**3. Potentials**

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 compo-

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

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

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

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

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

� ½ � *h z*ð Þ� � *vet h z*ð Þ � ð Þ *vet* <sup>þ</sup> *<sup>L</sup>* (1)

*ρ*ð Þ *t*, *r ε***0**

are Dirac delta functions of

!

, (2)

ð Þ *t*, *r* , taking

edges forms a quasi-static electromagnetic field in the near zone.

nents of electric field, are subtracted.

point and the observation point [3].

as the electromagnetic energy flux.

**2. Formulation of the problem**

*ρ t*, *r x*, *y*, *z*

**58**

REB segment may be written as follows:

*<sup>L</sup>* <sup>¼</sup> *<sup>Q</sup>*

*<sup>L</sup> <sup>δ</sup>*ð Þ� *<sup>x</sup> <sup>δ</sup> <sup>y</sup>*

coordinates. The electric scalar potential *ψ*ð Þ *t*, *r* and vector potential *A*

*c***2**

*∂***2** *∂t***<sup>2</sup>**

*ψ*ð Þ¼� *t*, *r*

where *h x*ð Þ is Heaviside step function; *δ*ð Þ *x* and *δ y*

*divgrad* � **<sup>1</sup>**

into account Eq. (1), satisfy the wave equations [3, 7]:

deserves special attention.

*Progress in Relativity*

A potential part of the vector potential *A* !*p* ð Þ *t*, *r* is related to the scalar potential by the Lorentz calibration [3, 7]:

$$d\dot{v}\,\vec{A}^p(t,r) = -\frac{1}{c^2}\frac{\partial}{\partial t}\psi(t,r),\tag{4}$$

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

$$\begin{split} \psi(t', \mathbf{x}' = 0, \mathbf{y}' = 0, v\_{\varepsilon}t' < z' < v\_{\varepsilon}t' + L; t, r(\mathbf{x}, \mathbf{y}, \mathbf{z})) &= \\ = -\frac{\mathbf{Q}}{L4\pi\varepsilon\_{0}} \int\_{\mathbf{v}, \mathbf{t'}}^{\mathbf{v}, t' + L} \frac{d\mathbf{z}'}{\sqrt{\mathbf{x}^{2} + \mathbf{y}^{2} + \left(\mathbf{z} - \mathbf{z'}\right)^{2}}} \Big|\_{\begin{subarray}{c} \left|\vec{r} - \vec{r}\right| \end{subarray}}, & \tag{5} \\ \overset{\cdot}{A}(t', \mathbf{x'} = 0, \mathbf{y}' = 0, v\_{\varepsilon}t' < z' < v\_{\varepsilon}t' + L; t, r(\mathbf{x}, \mathbf{y}, \mathbf{z})) &= \\ = -\frac{\mathbf{Q}\mu\_{0}}{L4\pi} \int\_{\mathbf{v}, \mathbf{t'}}^{\mathbf{v}, t' + L} \frac{d\mathbf{z'}}{\sqrt{\mathbf{x}^{2} + \mathbf{y}^{2} + \left(\mathbf{z} - \mathbf{z'}\right)^{2}}} \Big|\_{\begin{subarray}{c} \left|\vec{r} - \vec{r}\right| \end{subarray}}, & \tag{6} \end{split}$$

where the hatched coordinates refer to the source point at the time instant *t* <sup>0</sup> of the field radiation, and the non-hatched coordinates refer to the observation point at the time instant *t*.

The formula for the scalar potential can be obtained in the closed form using the table integral ([8], p. 34):

$$\begin{aligned} \boldsymbol{\psi}(\mathbf{r}', \mathbf{x}' = \mathbf{0}, \mathbf{y}' = \mathbf{0}, \boldsymbol{\nu}\boldsymbol{\xi}' &< \mathbf{z}' < \boldsymbol{\nu}\_{\epsilon}\boldsymbol{\epsilon}' + L; \mathbf{t}, \boldsymbol{\tau}(\mathbf{x}, \mathbf{y}, \mathbf{z})) \\ = \frac{Q}{L\mathbf{4}\pi\epsilon\_{0}} \ln \left| \left( \mathbf{z} - (\boldsymbol{v}\_{\epsilon}\boldsymbol{\epsilon}' + L) \right) + \sqrt{\mathbf{x}^2 + \boldsymbol{y}^2 + \left( \mathbf{z} - (\boldsymbol{v}\_{\epsilon}\boldsymbol{\epsilon}' + L) \right)^2} \right|\_{\left| \mathbf{r}' = \mathbf{t} - \frac{\left[ \boldsymbol{\tilde{r}} - \boldsymbol{\tilde{r}}'(\boldsymbol{t}' \boldsymbol{x} = \mathbf{e}' + \boldsymbol{L}) \right]}{\boldsymbol{\varepsilon}} \right|} \\ - \frac{Q}{L\mathbf{4}\pi\epsilon\_{0}} \ln \left| \left( \mathbf{z} - \boldsymbol{v}\_{\epsilon}\boldsymbol{t}' \right) + \sqrt{\mathbf{x}^2 + \boldsymbol{y}^2 + \left( \mathbf{z} - \boldsymbol{v}\_{\epsilon}\boldsymbol{t}' \right)^2} \right|\_{\left| \mathbf{r}' = \mathbf{t} - \frac{\left[ \boldsymbol{\tilde{r}} - \boldsymbol{\tilde{r}}'(\boldsymbol{t}' \boldsymbol{x}' = \mathbf{e}' + \boldsymbol{L}) \right]}{\boldsymbol{\varepsilon}} \right|}} \end{aligned} \tag{7}$$

where the expressions in the first and second summands refer to the REB segment end and its beginning, respectively.

#### **4. The electromagnetic field strengths**

For estimation of the electric and magnetic fields, we use standard formulas ([7], p. 432):

$$\overrightarrow{E}(t', \mathbf{x}' = 0, \mathbf{y}' = 0, v\_{\epsilon}t' < z' < v\_{\epsilon}t' + L; t, r(\mathbf{x}, \mathbf{y}, z)) =$$

$$= -\frac{\overrightarrow{\partial A}(t', \mathbf{x}' = 0, \mathbf{y}' = 0, v\_{\epsilon}t' < z' < v\_{\epsilon}t' + L; t, r(\mathbf{x}, \mathbf{y}, z))}{dt}$$

$$- \text{grad}\_r \eta(t', \mathbf{x}' = 0, \mathbf{y}' = 0, v\_{\epsilon}t' < z' < v\_{\epsilon}t' + L; t, r(\mathbf{x}, \mathbf{y}, z)), \tag{8}$$

$$\overrightarrow{H}(t', \mathbf{x}' = 0, \mathbf{y}' = 0, v\_{\epsilon}t' < z' < v\_{\epsilon}t' + L; t, r(\mathbf{x}, \mathbf{y}, z)) =$$

$$= \frac{1}{\mu\_0} r \text{rot}\_r \overrightarrow{A}(t', \mathbf{x}' = 0, \mathbf{y}' = 0, v\_{\epsilon}t' < z' < v\_{\epsilon}t' + L; t, r(\mathbf{x}, \mathbf{y}, z)), \tag{9}$$

� *<sup>Q</sup>μ*0*ve*

2

<sup>þ</sup> *<sup>Q</sup>μ*0*ve*

where

*L*4*π*

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet*

*cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet*

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet*

and

*<sup>t</sup>*,*r x*ð ÞÞ , *<sup>y</sup>*, *<sup>z</sup>* and *<sup>E</sup><sup>p</sup>*

dynamic component.

*t*,*r x*ð ÞÞ , *y*, *z* and *Hy t*

are:

**61**

*L*4*π*

2

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

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

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet*

*cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet*

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> <sup>½</sup> ð Þ <sup>þ</sup> *<sup>L</sup>* � ¼ *<sup>x</sup>*

<sup>0</sup> ð Þ <sup>þ</sup> *<sup>L</sup>* � � <sup>¼</sup> *<sup>y</sup>*

*κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

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

ð*vet* 0 þ*L*

*vet*<sup>0</sup>

<sup>0</sup> ½ �¼ ð Þ *<sup>x</sup>*

*r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup>

<sup>0</sup> ð Þ � � <sup>¼</sup> *<sup>y</sup>*

*r* ! � *r* !0 *t*<sup>0</sup>

<sup>0</sup> ½ �¼ ð Þ *<sup>z</sup>* � *vet*

*r* ! � *r* !0 *t*<sup>0</sup>

� � �

<sup>0</sup> ð Þ¼½<sup>1</sup> � *ve*

The transverse components of the electric field strength *E<sup>p</sup>*

The transverse components of the magnetic field strength *Hx t*

<sup>0</sup> ð Þ¼½ <sup>þ</sup> *<sup>L</sup>* <sup>1</sup> � *ve*

� � �

<sup>0</sup> <sup>½</sup> ð Þ <sup>þ</sup> *<sup>L</sup>* � ¼ *<sup>z</sup>* � *vet* ð Þ ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup> r* ! � *r* !0 *t*<sup>0</sup>

*κ z*<sup>0</sup> ¼ *vet*

*κ z*<sup>0</sup> ¼ *vet*

are the retardation factors ([3], p. 246).

coordinates, and the longitudinal component *Ez t*

*<sup>y</sup> t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t*

> 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t*

� � �

� � �

*κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

1

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

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

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

*c*

*c*

consists of both a potential component relative to the space coordinates and a

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� � � *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*

� � � *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*

� � � *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*

� � � *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*

� � � *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*

� � � *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*

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* are potential relative to the space

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

<sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* , according to the Eq. (6) and (9),

� � �

<sup>0</sup> ½ � ð Þ , (19)

<sup>0</sup> ½ � ð Þ þ *L* (20)

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

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

� � �

� � �

� � �

� � �

� � �

� � � � � � *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*

� � �

> � � � *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*

� � � þ

, *dz*<sup>0</sup> (12)

