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The Electromagnetic Mass



The Electromagnetic Mass

When you are inter­ested in physics you must read “Unbe­liev­able”!

In the chap­ter “The mag­netic energy of a sin­gle mov­ing charge” we demon­strated, with the help of the Elec­tron The­ory of Lorentz, that the mag­netic energy (Wm) of a sin­gle mov­ing bulb charge is equal to:

, where Mp is the mass equiv­a­lence of the elec­tro­sta­tic energy of the charged bulb accord­ing to .

In “The mov­ing elec­tron and mag­netic energy” we cal­cu­lated the­o­ret­i­cally the mag­netic energy of the “clas­si­cal elec­tron”. We derived the fol­low­ing equa­tion for the mag­netic energy of the mov­ing electron:

Let us sug­gest that the elec­tron, a charged mass, can be pre­sented by the energy/​mass of a charged bulb and a, not yet iden­ti­fied (mechan­i­cal) part of the mass.

The elec­tro­sta­tic mass of a charged bulb with the Comp­ton radius explains exactly half the energy/​mass of the elec­tron. The mag­netic energy of the mov­ing charge equals the kinetic energy of the mov­ing elec­tron. The ques­tion to be answered is: “What kind of energy/​mass presents the other half of the intrin­sic energy of the elec­tron?” We refer to the chap­ter “The Elec­tron”, where the mag­netic energy of the spin of the elec­tron is cal­cu­lated at exactly half the intrin­sic energy of the electron.

The mag­netic spin energy is respon­si­ble for and explains the physics why the charge of an elec­tron is con­fined. The expand­ing force of the charged bulb of the elec­tron is com­pen­sated by means of the con­tract­ing force of the spin­ning mag­netic field sur­round­ing the elec­tron. The charge of the elec­tron is trapped.

The above pre­sented E&M physics for the elec­tron is how­ever incon­sis­tent with the QM per­spec­tives and there­fore must be addressed. We refer to the QM per­spec­tives in “Lec­tures on Physics” part II chap­ter 28 “The Elec­tro­mag­netic Mass” by Feyn­man. In this chap­ter the elec­tro­mag­netic mass of the elec­tron is derived by means of the momen­tum den­sity g (Equa­tion 27.21 Lec­tures on Physics Vol­ume II, Feyn­man).

g=εo ExB

Accord­ing to the QM approach in (282) the mag­ni­tude of vec­tor g is:

because the momen­tum den­sity vec­tor is directed obliquely toward the line of motion (*).

Fur­ther­more, and I quote (282): “The fields are sym­met­ric about the line of motion, so when we inte­grate over space, the trans­verse com­po­nents will sum to zero, giv­ing a resul­tant momen­tum par­al­lel to v. The com­po­nent of g in this direc­tion is g sin(θ), which we must inte­grate all over space.” (**)

In the above argu­men­ta­tion (*) and (**) vec­tor p is thought to be partly com­pen­sated by the oppo­site vec­tor p. Vec­tor sum­ma­tion is allowed in sta­tic sit­u­a­tions, where the vec­tors for exam­ple express the mag­ni­tude and direc­tion of a sta­tic force. p is how­ever a dynamic vec­tor, pre­sent­ing the impulse of the mov­ing mass/​energy den­sity at a cer­tain point.

(For * and ** see Lec­tures on Physics Vol­ume II, Feyn­man)

The total momen­tum p, accord­ing to the QM approach is then:

where is the vol­ume ele­ment. The inte­gra­tion over all space gives:

(Equa­tion 28.3 Feynman).

This equa­tion, expressed in the sym­bols used in this arti­cle, gives the impulse:

The cal­cu­lated elec­tro­mag­netic mass accord­ing to equa­tion 28.3 is , which is 43 the mass equiv­a­lence of the elec­tric field or 23 the mass of the elec­tron Me.

Although the QM approach dif­fers from the approach in this arti­cle, the out­come should be con­sis­tent with each other. Because there is no con­sis­tency between the out­come of both approaches there must be an omis­sion. In the QM approach the momen­tum den­sity mag­ni­tude of vec­tor g is dimin­ished with the fac­tor , because of the pre­vi­ous men­tioned and marked argu­ments (*) and (**).

