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The magnetic energy of a single moving charge



The magnetic energy of a single moving charge

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

An elec­tric cur­rent induces a mag­netic field in the sur­round­ing space. The mag­netic energy of an elec­tric cur­rent is described by the for­mula (Joule), where L is the mag­netic induc­tion coef­fi­cient of the elec­tric cir­cuit and I the elec­tric current.

The mag­netic energy Wm of an elec­tric cur­rent tends to con­serve the elec­tric cur­rent. Only when there is elec­tric or mag­netic resis­tance the cur­rent I will decline in time and the mag­netic energy Wm will be “lost” and the elec­tric cur­rent I will even­tu­ally dis­ap­pear completely.

An elec­tric cur­rent nor­mally con­sists of an infi­nite num­ber of mov­ing elec­trons. There are how­ever no the­o­ret­i­cal objec­tions to an elec­tric cur­rent con­sist­ing of one sin­gle mov­ing charge. In the elec­tron the­ory of the Dutch sci­en­tist H.A. Lorentz an elec­tric cur­rent Ids (amp.m) induces a mag­netic field dH (ampere/​m) at a dis­tance R (m) equal to:

Fig­ure 16. The mag­netic field of a cur­rent IdS.

The total mag­netic field an elec­tric cur­rent induces at P is the sum­ma­tion (inte­gra­tion) of all the mag­netic fields dH each mov­ing indi­vid­ual elec­tron in the elec­tric cir­cuit induces at P.

The­o­ret­i­cally the cur­rent IdS can exist of one mov­ing charge Qe, because the total mag­netic field H at P is the sum­ma­tion (and by approx­i­ma­tion, when there are infi­nite elec­trons, the inte­gra­tion) of the mag­netic field of all indi­vid­ual elec­trons pass­ing through the elec­tric cir­cuit at the same moment.

When the cur­rent IdS con­sists of only one mov­ing charge than:

IdS=QeVe [charge.m/sec] IVedt=QeVe Idt=Qe [charge]

In the case of a sin­gle mov­ing charge Idt=Qe, where Qe is the charge of that sin­gle elec­tron. The cur­rent I is no longer divid­able, so IdS=QeVe is the dif­fer­en­tial limit of an elec­tric current.

When the elec­tric cur­rent IdS is pre­sented by a sin­gle charge Qe, mov­ing rel­a­tively to P(x,y,z) with speed Ve, the mag­netic field H at P(x,y,z), due to cur­rent IdS=QeVe, is accord­ing to the elec­tron the­ory of Lorentz:

Let us con­sider a bulb shaped charge Qe with radius Re. Because in nature energy always tries to min­i­mize the energy level, the charge Qe will be sit­u­ated at the sur­face of the bulb (Re). In the fig­ure below, the sit­u­a­tion is sketched, where charge Qe is at rest and the move­ment of the charge is revealed by the rel­a­tive speed V.

Fig­ure 17. The mag­netic field of a mov­ing charge.

When an observer moves rel­a­tively to Qe with speed Ve and wants to deter­mine the mag­netic field Qe is induc­ing in the sur­round­ing space, the observer can choose any coor­di­nate P(x,y,z), com­pared to the posi­tion of charge Qe (0,0,0).

Because there is only one mov­ing charge the mag­netic field H in P(x,y,z) is sim­ply deter­mined by means of the elec­tron the­ory of Lorentz and IdS=QeVe:

and

The energy den­sity of an mag­netic field is given by the exper­i­men­tal formula:

Sub­sti­tu­tion of the derived induced mag­netic field at P:

in the exper­i­men­tal derived for­mula for the energy den­sity Em gives:

The mag­netic energy dWm, for the observer, in vol­ume is:

Inte­grat­ing for dα and dβ gives:

This is the energy of the induced mag­netic field in the bulb shell at a radius R from the cen­ter of the charge Qe and a rel­a­tive speed Ve.

When the radius of the charge Qe is Re the total energy of the induced mag­netic field sur­round­ing Qe, becomes:

Wm is the mag­netic energy the rel­a­tively mov­ing (Ve) bulb shaped charge (Qe) with radius Re induces in the sur­round­ing (vac­uum) space of the observer.

We men­tioned a cur­rent that con­sists of only one elec­tron. The above men­tioned is how­ever valid for any sin­gle rel­a­tive mov­ing charged bulb. The sin­gle charge can be any (metal­lic) charged bulb. The induced mag­netic field B at R(x,y,z) can there­fore be ver­i­fied in an exper­i­ment accord­ing to the equation:

Qe is then, in the above equa­tion, the total charge of the bulb, Ve the rel­a­tive speed of the charge to the mag­ne­tome­ter and R the dis­tance to the cen­ter of the charge.