þ

, (13)

, (14)

, (15)

, (16)

, (17)

, (18)

! � *r* !0 *t*<sup>0</sup>

1

*cos α<sup>z</sup> z*<sup>0</sup> ½ � ð Þ

! � *<sup>r</sup>* !0 *t*<sup>0</sup>

> *r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ

*r* ! � *r* !0 *t*<sup>0</sup>

*r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup>

� � �

� � �

� � �

� � �

� � �

where it is necessary to perform the differentiation over the coordinates of the observation point, taking into account the retardation effect ([7], p. 432) and ([4], p. 43) as well as the differentiation of integrals by the integration limits and by the parameter ([9], p. 58). Using Eqs. (5), (6), and (8), we get:

*Ep <sup>x</sup> t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* <sup>0</sup> , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ <sup>¼</sup> *Qve L*4*πε*0*c cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ *κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � *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* � � � � � *Qve L*4*πε*0*c cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � *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* � � � þ <sup>þ</sup> *<sup>Q</sup> L*4*πε*<sup>0</sup> ð*vet* 0 þ*L vet*<sup>0</sup> *cos α<sup>x</sup> z*<sup>0</sup> ½ � ð Þ *r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ � � � � � � 2 *t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 *<sup>t</sup>*0,*z*<sup>0</sup> j j ð Þ *c* � � � , *dz*<sup>0</sup> (10) *Ep <sup>y</sup> t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ <sup>¼</sup> *Qve L*4*πε*0*c cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � � *κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � *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* � � � � � *Qve L*4*πε*0*c cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � � *κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � *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* � � � þ <sup>þ</sup> *<sup>Q</sup> L*4*πε*<sup>0</sup> ð*vet* 0 þ*L vet*<sup>0</sup> *cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> ð Þ � � *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ � � � � � � 2 *t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 *<sup>t</sup>*0,*z*<sup>0</sup> j j ð Þ *c* � � � , *dz*<sup>0</sup> (11) *Ez t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ <sup>¼</sup> *Qve L*4*πε*0*c cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ *κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � *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* � � � � � *Qve L*4*πε*0*c cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � *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* � � � �

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

� *<sup>Q</sup>μ*0*ve* 2 *L*4*π* 1 *κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � *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* � � � þ <sup>þ</sup> *<sup>Q</sup>μ*0*ve* 2 *L*4*π* 1 *κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � *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* � � � þ <sup>þ</sup> *<sup>Q</sup> L*4*πε*<sup>0</sup> ð*vet* 0 þ*L vet*<sup>0</sup> *cos α<sup>z</sup> z*<sup>0</sup> ½ � ð Þ *r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ � � � � � � 2 *t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 *<sup>t</sup>*0,*z*<sup>0</sup> j j ð Þ *c* � � � , *dz*<sup>0</sup> (12)

where

*E* ! *t* 0

*Progress in Relativity*

¼ � *<sup>∂</sup><sup>A</sup>* ! *t* 0

> *H* ! *t* 0

*Ep <sup>x</sup> t* 0

<sup>¼</sup> *Qve L*4*πε*0*c*

> *Ep <sup>y</sup> t* 0

<sup>¼</sup> *Qve L*4*πε*0*c*

� *Qve L*4*πε*0*c*

� *Qve L*4*πε*0*c*

¼ 1 *μ*0 *rotrA* ! *t* 0

�*gradrψ t*

0

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

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

parameter ([9], p. 58). Using Eqs. (5), (6), and (8), we get:

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

*κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

ð*vet* 0 þ*L*

*vet*<sup>0</sup>

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

*κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

ð*vet* 0 þ*L*

*vet*<sup>0</sup>

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

*κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

*κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

*κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

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

*Ez t* 0

<sup>¼</sup> *Qve L*4*πε*0*c*

� *Qve L*4*πε*0*c*

**60**

*κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

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

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

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

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

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

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

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

, *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z* , (8)

, *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z* , (9)

*∂t* �

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

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

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

where it is necessary to perform the differentiation over the coordinates of the observation point, taking into account the retardation effect ([7], p. 432) and ([4], p. 43) as well as the differentiation of integrals by the integration limits and by the

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

*cos α<sup>x</sup> z*<sup>0</sup> ½ � ð Þ

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

*cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � �

> ! � *r* !0 *t*<sup>0</sup>

*cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � �

*cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> ð Þ � �

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

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ

> ! � *r* !0 *t*<sup>0</sup>

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L*

! � *r* !0 *t*<sup>0</sup>

! � *r* !0 *t*<sup>0</sup>

> *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ

� � �

� � �

� � �

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ

> ! � *r* !0 *t*<sup>0</sup>

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L*

! � *<sup>r</sup>* !0 *t*<sup>0</sup>

> *r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ

� � �

� � �

� � � � � �

� � �

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

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

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

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

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

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

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

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� � � *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*

� � �

> � � � *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*

� � �

� � � � � � *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*

� � �

> � � � *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*

� � �

� � � � � � *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*

� � �

> � � � *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*

� � � �

, *dz*<sup>0</sup> (10)

�

, *dz*<sup>0</sup> (11)

�

�

þ

þ

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

$$\cos\left[a\_{\mathbf{x}}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}t')\right] = \frac{\mathbf{x}}{\left|\overrightarrow{r} - \overrightarrow{r}'(t',\mathbf{z}'=\mathbf{v}\_{\varepsilon}t')\right|}\bigg|\_{\left|t'=t-\frac{\left|\overrightarrow{r}-\overrightarrow{r}'(t',\mathbf{z}'=\mathbf{v}\_{\varepsilon}t'\right|}{\varepsilon}\right|},\tag{13}$$

$$\cos\left[a\_{\mathbf{x}}(\mathbf{z}'=\nu\_{\mathbf{c}}\mathbf{t}'+L)\right]=\frac{\mathbf{x}}{\left|\overrightarrow{r}-\overrightarrow{r}'(\mathbf{t}',\mathbf{z}'=\nu\_{\mathbf{c}}\mathbf{t}'+L)\right|}\bigg|\_{\left|\overrightarrow{r}-\overrightarrow{r}'(\mathbf{t}',\mathbf{z}'=\nu\_{\mathbf{c}}\mathbf{t}'+L)\right|\bigg|},\tag{14}$$

$$\cos\left[a\_{\mathcal{V}}(\mathbf{z}'=\boldsymbol{\nu}\_{\epsilon}\mathbf{t}')\right]=\frac{\mathcal{Y}}{\left|\overrightarrow{\boldsymbol{r}}-\overrightarrow{\boldsymbol{r}}'(\mathbf{t}',\mathbf{z}'=\boldsymbol{\nu}\_{\epsilon}\mathbf{t}')\right|}\Big|\_{\left|\overrightarrow{\boldsymbol{t}}'=\boldsymbol{t}-\frac{\overrightarrow{\boldsymbol{r}}\cdot\overrightarrow{\boldsymbol{r}}'(\mathbf{t}',\mathbf{z}'=\boldsymbol{\nu}\_{\epsilon}\mathbf{t}')}{\varepsilon}\right|},\tag{15}$$

$$\cos\left[a\_{\mathcal{V}}(\mathbf{z'} = v\_{\mathfrak{c}}\mathbf{t'} + L)\right] = \frac{\mathcal{Y}}{\left|\overrightarrow{r} - \overrightarrow{r'}(\mathbf{t'}, \mathbf{z'} = v\_{\mathfrak{c}}\mathbf{t'} + L)\right|}\bigg|\_{\left|\mathbf{z'} = t - \frac{\left|\overrightarrow{r} - \overrightarrow{r'}(\mathbf{t'}, \mathbf{z'} = v\_{\mathfrak{c}'}\mathbf{t} + L)\right|}{\varepsilon}\right|}},\tag{16}$$

$$\cos\left[a\_{\varepsilon}(\mathbf{z}' = \boldsymbol{\upsilon}\_{\varepsilon}t')\right] = \frac{(\mathbf{z} - \boldsymbol{\upsilon}\_{\varepsilon}t')}{\left|\overrightarrow{\boldsymbol{r}} - \overrightarrow{\boldsymbol{r}}'(t', \mathbf{z}' = \boldsymbol{\upsilon}\_{\varepsilon}t')\right|}\bigg|\_{\left|t' = t - \frac{\left|\overrightarrow{\boldsymbol{r}} - \overrightarrow{\boldsymbol{r}}'(t', \mathbf{z}' = \boldsymbol{\upsilon}\_{\varepsilon}t')\right|}{\varepsilon}\right|},\tag{17}$$

$$\cos\left[a\_{\mathbf{z}}(\mathbf{z}'=\boldsymbol{\nu}\_{\epsilon}\mathbf{z}'+L)\right]=\frac{(\mathbf{z}-(\boldsymbol{\nu}\_{\epsilon}\mathbf{z}'+L))}{\left|\overrightarrow{r}-\overrightarrow{r}'(\mathbf{t}',\mathbf{z}'=\boldsymbol{\nu}\_{\epsilon}\mathbf{t}'+L)\right|}\bigg|\_{\left|\overrightarrow{\boldsymbol{\nu}}'=\boldsymbol{t}-\frac{\left|\overrightarrow{r}-\overrightarrow{r}'(\boldsymbol{\nu}\_{\epsilon}\mathbf{z}'=\boldsymbol{\nu}\_{\epsilon}\mathbf{t}'+L)\right|}{\epsilon}\right|},\tag{18}$$

and

$$\kappa(\mathbf{z}' = \mathbf{v}\_{\epsilon}t') = [\mathbf{1} - \frac{\mathbf{v}\_{\epsilon}}{c}\cos\left[a\_{\mathbf{z}}(\mathbf{z}' = \mathbf{v}\_{\epsilon}t')\right],\tag{19}$$

$$\kappa(\mathbf{z'} = \upsilon\_{\mathbf{c}}t' + L) = \left[1 - \frac{\upsilon\_{\mathbf{c}}}{\mathbf{c}} \cos\left[a\_{\mathbf{z}}(\mathbf{z'} = \upsilon\_{\mathbf{c}}t' + L)\right] \tag{20}$$

are the retardation factors ([3], p. 246).

