**1. Introduction**

The physics of charged particle beam is an area where relativistic effects manifest themselves substantially. Here, one has to deal with a moving object, so both a fixed (laboratory) coordinate system and a moving coordinate system are to be used. A charged particle moves relative to the laboratory coordinate system, while in the moving coordinate system, it is at rest. Hence, in a laboratory coordinate system, the problem is to be considered as an electrodynamical one, and in a moving coordinate system, the problem belongs to the area of electrostatics. Thus, electrostatic phenomena in a charged particle set at rest are transformed into electrodynamic ones when it moves. Electromagnetic fields in these two inertial reference systems are tied via the Lorentz transform ([1], p. 79).

In the wave zone, the dynamic component of the electric field strength and the axially symmetric magnetic field form both a constant flux into a given solid angle, i.e., electromagnetic radiation, and a flux per time unit directed along the normal to the conical surface of the solid angle. The potential component of the electric field, directed along the radius, and the axially symmetric magnetic field form a flux oriented along the polar direction, i.e., along the normal to the above conical surface. The fluxes crossing the conical surface do not depend on the distance between the source point and the observation point. In the wave zone, the

radiations from the beginning and end of the REB segment are added up, while the fluxes through the above conical surface caused by dynamic and potential components of electric field, are subtracted.

*graddiv* � *rotrot* � **<sup>1</sup>**

A potential part of the vector potential *A*

respectively; and *k*

**3. Potentials**

!

*DOI: http://dx.doi.org/10.5772/intechopen.86980*

by the Lorentz calibration [3, 7]:

*ψ t* 0

*A* ! *t* 0

at the time instant *t*.

� *<sup>Q</sup> L*4*πε*<sup>0</sup>

<sup>¼</sup> *<sup>Q</sup> L*4*πε*<sup>0</sup>

([7], p. 432):

**59**

table integral ([8], p. 34):

*ψ t* 0

*ln z* � *vet*

�

� �

�

� �

<sup>0</sup> ð ð Þ þ *L* Þ þ

*ln z* � *vet* <sup>0</sup> ð Þþ

segment end and its beginning, respectively.

**4. The electromagnetic field strengths**

� <sup>q</sup>

� <sup>q</sup>

¼ � *<sup>Q</sup> L***4***πε***<sup>0</sup>**

> ¼ � *<sup>Q</sup> <sup>μ</sup>*<sup>0</sup> *L*4*π*

� �*<sup>A</sup>*

*Radiation and Energy Flux of Electromagnetic Fields by a Segment…*

*div A*!*<sup>p</sup>*

, *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet*

*vet* ð 0 þ*L*

*vet*<sup>0</sup>

, *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet*

*vet* ð 0 þ*L*

*vet*<sup>0</sup>

, *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet*

*c***2**

*∂***2** *∂t***<sup>2</sup>**

where *ε***<sup>0</sup>** and *μ***<sup>0</sup>** are the dielectric and magnetic permeability of vacuum,

ð Þ¼� *t*, *r*

Using the Green's function for the wave equation ([3], p. 243), we obtain:

0 , *z*<sup>0</sup>

0 , *z*<sup>0</sup>

q

q

!

<sup>0</sup> is the unit vector along the REB axis, the Oz axis.

!*p*

**1** *c***2** *∂ ∂t*

, *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼

> *dz*0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>***<sup>2</sup>** <sup>þ</sup> *<sup>y</sup>***<sup>2</sup>** <sup>þ</sup> *<sup>z</sup>* � *<sup>z</sup>*<sup>0</sup> ð Þ**<sup>2</sup>**

, *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼

> *dz*0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>* � *<sup>z</sup>*<sup>0</sup> ð Þ<sup>2</sup>

where the hatched coordinates refer to the source point at the time instant *t*

the field radiation, and the non-hatched coordinates refer to the observation point

0 , *z*<sup>0</sup>

where the expressions in the first and second summands refer to the REB

For estimation of the electric and magnetic fields, we use standard formulas

The formula for the scalar potential can be obtained in the closed form using the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>* � *vet*<sup>0</sup> ð Þ<sup>2</sup>

, *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z*

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>* � *vet* ð Þ ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* <sup>2</sup>

> > � � � � *t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 *<sup>t</sup>*0,*z*0¼*vet*<sup>0</sup> j j ð Þ *c*

� � �

ð Þ¼� *t*, *r μ***0***ρ*ð Þ *t*, *r vek*

*t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 j j *<sup>c</sup>*

� � �

*t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 j j *<sup>c</sup>*

> � � � � *t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 *<sup>t</sup>*0,*z*0¼*vet* j j ð Þ 0þ*<sup>L</sup> c*

� � �

� � � !

