.8.1)

Where m is

(3.8.2)

In most situations, magnetic energy EM is kept inside the object without our notice.

By the way, there is no gravitational wave can be emitted. So, there is also no such thing as graviton.

### 3.9. Entity Magnetic Field Is a Relative, Relativistic and Quantified Field

As we explained before that the source of a magnetic force field is an entity magnetic field inside a moving electron or proton. Since any motion is relative to the observer, an entity magnetic field is a relative field. Since any motion is related to a certain energy level, an entity magnetic field is a quantified field. Since any motion obeys special relativity, an entity magnetic field is a relativistic field.

There are two frames of reference O and O in figure (3.9.1). O is moving to the right with speed v along the positive X-axis of O system. Conversely, O is moving toward negative X-axis with the same speed v as measured in O system. If electrons e1 and e2 are stationary in O and O respectively, the two electrons have relative notion with respect to one another with the speed v. An observer in O system finds that the moving electron e2 carries magnetic field and the stationary electron e1 doesn’t. Meanwhile, another observer in O system finds e1 has magnetic field and e2 hasn’t.

One way to find out which electron really has relatively higher-level energy is to transfer electron from one system into the other. If an accelerating force is needed to transfer an electron from O to O, we know that e2 is at higher energy level in comparison with e1. Therefore, to the observer in O system, e2 has real kinetic energy and real entity magnetic field. Meanwhile, observer in O finds that e1 has imaginary kinetic energy and imaginary magnetic field. This is to say that an entity magnetic field is a relative field.

Since an entity magnetic field is a relative field, an observer can never detect any entity magnetic field if an object is at rest in his reference system. In fact, this resting object is indeed carrying entity magnetic field due to the relative motion of his reference system.

On the other hand, to move an electron, the accelerating force has to provide a kinetic energy to its particle and a magnetic energy to its entity field. The result of the kinetic energy is to increase the speed of the electron and the result of the magnetic energy (stored as entity magnetic field) is to increase the mass of the electron. The distribution of the two kinds of energy is decided by special relativity. Due to the involvement of the entity magnetic field, an electron can never be accelerated into speed c. This is to say that an entity magnetic field is a relativistic field.

Since entity magnetic field links electron directly with special relativity, entity field theory provides us the real reasons of special relativity. This means that the principles of special relativity can be better explained by entity field theory, so they are no longer necessary.

In ordinary situation, there is v<<c, the effect of special relativity can be ignored.

A well known fact in quantum-mechanics is that the speed of a moving electron is decided by its energy level distribution. This means that the entity magnetic field is also quantified according to its energy level. In low speed situation, the difference between two energy levels is too small to be calculated and the quantum effect of an entity magnetic field can be ignored.

### 3.10. Magnetic Force and Its Force Field

Since an entity magnetic field is a relative field, it has no absolute value. Once an electron has relative motion in a reference frame, its entity magnetic field becomes the source of a magnetic force and create a magnetic force field around it. This magnetic force field was misunderstood as magnetic field in classical physics. On the other hand, an observer moving with the same electron finds that there is no magnetic force at all.

Historically, magnetic phenomenon is studied by observing mature magnet. Our earth is considered to be a big magnet. If we hang up a piece of magnetic bar from its middle with a string, it automatically aligns with the magnetic force field of the earth. The end that points northward is called north-seeking pole. The end that points southward is south-seeking pole. A magnetic bar is shown in figure (3.10.1). Normally, they are simply called North Pole and South Pole, or N pole and S pole.

If two magnets are put near by, there is a magnetic interaction between them. The rule for this interaction is: like poles repel; unlike poles attract. As a tradition, the direction of magnetic force is defined from N pole pointing to S pole. So, the magnetic force is also shown in figure (3.10.1).

Although entity magnetic fields exist in every moving object, we can not detect any magnetic force if the forces are interacting within the object. Our study is only concentrated on the one can be detected and can be used – charged moving object. Since magnetic field is originated from a moving electron or proton, our study should also start from there.

Figure (3.10.2) shows a moving electron e1 in O system.

 Figure (3.10.2) Magnetic field of moving electron

Suppose at the time t=t0 the electron is passing point O with speed v along positive Z-axis. By setting up another magnetic force near by and checking the interaction between the two, we can get the result of its magnetic force. Testing results show that any point in X-Y plane gets maximum magnetic force; any point along both sides of Z-axis gets zero force. The magnetic force at any other point is between the maximum and zero.