The transverse components of the electric field strength *E<sup>p</sup> <sup>x</sup> t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð ð Þ ð Þ ; *<sup>t</sup>*,*r x*ð ÞÞ , *<sup>y</sup>*, *<sup>z</sup>* and *<sup>E</sup><sup>p</sup> <sup>y</sup> t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* are potential relative to the space coordinates, and the longitudinal component *Ez t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* consists of both a potential component relative to the space coordinates and a dynamic component.

The transverse components of the magnetic field strength *Hx t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð ð Þ ð Þ ; *t*,*r x*ð ÞÞ , *y*, *z* and *Hy t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* , according to the Eq. (6) and (9), are:

*Hx t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ ¼ � *Qve* 2 *L*4*πc cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � � *κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � *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* � � � <sup>þ</sup> *Qve* 2 *L*4*πc cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � � *κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � *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* � � � � *Qve L*4*π* ð*vet* 0 þ*L vet*<sup>0</sup> *dz*<sup>0</sup> *cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> ð Þ � � *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ � � � � � � 2 *t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 *<sup>t</sup>*0,*z*<sup>0</sup> j j ð Þ *c* � � � (21) *Hy t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ <sup>¼</sup> *Qve* 2 *L*4*πc cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ *κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � *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* � � � � � *Qve* 2 *L*4*πc cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � *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* � � � þ <sup>þ</sup> *Qve L*4*π* ð*vet* 0 þ*L vet*<sup>0</sup> *dz*<sup>0</sup> *cos <sup>α</sup><sup>x</sup> <sup>z</sup>*<sup>0</sup> ½ � ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ � � � � � � 2 *t*0 <sup>¼</sup>*t*� *<sup>r</sup>* !� *<sup>r</sup>* !0 *<sup>t</sup>*0,*z*<sup>0</sup> j j ð Þ *c* � � � (22)

� *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet*

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

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet*

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

! � *r* !0 *t*<sup>0</sup>

� �

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

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

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

*cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � �

� *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet*

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

<sup>¼</sup> �*Qve*

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

2 *L*4*πc*

! � *r* !0 *t*<sup>0</sup>

� � �

� *Qve L*4*π* � � �

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L*

<sup>þ</sup> *Qve* 2 *L*4*πc*

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

*κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

� *Qve*

<sup>¼</sup> *Qve* 2 *L*4*πc*

! � *r* !0 *t*<sup>0</sup>

� � �

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

<sup>þ</sup> *Qve* 2 *L*4*πc*

*κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

*j dz t* 0

**63**

<sup>þ</sup> *Qve L*4*π*

� <sup>2</sup>*Qve* 2 *L*4*πc*

� � �

<sup>þ</sup> *Qve L*4*π*

*j p dy t* 0

� <sup>1</sup> *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

� <sup>2</sup>*Qve* 2 *L*4*πc*

� *Qve* 3 *L*4*πc*<sup>2</sup>

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

� � � <sup>0</sup> ½ � ð Þ *z* � *vet*

! � *r* !0 *t*<sup>0</sup>

� � �

> � � �

<sup>0</sup> ½ � ð Þ *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ ð Þ <sup>0</sup> þ *L* � � *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L*

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

! � *r* !0 *t*<sup>0</sup>

> � � �

<sup>2</sup> <sup>þ</sup> *Qve*

<sup>0</sup> ð Þ h i<sup>þ</sup>

*cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ½ � ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>*

! � *r* !0 *t*<sup>0</sup>

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

<sup>0</sup> ½ ð Þ þ *L* � � *z* � *vet*

� � �

� � �

0

� � �

*<sup>L</sup>*4*<sup>π</sup>* � *cos <sup>α</sup><sup>x</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

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

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

*cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet*

� � �

� �

*cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet*

� � �

!� ¼ *vet*<sup>0</sup>

� � �

h i<sup>þ</sup>

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L*

! � *<sup>r</sup>* !0 *t*<sup>0</sup>

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

<sup>0</sup> ð Þ � � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

3 *L*4*πc*<sup>2</sup>

� � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

<sup>0</sup> ð Þ � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

<sup>0</sup> ½ � ð Þ

! � *r* !0 *t*<sup>0</sup>

<sup>0</sup> ð Þ�þ

<sup>0</sup> ½ � ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� � � 3

<sup>0</sup> ½ ð Þ þ *L* � � *z* � *vet*

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

*κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

<sup>0</sup> ½ � ð Þ *z* � *vet*

Þ� *z* � *vet*<sup>0</sup> ð Þ�ð*t*<sup>0</sup>

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

<sup>0</sup> ð ð Þ þ *L* Þ�

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� � �

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

� � �

*cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � �

! � *<sup>r</sup>* !0 *t*<sup>0</sup>

*cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � �

> ! � *<sup>r</sup>* !0 *t*<sup>0</sup>

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

> *sin* <sup>2</sup> *<sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ ð Þ��

<sup>3</sup> *r* ! � *r* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L*

h

� � �

<sup>0</sup> ½ � ð Þ

<sup>0</sup> ½ ð Þ��

� � �

� � � 2 �

> � � � 2 �

� � � <sup>2</sup> �

� � �

*cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � �

> ! � *r* !0 *t*<sup>0</sup>

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup>

� � � Þ

<sup>2</sup> � *Qve*

� � � 2 � � � 2

<sup>0</sup> ð Þ ð Þ þ *L*

<sup>2</sup> (24)

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

3 *L*4*πc*<sup>2</sup>

<sup>2</sup> (25)

� � � � � � 3

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

The strengths of the electric fields in Eqs. (10)–(12) and magnetic fields with Eqs. (21) and (22), formed by the ends and the main part of the beam, decrease inversely proportional to the first and second powers of the distance from the source point to the observation point.

#### **5. Displacement current**

We take into account that the displacement current density *j* ! *<sup>d</sup>*ð Þ *t*,*r* ([7], p. 87):

$$
\stackrel{\rightarrow}{j}\_d(t, r) = \frac{\partial}{\partial t} \stackrel{\rightarrow}{D}\_d(t, r) = \frac{\partial}{\partial t} \epsilon\_0 \stackrel{\rightarrow}{E}(t, r), \tag{23}
$$

where the *D* ! *<sup>d</sup>*ð Þ¼ *t*,*r ε*0*E* ! ð Þ *t*,*r* is the electric displacement vector. Taking into account the Eqs. (10)–(12) and (23), we get

$$\vec{j}\_{\rm dx}^{p}(t', \mathbf{x}'=0, \mathbf{y}'=0, \mathbf{v}\_{\epsilon}t' < \mathbf{z}' < \mathbf{v}\_{\epsilon}t' + L; t, r(\mathbf{x}, \mathbf{y}, \mathbf{z})) =$$

$$= \frac{Q\upsilon\_{\epsilon}}{L4\pi c} \cos\left[a\_{\mathbf{x}}(\mathbf{z}' = \mathbf{v}\_{\epsilon}t')\right] \cdot \cos\left[a\_{\mathbf{z}}(\mathbf{z}' = \mathbf{v}\_{\epsilon}t')\right] \cdot \frac{1}{\kappa^{2}(\mathbf{z}' = \mathbf{v}\_{\epsilon}t') \left|\overrightarrow{r} - \overrightarrow{r}'(t', \mathbf{z}' = \mathbf{v}\_{\epsilon}t')\right|^{2}}$$

$$+ \frac{Q\upsilon\_{\epsilon}\imath^{3}}{L4\pi c^{2}} \frac{\cos\left[a\_{\mathbf{x}}(\mathbf{z}' = \mathbf{v}\_{\epsilon}t')\right]}{\kappa^{3}(\mathbf{z}' = \mathbf{v}\_{\epsilon}t') \left|\overrightarrow{r} - \overrightarrow{r}'(t', \mathbf{z}' = \mathbf{v}\_{\epsilon}t')\right|^{3}} \left|(\overrightarrow{r} - \overrightarrow{r}'(t', \mathbf{z}' = \mathbf{v}\_{\epsilon}t')\right|$$

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

*Hx t* 0

¼ � *Qve*

*Hy t* 0

<sup>¼</sup> *Qve* 2 *L*4*πc*

� *Qve* 2 *L*4*πc*

<sup>þ</sup> *Qve* 2 *L*4*πc*

*Progress in Relativity*

2 *L*4*πc*

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

*κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

*κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

ð*vet* 0 þ*L*

*vet*<sup>0</sup>

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

*κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

ð*vet* 0 þ*L*

*vet*<sup>0</sup>

*κ z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

<sup>þ</sup> *Qve L*4*π*

source point to the observation point.

**5. Displacement current**

!

*j p dx t* 0

<sup>þ</sup> *Qve* 3 *L*4*πc*<sup>2</sup>

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet*

p. 87):

where the *D*

<sup>¼</sup> *Qve* 2 *L*4*πc*

**62**

� *Qve L*4*π* 0 , *z*<sup>0</sup>

*cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � �

> ! � *r* !0 *t*<sup>0</sup>

*cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � �

*r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ

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

� � �

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ

> ! � *r* !0 *t*<sup>0</sup>

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L*

*r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ð Þ

� � �

We take into account that the displacement current density *j*

*∂ ∂t D* !

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

*<sup>d</sup>*ð Þ¼ *t*,*r*

*∂ ∂t ε*0*E* !

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

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

<sup>0</sup> ½ �� ð Þ <sup>1</sup>

� � �

<sup>3</sup> ½ *r* ! � *r* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ

� � �

ð Þ *t*,*r* is the electric displacement vector. Taking into

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

! � *r* !0 *t*<sup>0</sup>

� � �

*j* !

!

