## 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 *Id s*

**(amp.m) induces a magnetic field**

*d*(ampere/m) at a distance

**H****(m) equal to:**

*R*

*Figure 16. The magnetic field of a current Id S*.

The total magnetic field an electric current induces at ** P** is the summation (integration) of all the magnetic fields

*d*

**H****each moving individual electron in the electric circuit induces at**

**.**

*P*Theoretically the current *Id S* can exist of one moving charge

*Qe*, because the total magnetic field

**at**

*H**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 *Id S* consists of only one moving charge than:

*Id S=QeVe* [charge.m/sec]

*I*[charge]

**V**edt=Qe**V**e Idt=QeIn 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 *Id S=QeVe* is the differential limit of an electric current.

When the electric current *Id S* is presented by a single charge

*Qe*, moving relatively to

*P**(x,y,z*) with speed

*V**e*, the magnetic field

**at**

*H*

*P**(x,y,z),*due to current

*Id*, is according to the electron theory of Lorentz:

**S**=Qe**V**eLet 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 *V**e* 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

*Id*:

**S**=Qe**V**eand

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 *d**α* and *d**β** *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**