If there is another moving electron e2 traveling in the same direction near e1, the two electrons will have magnetic force interaction. If we define the direction of magnetic force is along straight line connecting the two electrons, magnetic force of e1 could point to any direction around it.

Figure (3.10.3) shows that there is a magnetic force FB between two moving electrons. In the first picture, FB=0. In the third picture, there is a maximum magnetic force FB.

We now have to find out the direction of magnetic force FB between two moving electrons. From figure (3.10.3) we know that, for a moving electron, there is magnetic force around it except in its moving direction. We can imagine that this magnetic force builds up a magnetic force field around the moving electron as shown in figure (3.10.4).

Since this is a picture in two dimensions, the magnetic force field needs to be rotated about the axis of the electron’s moving direction to form a three dimensional magnetic force field. Classical physics failed to recognize this kind of magnetic force field.

If an electron does not move along straight line, it will form a different magnetic force field. For example, if an electron is moving in a circle, as it does in an atom, it will form a magnetic force field as shown in figure (3.10.5). It is this kind of magnetic force field we studied in classical physics as magnetic field by mistake.

The magnetic force field of a moving electron changes according to its moving track.

All experiments about magnetic force field show that N pole and S pole are symmetrical about a center plane. For the moving electron in figure (3.10.4), the center plane “A” and its N pole and S pole are shown in figure (3.10.6). We have to emphasize that the N pole and S pole are defined concepts to simplify our study. We cannot mix them with physical reality. Since classical physics failed to recognize the magnetic force field of a moving electron, its N pole and S pole can never be identified there. In modern physics, the two different magnetic properties of a moving electron are rediscovered as its spin of +1/2 and -1/2.

Figure (3.10.7) shows that two electrons e1 and e2 move in the same direction with the same speed v. If e2 is behind e1, the S pole of e2 and the N pole of e1 tend to attract the two together. If e2 is ahead of e1, the N pole of e2 and the S pole of e1 will attract the two together. As for e2 is at the side of e1, magnetic force still attracts the two together due to that the function of a magnetic force is opposite to the function of an electric force.

Suppose two electrons moving side by side, there are three forces between the two: an electric repelling force, a magnetic attracting force and a gravitation which is much weaker then the other two. These forces form perfect wave condition and force electrons move in wavelike track. This wave aspect of particles is the basic principle of quantum-mechanics.

Since a moving electron carries entity electric and magnetic fields, it has electric and magnetic force fields around it. In modern physics, especially in QED, the entity fields of a moving electron are treated as its virtual photon; the force fields of which are treated as its wave.

In order to study the magnitude of a magnetic force, a magnetic action B can be introduced and is defined in terms of the magnetic force F exerted on a positive test charge q0 whose velocity is v. That is

(3.10.1)

In which q is the angle between v and B as shown in figure (3.10.8).

As we noticed that, in classical physics, this B is believed to be representing the magnetic field by mistake. In fact, it reflects the strength of a magnetic force field. After correcting magnetic field B as magnetic action B, the magnetic theory of classical physics can be re-discussed based on the same structure but with entity meaning.

The directions of v, B and FB obey right-hand rule. It says:

Open your right hand so that the fingers are together and the thumb sticks out. When your thumb is in the direction of v and your palm faces in the direction of F, your fingers are in the direction of B.

In fact, test charge q0 in figure (3.10.8) is trying to form a magnetic action of its own to against the external magnetic action B.

Figure (3.10.9) shown that a test charge q0 travels in a circle under the influences of B, v and F, and forms its magnetic action B0 to against B.

The direction of magnetic action B0 can be worked out by using another right-hand rule:

Grasp the loop so that the curled fingers of the right hand point in the moving direction of the test charge, the thumb of that hand then points in the direction of B0.

On the other hand, if test charge q0 moves in a magnetic action B with speed v, a magnetic force FB on it is

(3.10.2)

Here the direction of FB obeys the right-hand rule for cross products of v0 and B.

If test charge q0 moves in both magnetic action B and electric action E with speed v, the force on it is

(3.10.3)

In the case of electric current in a conductor, every moving electron will build up a magnetic action around itself so that there is a magnetic action around the conductor. If a current-currying wire is put in a magnetic action B, there is a magnetic force FB on the wire due to the interaction of magnetic forces. This is shown in figure (3.10.10).