*<sup>d</sup>*ð Þ¼ *t*,*r ε*0*E*

account the Eqs. (10)–(12) and (23), we get

*<sup>d</sup>*ð Þ¼ *t*,*r*

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

<sup>0</sup> ½ �� ð Þ *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet*

� � �

*κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

*cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ

> ! � *r* !0 *t*<sup>0</sup>

*dz*<sup>0</sup> *cos <sup>α</sup><sup>x</sup> <sup>z</sup>*<sup>0</sup> ½ � ð Þ

The strengths of the electric fields in Eqs. (10)–(12) and magnetic fields with Eqs. (21) and (22), formed by the ends and the main part of the beam, decrease inversely proportional to the first and second powers of the distance from the

! � *r* !0 *t*<sup>0</sup>

� � �

� � � *dz*<sup>0</sup> *cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> ð Þ � �

! � *r* !0 *t*<sup>0</sup>

� � � � � �

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

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

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

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

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

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

� � �

� � �

> � � � *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*

� � �

> � � � *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*

� � �

� � � *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*

� � �

> � � � *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*

� � �

(21)

(22)

�

!

ð Þ *t*,*r* , (23)

*<sup>d</sup>*ð Þ *t*,*r* ([7],

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

� � � � � � 2

þ

� *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ *z* � *vet* <sup>0</sup> ð Þ�þ <sup>þ</sup> *Qve* 2 *L*4*πc cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � 2 � � <sup>2</sup>*Qve* 2 *L*4*πc cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ ð Þ <sup>0</sup> þ *L* � � *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � 2 � � *Qve* 3 *L*4*πc*<sup>2</sup> *cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � 3 � *r* ! � *r* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L* � � � � � � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ ð Þ þ *L* � � *z* � *vet* <sup>0</sup> ð Þ ð Þ þ *L* h i<sup>þ</sup> <sup>þ</sup> *Qve L*4*π cos α<sup>x</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � <sup>2</sup> � � *Qve <sup>L</sup>*4*<sup>π</sup>* � *cos <sup>α</sup><sup>x</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ � ð Þ *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � <sup>2</sup> (24) *j p dy t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ <sup>¼</sup> *Qve* 2 *L*4*πc cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ ð Þ�� � <sup>1</sup> *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � <sup>2</sup> <sup>þ</sup> *Qve* 3 *L*4*πc*<sup>2</sup> *cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � � *κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � 3 *r* ! � *<sup>r</sup>* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � � � � � � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ � ð Þ *z* � *vet* <sup>0</sup> ð Þ h i<sup>þ</sup> <sup>þ</sup> *Qve* 2 *L*4*πc cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ � ð Þ *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* !� ¼ *vet*<sup>0</sup> Þ� *z* � *vet*<sup>0</sup> ð Þ�ð*t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> Þ � � � � � � 2 � <sup>2</sup>*Qve* 2 *L*4*πc cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ½ � ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup> κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � <sup>2</sup> � *Qve* 3 *L*4*πc*<sup>2</sup> *cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � � *κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � <sup>3</sup> *r* ! � *r* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L* � � � � � � h � *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ ð Þ þ *L* � � *z* � *vet* <sup>0</sup> ð ð Þ þ *L* Þ� <sup>þ</sup> *Qve L*4*π cos <sup>α</sup><sup>y</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � � *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � 2 � *Qve L*4*π cos α<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � � *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *<sup>r</sup>* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � <sup>2</sup> (25) *j dz t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ <sup>¼</sup> �*Qve* 2 *L*4*πc sin* <sup>2</sup> *<sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ ð Þ��

� <sup>1</sup> *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � <sup>2</sup> <sup>þ</sup> *Qve* 3 *L*4*πc*<sup>2</sup> *cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ *κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � 3 � � *r* ! � *r* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ � � � � � � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ � ð Þ *z* � *vet* <sup>0</sup> ð Þ h i <sup>þ</sup> *Qve* 2 *L*4*πc cos* <sup>2</sup> *<sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ � ð Þ *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ � � � � � � 2 � *Qve* 2 *L*4*πc sin* <sup>2</sup> *<sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ½ � ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup> κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � <sup>2</sup> <sup>þ</sup> *Qve* 3 *L*4*πc*<sup>2</sup> � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ½ � ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup> κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � 3 � ½ *r* ! � *r* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L* � � � � � � � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ ð Þ þ *L* �� � *z* � *vet* <sup>0</sup> <sup>ð</sup> ð Þ <sup>þ</sup> *<sup>L</sup>* Þ� þ *Qve* 2 *L*4*πc cos* <sup>2</sup> *<sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ½ � ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup> κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r* ! � *r* !0 *t*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* � � � � � � <sup>2</sup> �

**6. Flux of electrical energy**

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

*Sψ <sup>x</sup> t* 0

¼ *ψ t* 0

¼ *ψ t* 0

¼ *ψ t* 0

The electrical energy flux density *S*

is determined by the formula ([3], p. 259)

*Sx t* 0

¼ �*Ez t* 0

> �*Hy t* 0

*Hy z*<sup>0</sup> ¼ *vet*

**65**

*j p dz t* 0

**7. Pointing vector**

�*j p dy t* 0

*Sψ <sup>z</sup> t* 0

*j p dx t* 0

*Sψ <sup>y</sup> t* 0

write

The electrical energy flux density per unit time *S*

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

p. 125) Eq. (15) and [11] Eqs. (7) and (8), has the form

*S* !*<sup>ψ</sup>*

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

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

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

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

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

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

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

*S* !

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

¼ �f*E<sup>p</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>E</sup><sup>r</sup>*

<sup>þ</sup>*E<sup>r</sup>*

<sup>0</sup> ð Þþ *Hy z*<sup>0</sup> ¼ *vet*

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

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>E</sup><sup>p</sup>*

<sup>0</sup> ð Þþþ <sup>þ</sup> *<sup>L</sup> <sup>H</sup><sup>c</sup>*

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

help of Eqs. (12) and (22) may be written as follows:

ð Þ¼ *t*,*r E* !

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

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

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

ð Þ¼ *t*,*r ψ*ð Þ� *t*,*r j*

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

0

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

0

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

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

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

!*<sup>ψ</sup>*

third power of the distance from the source point to the observation point. The electrical energy flux per unit time into a given solid angle decreases inversely proportional to the first power of the distance from the source point to the observation point. The flux takes place both in the near and the intermediate zones.

The Poynting vector or the flux density of electromagnetic energy per unit time

ð Þ� *t*,*r H* !

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

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

> *<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L*

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

, *vet* <sup>0</sup> ð þ *L*Þg�

> , *vet* <sup>0</sup> ð Þ þ *L*

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

The Poynting vector along the *Ox* axis estimated according to Eq. (31) with the

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

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

n o (32)

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

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

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

Taking into account the Eq. (5) or the Eq. (7) and the Eqs. (24)–(26), we can

!*<sup>ψ</sup>*

!

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

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

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

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

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

, *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z* (28)

, *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z* (29)

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

, *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z* (30)

ð Þ *t*,*r* , according to ([10],

*<sup>d</sup>*ð Þ *t*,*r* (27)

ð Þ *t*,*r* decreases inversely proportional to the

ð Þ *t*,*r* (31)

$$\begin{aligned} \frac{Q\nu\epsilon^{4}}{L4\pi^{3}}\frac{1}{\kappa^{3}(\mathbf{z}'=\nu\_{\epsilon}t')}\left|\overline{r}-\overline{r}'(\mathbf{z}',\mathbf{z}'=\nu\_{\epsilon}t')\right|^{3} & \left[-\cos\left[a\_{\epsilon}(\mathbf{z}'=\nu\_{\epsilon}t')\right](\mathbf{z}-\nu\_{\epsilon}t')\right] - \\\\ -\frac{Q\nu\_{\epsilon}\overline{\epsilon}^{3}}{L4\pi\kappa^{2}}\frac{\cos\left[a\_{\epsilon}(\mathbf{z}'=\nu\_{\epsilon}t')\right]}{\kappa^{2}(\mathbf{z}'=\nu\_{\epsilon}t')\left|\overline{r}-\overline{r}'(\mathbf{t}',\mathbf{z}'=\nu\_{\epsilon}t')\right|^{2}} \\\\ +\frac{Q\nu\_{\epsilon}\overline{\epsilon}^{4}}{L4\pi\kappa^{3}}\kappa^{3}(\mathbf{z}'=\nu\_{\epsilon}t'+L)\left|\overline{r}-\overline{r}'(\mathbf{t}',\mathbf{z}'=\nu\_{\epsilon}t'+L)\right|^{3} \\\\ \cdot\left[\left|\overline{r}-\overline{r}'(\mathbf{t}',\mathbf{z}'=\nu\_{\epsilon}t'+L)\right|-\cos\left[a\_{\epsilon}(\mathbf{z}'=\nu\_{\epsilon}t'+L)\right](\mathbf{z}-(\nu\_{\epsilon}t'+L))\right]+\\ +\frac{Q\nu\_{\epsilon}\overline{\epsilon}^{3}}{L4\pi\kappa^{2}}\kappa^{2}(\mathbf{z}'=\nu\_{\epsilon}t'+L)\left|\overline{r}-\overline{r}'(\mathbf{t}',\mathbf{z}'=\nu\_{\epsilon}t'+L)\right|^{2} \\\\ -\frac{Q\nu\_{\epsilon}}{L4\pi}\frac{\cos\left[a\_{\epsilon}(\mathbf{z}'=\nu\_{\epsilon}t')\right]}{\kappa(\mathbf{z}'-\nu\_{\epsilon}t')} \end{aligned} \tag{26}$$

The transverse components of the displacement current density *j p dx t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* and *j p dy t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* are potential with respect to space coordinates, and the longitudinal component *j dz t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* consists of potential and dynamic components. Displacement current densities are decreasing inversely proportional to the second power of the distance from the source point to the observation point.