To com­pre­hend the effect of the cor­rec­tion of the mag­ni­tude of the momen­tum den­sity g with the fac­tor we have to con­sider the QM equa­tion for the total momen­tum; the inte­gra­tion of the momen­tum den­sity g over space to p, in more detail (Feyn­man 282):

The motion of the charge v, in the above equa­tion, is inde­pen­dent of any vari­able in the equa­tion, so we are allowed to abstract v out of the integral.

Because presents the mass den­sity (kg/​m3) of the energy of the elec­tro­sta­tic field E and is the vol­ume ele­ment, the inte­gra­tion of the for­mula presents the cal­cu­la­tion of the mass equiv­a­lent of the elec­tro­sta­tic field sur­round­ing the charge. The cor­rec­tions men­tioned in (*) and (**) do not alter the implied physics of the inte­gra­tion. By inte­grat­ing the momen­tum den­sity g around the charge all over space, the phys­i­cal inter­pre­ta­tion of the inte­gra­tion is the cal­cu­la­tion of the mass of the elec­tro­sta­tic field sur­round­ing the charge. A mass is a scalar and there­fore the cor­rec­tion of the mag­ni­tude of the momen­tum den­sity vec­tor with fac­tor will be an omis­sion, by which the con­ser­va­tion law for energy is vio­lated. Part of the mass of the elec­tro­sta­tic field is unjustly ignored.

The mag­ni­tude of g presents the mag­ni­tude of the mass den­sity impulse of the mov­ing elec­tro­sta­tic field E in the direc­tion of v. The direc­tion of vec­tor g does not present the direc­tion of the move­ment of the elec­tric field and there­fore not the direc­tion of the impulse. Com­pen­sa­tion of the mag­ni­tude of g with fac­tor vio­lates the energy con­ser­va­tion law because part of the mass/​energy of the elec­tro­sta­tic field is then ignored.

The momen­tum den­sity g at P as a result of the mov­ing elec­tro­sta­tic field/​energy Er with v at r should be:

g=Mr.v

where presents the mass den­sity (kg/​m3) of the elec­tro­sta­tic field at r.

Inte­gra­tion of g over space gives:

Equa­tion (28.3 Feyn­man) is now:

This equa­tion, expressed in the sym­bols used in this book, is:

or or

(Me is the mass of the elec­tron and Mp is the mass equiv­a­lence of the elec­tro­sta­tic field).

The cor­rected total momen­tum of the mov­ing charge p, accord­ing to the QM approach, is now com­pletely con­sis­tent with the derived equa­tions in this chap­ter. The cor­rectly derived elec­tro­mag­netic mass, accord­ing to the QM approach, equals now the mass of the elec­tron Me. There­fore the dynamic energy of the elec­tron with the QM approach is also fully magnetic.

Elec­tro­mag­netic Mass Lec­tures on Physics Vol­ume II, Feynman

Next chap­ter: Dis­cus­sion on mag­netic and kinetic energy

When you are interested in physics you must read “Unbelievable“!

In the chapter “The magnetic energy of a single moving charge” we demonstrated, with the help of the Electron Theory of Lorentz, that the magnetic energy (Wm) of a single moving bulb charge is equal to:

, where Mp is the mass equivalence of the electrostatic energy of the charged bulb according to .

In “The moving electron and magnetic energy” we calculated theoretically the magnetic energy of the “classical electron”. We derived the following equation for the magnetic energy of the moving electron:

Let us suggest that the electron, a charged mass, can be presented by the energy/mass of a charged bulb and a, not yet identified (mechanical) part of the mass.

The electrostatic mass of a charged bulb with the Compton radius explains exactly half the energy/mass of the electron. The magnetic energy of the moving charge equals the kinetic energy of the moving electron. The question to be answered is: “What kind of energy/mass presents the other half of the intrinsic energy of the electron?” We refer to the chapter “The Electron”, where the magnetic energy of the spin of the electron is calculated at exactly half the intrinsic energy of the electron.