Suppose a charge dq is passing through the wire with average speed vd, magnetic force on it is

(3.10.4)

Considering equation (3.7.1), the above equation can be changed as

(3.10.5)

Because vdt is the distance that a charge is moved. Thus

(3.10.6)

If the length of the wire is l, the total force of the wire is

(3.10.7)

For normal situation, we have

(3.10.8)

By distinguishing magnetic action B from entity magnetic field itself, entity field physics not only unified electric field and magnetic field as entity field, but also ensured that entity magnetic field obeys conservation laws at all time.

### 3.11. Magnetic Action of a Current

Since a piece of current-currying wire has magnetic force field around it, is has magnetic action B.

In figure (3.11.1) we consider that a current element Idl is the source of magnetic action dB. If the distance between Idl and an arbitrary point is r, magnetic action B at that point can be defined as

(3.11.1)

In which m0=4p´10-7 T·m/A is permeability constant; r is vector distance pointing from Idl to the arbitrary point. The total magnetic action B is

(3.11.2)

Equation (3.11.2) can be used to solve many different kinds of current-currying problems.

For a long, straight current, we can get

(3.11.3)

The direction of this magnetic action B is decided by the right-hand as:

Grasp the wire with the right hand so that the thumb in the direction of the current, the curled fingers in the direction of the magnetic field. Please refer to figure (3.11.1).

For a solenoid or a toroid, we can get

(3.11.4)

In which N is the number of turns; l is the total length and n is the number of turns per unit length.

Figure (3.11.2) shows a long straight current I flowing form the screen. According to equation (3.11.3), the integral of its magnetic action B is

(3.11.5)

If we change reference system into polar one

(3.11.6)

Equation (3.11.5) then can be written as

(3.11.7)

The result of ∮dq for close path L1 is 2p, and for close path L2 is zero. So, we have

(3.11.8)

In which Ip means that the current must be within the close path.

From section 3.7 we know that a current can be expressed in terms of current density J and the cross-sectional area dA it through. That is

(3.7.2)

The integral of Ip is

(3.11.9)

Then, equation (3.11.8) can be written as

(3.11.10)

It implies that the magnetic action B could be produced by a changing electric flux.

On the other hand, the flux jm of magnetic action B is defined as

(3.11.11)

Since any magnetic action can be identified to have N pole at one end and S pole at the other end, the net magnetic flux jm through any closed surface is always zero

(3.11.12)

This means that there is no magnetic monopole in our physical world.

The magnetic action B is, from the viewpoint of magnetic force, the ability to interact with another magnetic action. For example, if two long straight parallel current I1 and I2 are separated by a distance a, magnetic action of wire 1 due to I2 is

(3.11.13)

 Figure (3.11.3) two long, straight parallel currents

If we consider only a certain length l of wire 1, magnetic force from I2 is decided by

(3.11.14)

Substituting B2 with equation (3.11.13), we have

(3.11.15)

Similarly, magnetic force on wire 2 from I1 has the same magnitude as F12 but to the opposite direction. Thus, we have

F21=−F12                                    (3.11.16)

Further more, from right-hand rule we can find out that the interaction of two long, straight current are: Parallel currents attract and anti-parallel current repel. Considering a current contains a lot of individual moving electrons, the same rule can be used for the magnetic force between moving electrons. This is another way of understanding the wave of moving electrons.

### 3.12. Induction

So far, we have got a clear picture of magnetic field and its interaction. The basic idea is that a magnetic field is an entity field caused by a moving entity electric field. In the language of unified entity field: an electric field refers to an entity field with electric property and a magnetic field refers to an entity field with magnetic property.

According to the principle of relative motion, the motion of an object is related to a certain reference system. Since a moving electric field can produce an additional entity magnetic field, a moving magnetic field should be able to produce an additional entity electric field, which is called induced electric field. This induced electric field then has interaction with another electric field and cause an induction.

Figure (3.12.1) shows a moving magnetic action B along negative Z-axis with speed v. The electric field produced by this motion causes an electric action to interact with the rest charge q. So, there is an electric force F on charge q.

To check out this induced electric action E, we can first change our reference system into a one fixed on the moving magnet. Then the problem becomes a moving charge q in a stationary magnetic action B, which is familiar to us.

An observer now finds that the charge q is moving along the positive Z-axis with speed v. This moving charge produces a magnetic field. Magnetic force on the charge is decided by equation (3.10.2) as

(3.10.2)

The observer then changes back to his original reference system in which the charge is stationary while the magnetic action B is moving along negative Z-axis.  He now finds that the magnetic action B has an additional electric action E interacting with the electric action of the charge q

F=qE                                        (3.12.1)

Since both reference systems are inertial system, the physical result must be the same. So, we have

F=qv×B=qE                             (3.12.2)

In which v is along positive Z-axis as equation (3.10.2) required. Considering B is actually moving along the negative Z-axis, from equation (3.12.2) we can get

E=B×v=−v×B                           (3.12.3)

This electric action E can only be caused by entity electric field. It means that a moving entity magnetic field produces an entity electric field.