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

#### **6. Flux of electrical energy**

� <sup>1</sup> *κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

*Progress in Relativity*

! � *r* !0 *t*<sup>0</sup>

> <sup>þ</sup> *Qve* 2 *L*4*πc*

� � �

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

� *Qve* 2 *L*4*πc*

� � � , *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

� ½ *r* ! � *r* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L*

<sup>0</sup> <sup>ð</sup> ð Þ <sup>þ</sup> *<sup>L</sup>* Þ� þ *Qve*

1

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

*κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

<sup>þ</sup> *Qve* 3 *L*4*πc*<sup>2</sup>

<sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* and *j*

� *Qve L*4*π*

! � *r* !0 *t*<sup>0</sup>

� � �

� *z* � *vet*

*κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

� *Qve* 3 *L*4*πc*<sup>2</sup>

<sup>þ</sup> *Qve* 4 *L*4*πc*<sup>3</sup>

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

� � �

*Qve* 4 *L*4*πc*<sup>3</sup>

*j p dx t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t*

*j dz t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t*

**64**

� � �

� *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ½ � ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>*

2 *L*4*πc*

> � � � <sup>3</sup> *r* ! � *r* !0 *t* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ

> ! � *r* !0 *t*<sup>0</sup>

> > � � �

> > > � �

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

*κ z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

� � �

1

! � *r* !0 *t*<sup>0</sup>

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

� � � � � �

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

� � �

> � �

> > � � �

<sup>2</sup> <sup>þ</sup> *Qve*

h i

*cos* <sup>2</sup> *<sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* <sup>0</sup> ½ � ð Þ

> ! � *r* !0 *t*<sup>0</sup>

*sin* <sup>2</sup> *<sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ½ � ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>*

! � *r* !0 *t*<sup>0</sup>

� � �

! � *r* !0 *t*<sup>0</sup>

3 *L*4*πc*<sup>2</sup>

� � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

*κ*<sup>3</sup> *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ *r*

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

*cos* <sup>2</sup> *<sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ½ � ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>*

� �

> � � � 3 �

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� � �

<sup>0</sup> ½ � ð Þ þ *L z* � *vet*

! � *r* !0 *t*<sup>0</sup>

� � �

� � � 2

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* ½ � ð Þ <sup>0</sup> þ *L*

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

! � *r* !0 *t*<sup>0</sup>

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� �

*κ*<sup>2</sup> *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L r*

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

� � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

h i

� � �

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ

> ! � *<sup>r</sup>* !0 *t*<sup>0</sup>

� � �

> *p dy t* 0 ,*r*<sup>0</sup> *x*<sup>0</sup> , *y*<sup>0</sup> , *z*<sup>0</sup> *t* <sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* are

potential with respect to space coordinates, and the longitudinal component

<sup>0</sup> ð Þ ð Þ ð Þ ; *t*,*r x*ð Þ , *y*, *z* consists of potential and dynamic components. Displacement current densities are decreasing inversely proportional to the second

The transverse components of the displacement current density

power of the distance from the source point to the observation point.

<sup>0</sup> ½ � ð Þ *z* � *vet*

� � � 2

� � � 3 � � �

<sup>0</sup> ½ ð Þ þ *L* ��

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L*

� � *cos <sup>α</sup><sup>z</sup> <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

<sup>0</sup> ð Þ ð Þ þ *L*

� � � 2

<sup>2</sup> (26)

h i

� � � <sup>2</sup> �

<sup>0</sup> ½ � ð Þ *z* � *vet*

<sup>0</sup> ð Þ

þ

�

<sup>2</sup> <sup>þ</sup> *Qve*

3 *L*4*πc*<sup>2</sup> �

*cos α<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ½ � ð Þ

> ! � *r* !0 *t*<sup>0</sup>

<sup>0</sup> ð Þ

� � �

, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ð Þ

� � � 3 �

The electrical energy flux density per unit time *S* !*<sup>ψ</sup>* ð Þ *t*,*r* , according to ([10], p. 125) Eq. (15) and [11] Eqs. (7) and (8), has the form

$$
\overrightarrow{\boldsymbol{S}}^{\psi}(t, r) = \boldsymbol{\psi}(t, r) \cdot \overrightarrow{\boldsymbol{j}}\_d(t, r) \tag{27}
$$

Taking into account the Eq. (5) or the Eq. (7) and the Eqs. (24)–(26), we can write

*Sψ <sup>x</sup> t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ ¼ *ψ t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ� *j p dx t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z* (28) *Sψ <sup>y</sup> t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ ¼ *ψ t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ� �*j p dy t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z* (29) *Sψ <sup>z</sup> t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ ¼ ¼ *ψ t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*; *t*,*r x*ð Þ , *y*, *z* Þ� *j p dz t* 0 , *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð Þ þ *L*; *t*,*r x*ð Þ , *y*, *z* (30)

The electrical energy flux density *S* !*<sup>ψ</sup>* ð Þ *t*,*r* decreases inversely proportional to the third power of the distance from the source point to the observation point. The electrical energy flux per unit time into a given solid angle decreases inversely proportional to the first power of the distance from the source point to the observation point. The flux takes place both in the near and the intermediate zones.

#### **7. Pointing vector**

The Poynting vector or the flux density of electromagnetic energy per unit time is determined by the formula ([3], p. 259)

$$
\overrightarrow{S}(t, r) = \overrightarrow{E}(t, r) \times \overrightarrow{H}(t, r) \tag{31}
$$

The Poynting vector along the *Ox* axis estimated according to Eq. (31) with the help of Eqs. (12) and (22) may be written as follows:

$$S\_x(t', \mathbf{x}'=0, \mathbf{y}'=0, \mathbf{v}\_\epsilon t' < \mathbf{z}' < \mathbf{v}\_\epsilon t' + L; t, r(\mathbf{x}, \mathbf{y}, \mathbf{z})) = $$

$$= -E\_x(t', \mathbf{x}'=0, \mathbf{y}'=0, \mathbf{v}\_\epsilon t' < \mathbf{z}' < \mathbf{v}\_\epsilon t' + L; t, r(\mathbf{x}, \mathbf{y}, \mathbf{z})) \cdot$$

$$\cdot H\_\mathcal{Y}(t', \mathbf{x}'=0, \mathbf{y}'=0, \mathbf{v}\_\epsilon t' < \mathbf{z}' < \mathbf{v}\_\epsilon t' + L; t, r(\mathbf{x}, \mathbf{y}, \mathbf{z}))$$

$$= -\left\{ E\_x^p(\mathbf{z}'=v\_\epsilon t') + E\_x^p(\mathbf{z}'=v\_\epsilon t' + L) \right.$$

$$\left. \quad + E\_x^r(\mathbf{z}'=v\_\epsilon t') + E\_x^r(v\_\epsilon t' < \mathbf{z}' < v\_\epsilon t' + L) \right\}.$$

$$\left\{ H\_\mathcal{Y}(\mathbf{z}'=v\_\epsilon t') + H\_\mathcal{Y}(\mathbf{z}'=v\_\epsilon t' + L) + \left. + H\_\mathcal{Y}^c(v\_\epsilon t' < \mathbf{z}' < v\_\epsilon t' + L) \right| \right\} \tag{32}$$

�*E<sup>p</sup>*

where the summands in curly brackets are defined by Eq. (12) and Eq. (22), respectively. Rewriting the Eq. (32) in the following form:

$$\begin{split} \mathbf{S}\_{\mathbf{x}}(\mathbf{t}', \mathbf{x}'=0, \mathbf{y}'=0, \mathbf{y}, \mathbf{t}' \leq \mathbf{z}' \leq \mathbf{v}\_{\epsilon}\mathbf{t}'+L; \mathbf{t}, \mathbf{r}(\mathbf{x}, \mathbf{y}, \mathbf{z})) \\ = \mathbf{^{i}S\_{\mathbf{x}}}(\mathbf{z}' = \mathbf{v}\_{\epsilon}\mathbf{z}', \mathbf{z}' = \mathbf{v}\_{\epsilon}\mathbf{t}'+L) + \mathbf{^{pi}S\_{\mathbf{x}}}(\mathbf{z}' = \mathbf{v}\_{\epsilon}\mathbf{t}', \mathbf{z}' = \mathbf{v}\_{\epsilon}\mathbf{t}' + L, \mathbf{v}\_{\epsilon}\mathbf{z}' \leq \mathbf{z}' \leq \mathbf{v}\_{\epsilon}\mathbf{t}' + L) \\ + \mathbf{^{f}S\_{\mathbf{x}}}(\mathbf{v}\_{\epsilon}\mathbf{t}' < \mathbf{z}' < \mathbf{v}\_{\epsilon}\mathbf{t}' + L), \end{split} \tag{33}$$

where the *<sup>i</sup> Sx z*<sup>0</sup> ¼ *vet*<sup>0</sup> , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* there is a flux of electromagnetic energy in a unit time that goes into the wave zone, the *piSx <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> , *z* ð <sup>0</sup> ¼ *vet*<sup>0</sup> þ *L*, *vet*<sup>0</sup> , *z*<sup>0</sup> , *vet*<sup>0</sup> þ *L*Þ there is a flux of electromagnetic energy in the intermediate zone, the *<sup>f</sup> Sc <sup>x</sup> vet*<sup>0</sup> , *z*<sup>0</sup> , *vet* ð Þ <sup>0</sup> þ *L* there is a flux of electromagnetic energy in the near zone. As this takes place

$$\begin{aligned} \ ^i\mathbf{S}\_{\mathbf{x}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{c}}t',\mathbf{z}'=\mathbf{v}\_{\mathbf{c}}t'+L) &= \ ^i\mathbf{S}\_{\mathbf{x}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{c}}t')+\\ \ + \ ^i\mathbf{S}\_{\mathbf{x}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{c}}t'+L) + \ ^i\mathbf{S}\_{\mathbf{x}}^{yA}(\mathbf{z}'=\mathbf{v}\_{\mathbf{c}}t',\mathbf{z}'=\mathbf{v}\_{\mathbf{c}}t'+L), \end{aligned} \tag{34}$$

