The magnetic energy of a single moving charge

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An electric current induces a magnetic field in the surrounding space. The magnetic energy of an electric current is described 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 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]                I Vedt=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 divisible, so  IdS=QeVe  is the differential limit of an electric current.

When the electric current  IdS  is presented 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 minimise 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 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 means of the electron theory of Lorentz and IdS=QeVe:

 and 

The energy density of a magnetic field is given 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 centre 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  Qebecomes:

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.</assistant

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 centre 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 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:  the moving electron and magnetic energy