If we try to limit the above discuss to an electron, this electron now has two relative motions. The structure of this moving electron now has four entities: the particle and the unit entity electric field of a stationary electron, an entity magnetic field caused by the first relative motion, and an entity electric field caused by the second relative motion. Each of the four entities has its force and forms its force field.

Since a moving magnetic action can force an electron move, it can cause current flow in a circuit. Figure (3.12.2) is an experiment to demonstrate how this can be done.

The circuit is connected with an ampere meter to check any current flow in it.

As the magnet approaches the center of the coil from left, an induced current in the coil is established to against the approaching N pole. The direction of the current is in the same direction of the arrows in figure (3.12.2a) according to right hand rule. And the current is plotted as the left half of figure (3.12.2b)

When the middle of the magnet reaches the center of the coil, magnetic actions form N pole and S pole are counteracted. There is no current in the coil.

As the magnet passes the center of the coil, an induced current is established again to hold the leaving S pole. The direction of this current is opposite to the arrows in figure (3.12.2a). The current is show as the right half of figure (3.12.2b).

Figure (3.12.3) is two parallel coils. Coil 1 is connected with a battery V and a switch K. Coil 2 has an ampere meter to check it’s current.

When switch K is turned on, coil 1 has an increasing N pole at its right due to the increasing current in it. The induced current in coil 2 responds with N pole at its left to against the increasing N pole from coil 1. The direction of induced current is shown by the arrows in coil 2 according to right-hand rule.

At the moment switch K is turned off, coil 1 has a decreasing N pole at its right due to the decreasing current in it. The induced current in coil 2 then sets up S pole at its left to hold the decreasing N pole form coil 1. The direction of this induced current is opposite to the arrows in coil 2.

The above two experiments show that both moving magnetic action and changing magnetic action can produce entity electric field, and then, can cause induced current in other circuit. This means that the real source of an induced current is the electric action E. We can define an electromotive force e (also called emf) of induced current as

(3.12.4)

From equation (3.12.3) we can change the above equation into

(3.12.5)

In which n is normal vector of area A. Combining it with equation (3.11.11) we have

(3.12.6)

This means that an induced current can also be understood as the result of the rate of changing magnetic flux jm through a coil.

As the same entity field, if a time-varying magnetic field can produce an electric field, a time-varying electric field can surely produce a magnetic field. This is the principle of a capacitor.

Let’s consider a RC circuit in figure (3.12.4). At the moment the switch is turned on, the battery starts charging the capacitor C. the surface charges on both plates are time-varying quantities ±Q(t) and the charges increase from zero to the maximum. Correspondingly, the current in the circuit decreases from the maximum to zero. So, the current can be expressed as

(3.12.7)

The time-varying charges ±Q(t) indicate that there are time-varying electric actions between the two plates. By using electric displacement D and equation (3.6.17), Q(t) can be expressed as

(3.12.8)

This means that a time-varying electric displacement, which marks the present of an entity electric field, provides substantial linkage in the broken space of a capacitor.

For the complete RC circuit, the existence of current I(t) during charge and discharge periods means that there is a current flowing through capacitor. As we know that electron cannot pass through capacitor. This current is formed by time-varying entity electric field, and can be described as electric displacement current Ic(t)

Combining equation (3.12.8) and equation (3.7.1), we can get

(3.12.9)

For free space situation, there is D=e0E. The above equation can be changed into

(3.12.10)

Figure (3.12.5) is a circuit unit of a capacitor in a long, straight wire. To simplify the problem, we suppose there is a free space in the capacitor.

If there is a time-varying current I(t) in the wire, there is also a time-varying displacement current Ic(t) in the capacitor. The total current is the sum of the two. The magnetic action B of this circuit is contributed by both I(t) and Ic(t). Thus, equation (3.11.8) should be expressed as

(3.12.11)

Combining equation (3.7.2) and equation (3.12.10) we have

(3.12.12)

In which time-varying electric action dE/dt becomes a source of magnetic action B. In other words: time-varying entity electric field produces entity magnetic field.

### 3.13. Inductance and Magnetic Energy

For two parallel coils, a changing current in coil 1 causes an induced current in coil 2. Since the induced current is also a changing current, it can cause induced current in coil 1 again. This is called mutual induction.