*i*

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

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*Sy z*<sup>0</sup> ¼ *vet*

<sup>¼</sup> *<sup>E</sup><sup>p</sup>*

*i*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þþ *<sup>E</sup><sup>r</sup>*

0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ ¼ *<sup>E</sup><sup>p</sup>*

<sup>¼</sup> *<sup>E</sup><sup>p</sup>*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

�*Hx z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup>*

> *<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup>*

*i S<sup>ψ</sup><sup>A</sup>*

*Ep*

<sup>þ</sup>*E<sup>c</sup> <sup>z</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ � *Hx z*<sup>0</sup> ¼ *vet*

*Sz t* 0

<sup>¼</sup> *<sup>E</sup><sup>p</sup>*

*Ep*

<sup>þ</sup>*E<sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup>

�*E<sup>p</sup>*

�*E<sup>c</sup> <sup>y</sup> vet* 0 , *z*<sup>0</sup>

**67**

*f Sc <sup>z</sup> vet* 0

*<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup>*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup>*

<sup>þ</sup>*piSz <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

<sup>¼</sup> *<sup>E</sup><sup>p</sup>*

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

*i S<sup>ψ</sup><sup>A</sup>*

<sup>þ</sup>*E<sup>p</sup>*

*f Sc <sup>y</sup> vet* 0 , *z*<sup>0</sup>

<sup>þ</sup>*E<sup>r</sup>*

*Sy z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼ *<sup>i</sup>*

<sup>0</sup> ð Þ¼ <sup>þ</sup> *<sup>L</sup> <sup>i</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*piSy <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

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

(22), and (31), may be written as follows:

*i*

þ*i*

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� þ *L Hy z*<sup>0</sup> ¼ *vet*

*<sup>y</sup> vet* 0

<sup>0</sup> ð þ *L*Þ � *Hy z*<sup>0</sup> ¼ *vet*

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

<sup>0</sup> ð þ *L*Þ � *Hx z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

, *vet*

, *vet*

, *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ ¼ *<sup>E</sup><sup>c</sup>*

> �*Ec <sup>e</sup> vet* 0

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

*<sup>x</sup> vet* 0

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*޼�*E<sup>c</sup>*

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

0 , *z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� þ *L Hx z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet*

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

*S<sup>ψ</sup><sup>A</sup>*

<sup>0</sup> ð Þþ *<sup>E</sup><sup>r</sup>*

<sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>r</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

*S<sup>ψ</sup><sup>A</sup>*

<sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet*

0 , *z*<sup>0</sup> ¼ *vet*

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *<sup>E</sup><sup>r</sup>*

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

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

> 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ ¼ *<sup>i</sup>*

<sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>i</sup>*

*Sz z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼ *<sup>i</sup>*

<sup>0</sup> ð Þ¼ <sup>þ</sup> *<sup>L</sup> <sup>i</sup>*

<sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>p</sup>*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þþ *<sup>E</sup><sup>c</sup>*

<sup>0</sup> ð Þ� *<sup>E</sup><sup>c</sup>*

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

<sup>0</sup> ð Þ� *<sup>E</sup><sup>p</sup>*

<sup>0</sup> ð Þ� *<sup>E</sup><sup>p</sup>*

<sup>0</sup> þ *L*, *vet* 0 , *z*<sup>0</sup>

*Sz z*<sup>0</sup> ¼ *vet*

*i*

<sup>0</sup> ð Þ� *Hy z*<sup>0</sup> ¼ *vet*

*Sz z*<sup>0</sup> ¼ *vet*

*i*

0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ ¼ *<sup>E</sup><sup>p</sup>*

<sup>0</sup> ð Þ� þ *L Hy z*<sup>0</sup> ¼ *vet*

*piSz <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

�*E<sup>p</sup>*

, *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *<sup>E</sup><sup>p</sup>*

0 , *z*<sup>0</sup> ¼ *vet*

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>E</sup><sup>p</sup>*

> , *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>H</sup><sup>c</sup>*

*Sz z*<sup>0</sup> ¼ *vet*

<sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *<sup>f</sup>*

<sup>0</sup> ð Þþ *<sup>E</sup><sup>c</sup>*

, *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *<sup>E</sup><sup>p</sup>*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>i</sup>*

*SA*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>i</sup>*

<sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>r</sup>*

<sup>0</sup> þ *L*, *vet* 0

*<sup>z</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ � *Hx z*<sup>0</sup> ¼ *vet*

, *vet*

*Sψ<sup>A</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

*Sψ<sup>A</sup>*

*Sψ<sup>A</sup>*

<sup>0</sup> ð Þ� þ *L Hx z*<sup>0</sup> ¼ *vet*

<sup>0</sup> þ *L*, *vet* 0 , *z*<sup>0</sup>

<sup>0</sup> ð þ *L*Þ ¼

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

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

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

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>H</sup><sup>c</sup>*

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

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>H</sup><sup>c</sup>*

The Poynting vector along the *Oz* axis, taking into account Eqs. (10), (11), (21),

<sup>0</sup> ð þ *L*Þ ¼

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� þ *L Hx z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>p</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

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

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

0 , *z*<sup>0</sup> ¼ *vet*

*Sz z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ

<sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� þ *L Hx z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ¼ þ *L*

<sup>0</sup> ð Þ� *Hy z*<sup>0</sup> ¼ *vet*

, *vet*

, *vet*

, *vet*

<sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>H</sup><sup>c</sup>*

<sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>H</sup><sup>c</sup>*

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

<sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>H</sup><sup>c</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup>*

<sup>0</sup> ð Þ¼ þ *L*

<sup>0</sup> ð Þ (42)

<sup>0</sup> ð Þ þ *L* ,

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð þ *L*Þ�

<sup>0</sup> , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þþ

, *vet* <sup>0</sup> ð þ *L*Þþ

<sup>0</sup> ð Þ þ *L* , (45)

<sup>0</sup> ð Þ� þ *L Hx z*<sup>0</sup> ¼ *vet*

, *vet* <sup>0</sup> ð Þ þ *L :* (46)

0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð þ *L*Þþ

, *vet* <sup>0</sup> ð Þ þ *L* , (47)

<sup>0</sup> ð þ *L*Þþ

<sup>0</sup> ð Þ (51)

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

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

<sup>0</sup> ð Þ þ *L :* (53)

, *vet* <sup>0</sup> ð þ *L*Þ�

<sup>0</sup> ð þ *L*Þ � *Hy z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð þ *L*Þ � *Hx z*<sup>0</sup> ¼ *vet*

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

, *vet*

<sup>0</sup> ð þ *L*Þ�

<sup>0</sup> ð Þ (49)

<sup>0</sup> ð Þ þ *L* (50)

, *vet* <sup>0</sup> ð þ *L*Þþ

<sup>0</sup> ð þ *L*Þ�

<sup>0</sup> ð Þ þ *L* ,

(52)

, *vet* <sup>0</sup> ð þ *L*Þ�

*<sup>x</sup> vet*

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

*Sz z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ þ *L* (48)

(43)

<sup>0</sup> ð Þ, (44)

<sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet*

*SA*

$$\begin{aligned} \ ^i\mathbf{S}\_x(\mathbf{z}'=\mathbf{v}\_\epsilon t') &= \, ^i\mathbf{S}\_x^{yA}(\mathbf{z}'=\mathbf{v}\_\epsilon t') + \, ^i\mathbf{S}\_x^A(\mathbf{z}'=\mathbf{v}\_\epsilon t') = \\ \ = -\mathbf{E}\_x^p(\mathbf{z}'=\mathbf{v}\_\epsilon t') \cdot \mathbf{H}\_p(\mathbf{z}'=\mathbf{v}\_\epsilon t') - \mathbf{E}\_x^r(\mathbf{z}'=\mathbf{v}\_\epsilon t') \cdot \mathbf{H}\_p(\mathbf{z}'=\mathbf{v}\_\epsilon t'), \end{aligned} \tag{35}$$

$${}^{i}\mathbf{S}\_{\mathbf{x}}(\mathbf{z}^{\prime}=\mathbf{v}\_{\epsilon}\mathbf{t}^{\prime}+L)={}^{i}\mathbf{S}\_{\mathbf{x}}^{\prime A}(\mathbf{z}^{\prime}=\mathbf{v}\_{\epsilon}\mathbf{t}^{\prime}+L)+{}^{i}\mathbf{S}\_{\mathbf{x}}^{A}(\mathbf{z}^{\prime}=\mathbf{v}\_{\epsilon}\mathbf{t}^{\prime}+L)=$$

$${}^{i}\mathbf{S}\_{\mathbf{z}}^{\prime}(\mathbf{z}^{\prime}=\mathbf{v}\_{\epsilon}\mathbf{t}^{\prime}+L)\cdot{H}\_{\mathbf{y}}(\mathbf{z}^{\prime}=\mathbf{v}\_{\epsilon}\mathbf{t}^{\prime}+L)-{}^{i}\mathbf{E}\_{\mathbf{z}}^{\prime}(\mathbf{z}^{\prime}=\mathbf{v}\_{\epsilon}\mathbf{t}^{\prime}+L)\cdot{H}\_{\mathbf{y}}(\mathbf{z}^{\prime}=\mathbf{v}\_{\epsilon}\mathbf{t}^{\prime}+L),\tag{36}$$

$$\begin{cases} ^i\mathbf{S}\_x^{\rm va}(\mathbf{z}'=\mathbf{v}\_\varepsilon t', \mathbf{z}'=\mathbf{v}\_\varepsilon t'+L) = -\mathbf{E}\_x^p(\mathbf{z}'=\mathbf{v}\_\varepsilon t') \cdot \mathbf{H}\_y(\mathbf{z}'=\mathbf{v}\_\varepsilon t'+L) - \mathbf{E}\_x^p(\mathbf{z}'=\mathbf{v}\_\varepsilon t'+L) \\ \qquad \cdot \mathbf{H}\_y(\mathbf{z}'=\mathbf{v}\_\varepsilon t') - \mathbf{E}\_x^r(\mathbf{z}'=\mathbf{v}\_\varepsilon t') \cdot \mathbf{H}\_y(\mathbf{z}'=\mathbf{v}\_\varepsilon t'+L) \\ \qquad - \mathbf{E}\_x^r(\mathbf{z}'=\mathbf{v}\_\varepsilon t'+L) \cdot \mathbf{H}\_y(\mathbf{z}'=\mathbf{v}\_\varepsilon t'). \end{cases} \tag{37}$$