The magnetic spin energy is responsible for and explains the physics why the charge of an electron is confined. The expanding force of the charged bulb of the electron is compensated by means of the contracting force of the spinning magnetic field surrounding the electron. The charge of the electron is trapped.

The above presented E&M physics for the electron is however inconsistent with the QM perspectives and therefore must be addressed. We refer to the QM perspectives in “Lectures on Physics” part II chapter 28 “The Electromagnetic Mass” by Feynman. In this chapter the electromagnetic mass of the electron is derived by means of the momentum density (Equation 27.21 Lectures on Physics Volume II, Feynman).

g=εo ExB 

According to the QM approach in (28-2) the magnitude of vector g is:

because the momentum density vector is directed obliquely toward the line of motion (*).

Furthermore, and I quote (28-2): “The fields are symmetric about the line of motion, so when we integrate over space, the transverse components will sum to zero, giving a resultant momentum parallel to v. The component of g in this direction is g sin(θ), which we must integrate all over space.” (**)

In the above argumentation (*) and (**) vector p is thought to be partly compensated by the opposite vector p. Vector summation is allowed in static situations, where the vectors for example express the magnitude and direction of a static force. p is however a dynamic vector, presenting the impulse of the moving mass/energy density at a certain point.

(For * and ** see Lectures on Physics Volume II, Feynman)

The total momentum p, according to the QM approach is then:

where  is the volume element. The integration over all space gives:

 (Equation 28.3 Feynman).

This equation, expressed in the symbols used in this article, gives the impulse:

The calculated electromagnetic mass according to equation 28.3 is , which is 4/3 the mass equivalence of the electric field or 2/3 the mass of the electron Me.

Although the QM approach differs from the approach in this article, the outcome should be consistent with each other. Because there is no consistency between the outcome of both approaches there must be an omission. In the QM approach the momentum density magnitude of vector g is diminished with the factor , because of the previous mentioned and marked arguments (*) and (**).

To comprehend the effect of the correction of the magnitude of the momentum density g with the factor  we have to consider the QM equation for the total momentum; the integration of the momentum density g over space to p, in more detail (Feynman 28-2):

The motion of the charge v, in the above equation, is independent of any variable in the equation, so we are allowed to abstract v out of the integral.

Because  presents the mass density (kg/m3) of the energy of the electrostatic field E and  is the volume element, the integration of the formula presents the calculation of the mass equivalent of the electrostatic field surrounding the charge. The corrections mentioned in (*) and (**) do not alter the implied physics of the integration. By integrating the momentum density g around the charge all over space, the physical interpretation of the integration is the calculation of the mass of the electrostatic field surrounding the charge. A mass is a scalar and therefore the correction of the magnitude of the momentum density vector with factor  will be an omission, by which the conservation law for energy is violated. Part of the mass of the electrostatic field is unjustly ignored.

The magnitude of g presents the magnitude of the mass density impulse of the moving electrostatic field E in the direction of v. The direction of vector g does not present the direction of the movement of the electric field and therefore not the direction of the impulse. Compensation of the magnitude of g with factor  violates the energy conservation law because part of the mass/energy of the electrostatic field is then ignored.

The momentum density g at P as a result of the moving electrostatic field/energy Er with v at r should be:

g=Mr.v

where  presents the mass density (kg/m3) of the electrostatic field at r.

Integration of g over space gives:

Equation (28.3 Feynman) is now:

This equation, expressed in the symbols used in this book, is:

  or     or     

(Me is the mass of the electron and Mp is the mass equivalence of the electrostatic field).

The corrected total momentum of the moving charge p, according to the QM approach, is now completely consistent with the derived equations in this chapter. The correctly derived electromagnetic mass, according to the QM approach, equals now the mass of the electron Me. Therefore the dynamic energy of the electron with the QM approach is also fully magnetic.

Electromagnetic Mass Lectures on Physics Volume II, Feynman

Next chapter: Discussion on magnetic and kinetic energy

 

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