ð Þ *t*, *r* is related to the scalar potential

*ψ*ð Þ *t*, *r* , (4)

, (5)

, (6)

<sup>0</sup> of

(7)

**<sup>0</sup>**, (3)

To date, the issue of influence of the finite length of a charged particle beam, moving uniformly in vacuum on the radiation of electromagnetic fields remains poorly studied, with an exception of publication [2], where its experimental part deserves special attention.

This chapter presents the results of our theoretical analysis of the electromagnetic field radiated by a finite-length segment of filamentous relativistic electron beam (REB). The REB segment moves uniformly in vacuum along its own axis which we will address as the *longitudinal direction*. The stepped varying of the charge density at the edges of the REB segment creates point-like sources of the potential electric field; the strength of which is inversely proportional to the distance between the source point and the observation point. In addition, the time variation of the REB current density forms at the REB edges the point-like sources of both potential and vortex electric fields, as well as the vortex magnetic field, with their strengths being also inversely proportional to the distance between the source point and the observation point [3].

The filamentary REB edges are considered as relativistic point-like radiators of the electromagnetic energy propagating to the wave zone. The presence of a potential electric field in the wave zone is due to the fact that the electric scalar potential in the wave zone is proportional to the electric *monopole* moment ([4], p. 51), which equals to the total charge in the selected volume ([5], p. 280). As follows from the Jefimenko's generalization of the Coulomb law ([3], p. 246), the potential electric field strength in the wave zone is proportional to the time derivative of the electric monopole moment.

In the intermediate zone, there is a flow of electrical field energy, due to the electric potential field and the field of the displacement current. The electrical energy flux in the intermediate zone is due to the electric potential field and field of the displacement current. The REB part with a constant charge density between its edges forms a quasi-static electromagnetic field in the near zone.

Note that a similar problem has been considered in [6], but it was devoted to similarity of the solutions obtained with the help of two different methods: retarded field integral and transformation equations of the special theory of relativity. Unlike our work, it does not contain expressions for scalar and vector potentials, as well as the electromagnetic energy flux.

#### **2. Formulation of the problem**

Consider a filamentary REB segment of length *L* and electric charge density *Q* moving uniformly along its axis direction with velocity *ve*. Charge density of the REB segment may be written as follows:

$$\rho\left(\mathfrak{t},r\left(\mathfrak{x},\mathfrak{y},\mathfrak{z}\right)\right)\_{L} = \frac{\mathbf{Q}}{L}\delta(\mathfrak{x})\cdot\delta\left(\mathfrak{y}\right)\cdot\left[h(\mathfrak{x}-\mathfrak{v}\_{\mathfrak{e}}\mathfrak{t}) - h(\mathfrak{x}-\left(\mathfrak{v}\_{\mathfrak{e}}\mathfrak{t}+L\right))\right] \tag{1}$$

where *h x*ð Þ is Heaviside step function; *δ*ð Þ *x* and *δ y* are Dirac delta functions of coordinates. The electric scalar potential *ψ*ð Þ *t*, *r* and vector potential *A* ! ð Þ *t*, *r* , taking into account Eq. (1), satisfy the wave equations [3, 7]:

$$
\left[ \operatorname{div} \mathbf{grad} - \frac{\mathbf{1}}{c^2} \frac{\partial^2}{\partial t^2} \right] \boldsymbol{\psi}(\mathbf{t}, \mathbf{r}) = -\frac{\rho(\mathbf{t}, \mathbf{r})}{\mathbf{e\_0}},\tag{2}
$$

*Radiation and Energy Flux of Electromagnetic Fields by a Segment… DOI: http://dx.doi.org/10.5772/intechopen.86980*

$$
\left[\vec{grad}\,d\vec{v} - \vec{rot}\,t - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right]\vec{A}(t,r) = -\mu\_0\rho(t,r)\nu\_e\vec{k}\_0,\tag{3}
$$

where *ε***<sup>0</sup>** and *μ***<sup>0</sup>** are the dielectric and magnetic permeability of vacuum, respectively; and *k* ! <sup>0</sup> is the unit vector along the REB axis, the Oz axis.