The energy fluxes, *<sup>i</sup> Sx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ, *<sup>i</sup> Sx <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* , *<sup>i</sup> Sψ<sup>A</sup> <sup>x</sup> z*<sup>0</sup> ¼ *vet* 0 , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* , are determined by point sources of radiation at the REB segment beginning, the REB segment end, and the REB segment interference, respectively.

$$\mathbb{P}^{\mathbb{P}}\mathbb{S}\_{\mathbf{x}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{t}}t',\mathbf{z}'=\mathbf{v}\_{\mathbf{t}}t'+L,\mathbf{v}\_{\mathbf{t}}\mathbf{z}'<\mathbf{z}'<\mathbf{v}\_{\mathbf{t}}t'+L)=$$

$$-E\_{\mathbf{z}}^{\mathbb{P}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{t}}t')\cdot H\_{\mathbf{y}}^{\epsilon}(\mathbf{v}\_{\mathbf{t}}t'<\mathbf{z}'<\mathbf{v}\_{\mathbf{t}}t'+L)-E\_{\mathbf{z}}^{\mathbb{P}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{t}}t'+L)\cdot H\_{\mathbf{y}}^{\epsilon}(\mathbf{v}\_{\mathbf{t}}t'<\mathbf{z}'<\mathbf{v}\_{\mathbf{t}}t'+L))$$

$$-E\_{\mathbf{z}}^{\epsilon}(\mathbf{z}'=\mathbf{v}\_{\mathbf{t}}t')\cdot H\_{\mathbf{y}}^{\epsilon}(\mathbf{v}\_{\mathbf{t}}t'<\mathbf{z}'<\mathbf{v}\_{\mathbf{t}}t'+L)-E\_{\mathbf{z}}^{\epsilon}(\mathbf{z}'=\mathbf{v}\_{\mathbf{t}}t')\cdot H\_{\mathbf{y}}^{\epsilon}(\mathbf{v}\_{\mathbf{t}}t'<\mathbf{z}'<\mathbf{v}\_{\mathbf{t}}t'+L)$$

$$E\_{\mathbf{z}}^{\epsilon}(\mathbf{v}\_{\mathbf{t}}t'<\mathbf{z}'<\mathbf{v}\_{\mathbf{t}}t'+L)\cdot H\_{\mathbf{y}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{t}}t')-E\_{\mathbf{z}}^{\epsilon}(\mathbf{v}\_{\mathbf{t}}t'<\mathbf{z}'<\mathbf{v}\_{\mathbf{t}}t'+L)\cdot H\_{\mathbf{y}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{t}}t'+L). \tag{38}$$

$$\mathbf{u}^{\ell} \cdot \mathbf{S}\_{\mathbf{x}}^{\varepsilon} (v\_{\varepsilon} t' \le \mathbf{z}' \le v\_{\varepsilon} t' + L) = -E\_{\mathbf{z}}^{\varepsilon} (v\_{\varepsilon} t' \le \mathbf{z}' \le v\_{\varepsilon} t' + L) \cdot H\_{\mathbf{y}}^{\varepsilon} (v\_{\varepsilon} t' \le \mathbf{z}' \le v\_{\varepsilon} t' + L). \tag{39}$$

The Poynting vector along the *Oy* axis, taking into account Eqs. (12), (21), (31), similarly to Eqs. (33)–(39), is represented by:

$$\mathbf{S}\_{\mathcal{V}}(\mathbf{z}',\mathbf{x}'=0,\mathbf{y}'=0,\mathbf{v}\_{\mathbf{z}}\mathbf{t}'<\mathbf{z}'<\mathbf{v}\_{\mathbf{z}}\mathbf{t}'+L;\mathbf{i};\mathbf{r}(\mathbf{x},\mathbf{y},\mathbf{z}))=\mathbf{i}^{\dagger}\mathbf{S}\_{\mathcal{V}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}',\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}'+L)+\cdots$$

$$+\mathbf{P}\_{\mathcal{V}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}',\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}'+L,\mathbf{v}\_{\mathbf{z}}\mathbf{t}'<\mathbf{z}'<\mathbf{v}\_{\mathbf{z}}\mathbf{t}'+L)+\mathbf{j}^{\dagger}\mathbf{S}\_{\mathcal{V}}^{\prime}(\mathbf{v}\_{\mathbf{z}}\mathbf{t}'<\mathbf{z}'<\mathbf{v}\_{\mathbf{z}}\mathbf{t}'+L),\tag{40}$$

$$\mathbf{i}^{\dagger}\mathbf{S}\_{\mathcal{V}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}',\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}'+L)=\mathbf{i}^{\dagger}\mathbf{S}\_{\mathcal{V}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}')+\mathbf{i}^{\dagger}\mathbf{S}\_{\mathcal{V}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}'+L)+$$

$$+\mathbf{i}^{\dagger}\mathbf{S}\_{\mathcal{V}}^{\mathrm{q}\mathbf{A}}(\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}',\mathbf{z}'=\mathbf{v}\_{\mathbf{z}}\mathbf{t}'+L)\tag{41}$$

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

where the summands in curly brackets are defined by Eq. (12) and Eq. (22),

*L*, *vet*<sup>0</sup> , *z*<sup>0</sup> , *vet*<sup>0</sup> þ *L*Þ there is a flux of electromagnetic energy in the intermediate

*Sψ<sup>A</sup>*

*<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>i</sup>*

*Sψ<sup>A</sup>*

<sup>0</sup> ð Þ� *<sup>E</sup><sup>r</sup>*

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

are determined by point sources of radiation at the REB segment beginning, the

<sup>0</sup> þ *L*, *vet* 0 , *z*<sup>0</sup>

<sup>0</sup> ð þ *L*Þ ¼

*<sup>z</sup> vet*

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>H</sup><sup>c</sup>*

The Poynting vector along the *Oy* axis, taking into account Eqs. (12), (21), (31),

*Sy z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>i</sup>*

, *vet*

0 , *z*<sup>0</sup> ¼ *vet*

*<sup>y</sup> z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup>*

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

*Sc <sup>y</sup> vet* 0 , *z*<sup>0</sup>

<sup>0</sup> ð Þ� *<sup>E</sup><sup>r</sup>*

� *Hy z*<sup>0</sup> ¼ *vet*

*Sψ<sup>A</sup>*

<sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>r</sup>*

� *<sup>E</sup><sup>r</sup>*

REB segment end, and the REB segment interference, respectively.

*Sx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ, *<sup>i</sup>*

> 0 , *z*<sup>0</sup> ¼ *vet*

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>E</sup><sup>p</sup>*

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>E</sup><sup>r</sup>*

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

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

> <sup>0</sup> þ *L*, *vet* 0 , *z*<sup>0</sup>

> > þ*i S<sup>ψ</sup><sup>A</sup>*

<sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *<sup>f</sup>*

<sup>0</sup> ð Þ� *<sup>E</sup><sup>c</sup>*

*piSx <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

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

similarly to Eqs. (33)–(39), is represented by:

0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ ¼ *<sup>i</sup>*

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

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

<sup>0</sup> ð þ *L*Þ � *Hy z*<sup>0</sup> ¼ *vet*

0 , *z*<sup>0</sup> ¼ *vet*

, *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* there is a flux of electromagnetic energy

*Sx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ

> *<sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼

> > *<sup>x</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>p</sup>*

<sup>0</sup> ð Þ*:*

<sup>0</sup> ð Þ� *Hy z*<sup>0</sup> ¼ *vet*

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

0 , *z*<sup>0</sup> ¼ *vet* ð Þ <sup>0</sup> þ *L* ,

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

, *vet* <sup>0</sup> ð Þ þ *L :* (39)

0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð þ *L*Þþ

, *vet* <sup>0</sup> ð Þ þ *L* , (40)

<sup>0</sup> ð þ *L*Þþ

*<sup>y</sup> vet* 0

<sup>0</sup> ð þ *L*Þ � *Hy z*<sup>0</sup> ¼ *vet*

*Sy z*<sup>0</sup> ¼ *vet*

*Sy z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ þ *L* (41)

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

<sup>0</sup> ð Þ� *Hy z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� þ *L Hy z*<sup>0</sup> ¼ *vet*

*Sψ<sup>A</sup>*

, *vet*

<sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>H</sup><sup>c</sup>*

*SA*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

0 , *z*<sup>0</sup> ¼ *vet*

*SA*

*<sup>x</sup> vet*<sup>0</sup> , *z*<sup>0</sup> , *vet* ð Þ <sup>0</sup> þ *L* there is a flux of electromagnetic energy in the

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>i</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� *Hy z*<sup>0</sup> ¼ *vet*

*Sx <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* ð Þ <sup>0</sup> <sup>þ</sup> *<sup>L</sup>* , *<sup>i</sup>*

<sup>0</sup> ð Þ� þ *L Hy z*<sup>0</sup> ¼ *vet*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> þ *L*, *vet* 0 , *z*<sup>0</sup>

, *z* ð <sup>0</sup> ¼ *vet*<sup>0</sup> þ

<sup>0</sup> ð Þ þ *L* , (34)

<sup>0</sup> ð Þ¼ þ *L*

<sup>0</sup> ð Þ, (35)