To be able to relate the mag­netic energy Wm of the mov­ing charge to the elec­tro­sta­tic energy of Qe, we have to con­sider the poten­tial elec­tro­sta­tic energy of a bulb (Re) shaped charge Qe. The elec­tro­sta­tic energy of a charged (Qe) bulb (Re) in vac­uum is given by the formula:

For the observer, mov­ing rel­a­tive to charge Qe with speed Ve, the total energy (Wt) the charge presents is the sum of mag­netic (Wm) and elec­tro­sta­tic energy (Wp):

Wt=Wm+Wp

Con­sid­er­ing we derive:

Wt is the total energy the mov­ing charge presents to an observer: the elec­tro­sta­tic energy and the dynamic energy. Con­sid­er­ing the mass Mp the elec­tro­sta­tic energy Wp presents:

Sub­sti­tut­ing the equa­tion for the elec­tro­sta­tic mass Mp with the for­mula for the total energy Wt of the mov­ing charge we derive:

The mag­netic energy (Wm) of the mov­ing charge, expressed in the mass equiv­a­lence (Mp) of the elec­tro­sta­tic energy, becomes:

This derived for­mula for the mag­netic energy of a mov­ing charge is remark­able con­sid­er­ing the kinetic energy (Wk) of a “nor­mal” mass Mp, mov­ing with rel­a­tive speed Ve, is:

Next chap­ter: he mov­ing elec­tron and mag­netic energy

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

An electric current induces a magnetic field in the surrounding space. The magnetic energy of an electric current is described by the formula  (Joule), where L is the magnetic induction coefficient of the electric circuit and I the electric current.

The magnetic energy Wm of an electric current tends to conserve the electric current. Only when there is electric or magnetic resistance the current will decline in time and the magnetic energy Wm will be “lost” and the electric current I will eventually disappear completely.

An electric current normally consists of an infinite number of moving electrons. There are however no theoretical objections to an electric current consisting of one single moving charge. In the electron theory of the Dutch scientist H.A. Lorentz an electric current Ids (amp.m) induces a magnetic field dH (ampere/m) at a distance R (m) equal to:

Figure 16.  The magnetic field of a current IdS.

The total magnetic field an electric current induces at P is the summation (integration) of all the magnetic fields dH each moving individual electron in the electric circuit induces at P.

Theoretically the current IdS can exist of one moving charge Qe, because the total magnetic field H at P is the summation (and by approximation, when there are infinite electrons, the integration) of the magnetic field of all individual electrons passing through the electric circuit at the same moment.

When the current IdS consists of only one moving charge than:

IdS=QeVe   [charge.m/sec]        IVedt=QeVe        Idt=Qe   [charge]

In the case of a single moving charge Idt=Qe, where Qe is the charge of that single electron. The current I is no longer dividable, so IdS=QeVe is the differential limit of an electric current.

When the electric current IdS is presented by a single charge Qe, moving relatively to P(x,y,z) with speed Ve, the magnetic field H at P(x,y,z), due to current IdS=QeVe, is according to the electron theory of Lorentz:

Let us consider a bulb shaped charge Qe with radius Re. Because in nature energy always tries to minimize the energy level, the charge Qe will be situated at the surface of the bulb (Re). In the figure below, the situation is sketched, where charge Qe is at rest and the movement of the charge is revealed by the relative speed V.

Figure 17. The magnetic field of a moving charge.

When an observer moves relatively to Qe with speed Ve and wants to determine the magnetic field Qe is inducing in the surrounding space, the observer can choose any coordinate P(x,y,z), compared to the position of charge Qe (0,0,0).

Because there is only one moving charge the magnetic field H in P(x,y,z) is simply determined by means of the electron theory of Lorentz and IdS=QeVe:

and

The energy density of an magnetic field is given by the experimental formula:

Substitution of the derived induced magnetic field at P:

in the experimental derived formula for the energy density Em gives:

The magnetic energy dWm, for the observer, in volume  is:

Integrating for dα and dβ gives:

This is the energy of the induced magnetic field in the bulb shell  at a radius R from the center of the charge Qe and a relative speed Ve.

When the radius of the charge Qe is Re the total energy of the induced magnetic field surrounding Qe, becomes:

Wm is the magnetic energy the relatively moving (Ve) bulb shaped charge (Qe) with radius Re induces in the surrounding (vacuum) space of the observer.

We mentioned a current that consists of only one electron. The above mentioned is however valid for any single relative moving charged bulb. The single charge can be any (metallic) charged bulb. The induced magnetic field B at R(x,y,z) can therefore be verified in an experiment according to the equation:

Qe is then, in the above equation, the total charge of the bulb, Ve the relative speed of the charge to the magnetometer and R the distance to the center of the charge.

To be able to relate the magnetic energy Wm of the moving charge to the electrostatic energy of Qe, we have to consider the potential electrostatic energy of a bulb (Re) shaped charge Qe. The electrostatic energy of a charged (Qe) bulb (Re) in vacuum is given by the formula:

For the observer, moving relative to charge Qe with speed Ve, the total energy (Wt) the charge presents is the sum of magnetic (Wm) and electrostatic energy (Wp):

Wt=Wm+Wp

Considering we derive:

Wt is the total energy the moving charge presents to an observer: the electrostatic energy and the dynamic energy. Considering the mass Mp the electrostatic energy Wp presents:

Substituting the equation for the electrostatic mass Mp with the formula for the total energy Wt of the moving charge we derive:

The magnetic energy (Wm) of the moving charge, expressed in the mass equivalence (Mp) of the electrostatic energy, becomes:

This derived formula for the magnetic energy of a moving charge is remarkable considering the kinetic energy (Wk) of a “normal” mass Mp, moving with relative speed Ve, is:

Next chapter: he moving electron and magnetic energy

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