Suppose coil 1 has number of turns N1, current I1 and magnetic action B, it produces flux  to each turn of coil 2. If coil 2 has number of turns N2, its flux linkage is N2j21. An induced emf in coil 2 is given by equation (3.12.6) as

(3.13.1)

For a given coil, its flux linkage is in proportion to the current of causing the induction. So, we can change the above equation into

(3.13.2)

In which M21 is mutual inductance of coil 2 in respect to coil 1. It is defined as

(3.13.3)

In return, the induced current I2 in coil 2 will cause an induced emf in coil 1 as e12. With similar procedure we can get

(3.13.4)

In which M12 is mutual inductance of coil 1 with respect to coil 2.

Experiments show that constant M21 equals M12. That is

M12=M21=M                                  (3.13.5)

A changing current not only causes induced current to other coils, but also causes induced current in its own coil. This is called self-induction. If a coil has number of turns N, its self-induced emf is

(3.13.6)

Where flux linkage Njm is in proportion to its current. So, we have

(3.13.7)

In which L is self-inductance. It is defined as

(3.13.8)

The power of induced emf and induced current is

(3.13.9)

Its total energy is

(3.13.10)

If we define magnetic energy density as wm=Wm/V, by studying a solenoid (L=m0n2V and B=m0In) we can get

(3.13.11)

### 3.14. Magnetic Properties of Matter

Since the entity magnetic field is in the structure of every moving electron and moving electrons exist in atoms of every object, so magnetic field exists in any object. In normal situation, magnetic fields are randomly oriented inside an object so that no magnetic force can be detected from outside.

When an external magnetic action is applied on an object, its atomic magnetic actions tend to be aligned with the external one. For some materials, this alignment remains even after external magnetic action is withdrawn.

Different names are used for magnetisms with different magnetic effects. A magnetism has internal magnetic action against the external one is called diamagnetism; The one has internal magnetic action along with the external one is called paramagnetism; The one has very strong magnetic action along with the external one is called ferromagnetism.

If a magnetism is put in an external magnetic action B0, the total magnetic action will be changed to B due to the contribution of the magnetism. The ratio between B and B0 is called relative permeability and is denoted by mr. So, we have

(3.14.1)

For diamagnetism, there is mr<1. For paramagnetism or ferromagnetism, there is mr>1.

A magnetic permeability m can be defined as

m=m0mr                               (3.14.2)

Because mr is a constant for most magnetism, it can be expressed with another constant called magnetic susceptibility cm

(3.14.3)

Comparing equation (3.14.2) and equation (3.14.3) we can get

m = (1+cm)m0                        (3.14.4)

To study the internal magnetic action of magnetism, a magnetic dipole moment m can be used to describe its atomic magnetic action. If we define a magnetization M as density of magnetic dipole moments, M can be expressed as

M=nm                                 (3.14.5)

In which n is the number density of m.

For magnetism, the contribution of its internal magnetic action can be generally expressed as

m0M=B-B0                         (3.14.6)

And the relationships among M, B and B0 are shown in figure (3.14.1).

 Figure (3.14.1) Relations of B0, B and μ0M

Considering equations (3.14.1) and (3.14.3), equation (3.14.6) can be changed as

(3.14.7)

For a long, straight current described by equation (3.11.8), we have

(3.14.8)

To simplify above equation, a magnetic force field strength H is introduced and is defined as

(3.14.9)

Substituting equation (3.14.9) to equation (3.14.8) yields

(3.14.10)

Where IT is the sum of electric current I and displacement current Ic. Substituting equation (3.11.9) and equation (3.12.9) into above equation gets

(3.14.11)

Combining equation (3.14.1) and equation (3.14.2) with equation (3.14.9) can get

(3.14.12)

For a solenoid with magnetism inside, its magnetic energy density is given by equation (3.13.11) as

(3.14.13)

### 3.15. Maxwell’s Equations

So far, we have successfully restructured the whole electromagnetism by using entity field theory. Based on the correct understanding of entity fields and their force fields, Maxwell’s equations can be written in the same form but with entity meaning

(3.15.1)

(3.15.2)

(3.15.3)

(3.15.4)

Equations (3.15.1) – (3.15.4) can also be expressed as

(3.15.5)

(3.15.6)

(3.15.7)

(3.15.8)

There are three supplementary equations

(3.15.9)

(3.15.10)

(3.15.11)

And the density of electromagnetic energy

(3.15.12)