<sup>0</sup> ð Þ þ *L* , (36)

<sup>0</sup> ð Þ þ *L*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L*

, *vet* <sup>0</sup> ð Þ þ *L*

<sup>0</sup> ð Þ þ *L :*

(38)

, *z*<sup>0</sup> , *vet* <sup>0</sup> ð Þ þ *L*

(37)

<sup>0</sup> ð Þ þ *L*

, *vet*

(33)

respectively. Rewriting the Eq. (32) in the following form:

<sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *piSx <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

in a unit time that goes into the wave zone, the *piSx <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup>

0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ ¼ *<sup>i</sup>*

<sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>i</sup>*

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

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

, *vet* <sup>0</sup> ð Þ þ *L* ,

*Sx t* 0

zone, the *<sup>f</sup>*

�*E<sup>p</sup>*

�*E<sup>p</sup>*

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

*f Sc <sup>x</sup> vet* 0 , *z*<sup>0</sup>

*Sy t* 0

**66**

<sup>þ</sup>*piSy <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*

*i*

�*E<sup>r</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup>*

> *<sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup>*

> > , *vet*

, *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*޼�*E<sup>c</sup>*

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

0 , *z*<sup>0</sup> ¼ *vet*

*Sy z*<sup>0</sup> ¼ *vet*

*i Sψ<sup>A</sup>*

¼ *i*

þ *f Sc <sup>x</sup> vet* 0 , *z*<sup>0</sup>

*Progress in Relativity*

where the *<sup>i</sup>*

*Sc*

¼ �*E<sup>p</sup>*

*i*

0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*޼�*E<sup>p</sup>*

The energy fluxes, *<sup>i</sup>*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

*<sup>x</sup> z*<sup>0</sup> ¼ *vet*

near zone. As this takes place

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

0 , *z*<sup>0</sup> ¼ *vet*

*Sx z*<sup>0</sup> ¼ *vet*<sup>0</sup>

*i*

þ*i*

*i*

*<sup>z</sup> z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� þ *L Hy z*<sup>0</sup> ¼ *vet*

*Sx z*<sup>0</sup> ¼ *vet*

*Sx z*<sup>0</sup> ¼ *vet*

<sup>0</sup> ð Þ� *Hy z*<sup>0</sup> ¼ *vet*

*Sx z*<sup>0</sup> ¼ *vet*

*Sx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼ *<sup>i</sup>*

<sup>0</sup> ð Þ¼ <sup>þ</sup> *<sup>L</sup> <sup>i</sup>*

*Sx z*<sup>0</sup> ¼ *vet*

*i Sy z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼ *<sup>i</sup> S<sup>ψ</sup><sup>A</sup> <sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>i</sup> SA <sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼ <sup>¼</sup> *<sup>E</sup><sup>p</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>E</sup><sup>r</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ (42) *i Sy z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼ <sup>þ</sup> *<sup>L</sup> <sup>i</sup> S<sup>ψ</sup><sup>A</sup> <sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>i</sup> SA <sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ¼ þ *L* <sup>¼</sup> *<sup>E</sup><sup>p</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� þ *L Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>r</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� þ *L Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L* , (43) *i S<sup>ψ</sup><sup>A</sup> <sup>y</sup> z*<sup>0</sup> ¼ *vet* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ ¼ *<sup>E</sup><sup>p</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>p</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð þ *L*Þ� �*Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>E</sup><sup>r</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ <sup>þ</sup> *<sup>L</sup> <sup>E</sup><sup>r</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� þ *L Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ, (44)

*piSy <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> þ *L*, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ ¼ *Ep <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *<sup>E</sup><sup>p</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>H</sup><sup>c</sup> <sup>x</sup> vet* <sup>0</sup> , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þþ <sup>þ</sup>*E<sup>r</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *<sup>E</sup><sup>r</sup> <sup>z</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þþ <sup>þ</sup>*E<sup>c</sup> <sup>z</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ � *Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>E</sup><sup>c</sup> <sup>z</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ � *Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L* , (45) *f Sc <sup>y</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*޼�*E<sup>c</sup> <sup>z</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>H</sup><sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð Þ þ *L :* (46)

The Poynting vector along the *Oz* axis, taking into account Eqs. (10), (11), (21), (22), and (31), may be written as follows:

$$\mathbf{S}\_{\mathbf{z}}(t', \mathbf{x}'=0, \mathbf{y}'=0, \mathbf{v}\_{\mathbf{c}}t' < \mathbf{z}' < \mathbf{v}\_{\mathbf{c}}t' + L; t, r(\mathbf{x}, \mathbf{y}, \mathbf{z})) = {}^{i}\mathbf{S}\_{\mathbf{z}}(\mathbf{z}' = \mathbf{v}\_{\mathbf{c}}t', \mathbf{z}' = \mathbf{v}\_{\mathbf{c}}t' + L) + \mathbf{L}$$

$$+ {}^{pi}\mathbf{S}\_{\mathbf{z}}(\mathbf{z}' = \mathbf{v}\_{\mathbf{c}}t', \mathbf{z}' = \mathbf{v}\_{\mathbf{c}}t' + L, \mathbf{v}\_{\mathbf{c}}t' < \mathbf{z}' < \mathbf{v}\_{\mathbf{c}}t' + L) + {}^{f}\mathbf{S}\_{\mathbf{z}}^{\prime}(\mathbf{v}\_{\mathbf{c}}t' < \mathbf{z}' < \mathbf{v}\_{\mathbf{c}}t' + L), \quad \text{(47)}$$

$${}^{i}\mathbf{S}\_{\mathbf{z}}(\mathbf{z}' = \mathbf{n}\_{\mathbf{z}}t', \mathbf{z}' = \mathbf{n}\_{\mathbf{z}}t' + L) = {}^{i}\mathbf{S}\_{\mathbf{z}}(\mathbf{z}' = \mathbf{n}\_{\mathbf{z}}t') +$$

$$\mathbf{^iS\_x(z'=v\_ct',z'=v\_ct'+L)} = \mathbf{^iS\_x(z'=v\_ct')} + \mathbf{^jS\_x(z'=v\_ct')} + \mathbf{^jS\_x(z'=v\_ct',z'=v\_ct'+L)}$$

$$\mathbf{^iS\_x(z'=v\_ct'+L)} + \mathbf{^iS\_x^{yA}(z'=v\_ct',z'=v\_ct'+L)} \tag{48}$$

$$\mathbf{^iS\_x(z'=v\_\varepsilon t') = ^iS\_x^{\nu A}(z'=v\_\varepsilon t') = }$$

$$\mathbf{J} = \mathbf{E\_x^p(z'=v\_\varepsilon t') \cdot H\_\mathcal{Y}(z'=v\_\varepsilon t') - \mathbf{E\_{\mathcal{Y}}^p}(z'=v\_\varepsilon t') \cdot H\_\mathcal{x}(z'=v\_\varepsilon t') \tag{49}$$

*Sψ<sup>A</sup>*

$${}^{i}S\_{x}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) = {}^{i}S\_{x}^{pA}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) = $$

$$=E\_{x}^{p}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) \cdot H\_{\mathcal{Y}}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) - E\_{\mathcal{Y}}^{p}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) \cdot H\_{\mathbf{x}}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) \tag{50}$$

$${}^{i}S\_{x}^{pA}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}', \mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) = E\_{\mathbf{x}}^{p}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}') \cdot H\_{\mathcal{Y}}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) +$$

$$+E\_{\mathbf{x}}^{p}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) \cdot H\_{\mathcal{Y}}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}') - E\_{\mathcal{Y}}^{p}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}') \cdot H\_{\mathbf{x}}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) -$$

$$-E\_{\mathcal{Y}}^{p}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}'+L) \cdot H\_{\mathbf{x}}(\mathbf{z}'=\mathbf{v}\_{\varepsilon}\mathbf{t}') \tag{51}$$

*i*

*piSz <sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet* 0 , *z*<sup>0</sup> ¼ *vet* <sup>0</sup> þ *L*, *vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ ¼ *Ep <sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup> <sup>y</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ þ *<sup>E</sup><sup>p</sup> <sup>x</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>H</sup><sup>c</sup> <sup>y</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þþ <sup>þ</sup>*E<sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ � *Hy z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þþ *<sup>E</sup><sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ � *Hy z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð þ *L*Þ� �*E<sup>p</sup> <sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>H</sup><sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> <sup>ð</sup> <sup>þ</sup> *<sup>L</sup>*Þ � *<sup>E</sup><sup>p</sup> <sup>y</sup> z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� <sup>þ</sup> *<sup>L</sup> <sup>H</sup><sup>c</sup> <sup>x</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ� �*E<sup>c</sup> <sup>y</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ � *Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ� *<sup>E</sup><sup>c</sup> <sup>y</sup> vet* 0 , *z*<sup>0</sup> , *vet* <sup>0</sup> ð þ *L*Þ � *Hx z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ þ *L* , (52)

$${}^{\ell}S\_{\mathbf{z}}^{\varepsilon}(v\_{\varepsilon}t' \le z' \le v\_{\varepsilon}t' + L) = E\_{\mathbf{x}}^{\varepsilon}(v\_{\varepsilon}t' \le z' \le v\_{\varepsilon}t' + L) \cdot H\_{\mathcal{I}}^{\varepsilon}(v\_{\varepsilon}t' \le z' \le v\_{\varepsilon}t' + L) - \varepsilon$$

$$-E\_{\varepsilon}^{\varepsilon}(v\_{\varepsilon}t' \le z' \le v\_{\varepsilon}t' + L) \cdot H\_{\mathbf{x}}^{\varepsilon}(v\_{\varepsilon}t' \le z' \le v\_{\varepsilon}t' + L). \tag{53}$$

#### **8. Numerical results**

We have considered the filamentary REB of the length *L* ¼ 3 *m*, moving along the *Oz* axis with velocity *ve* ¼ 0*:*94 *c* ( *c* is the speed of light) and having overall charge *<sup>Q</sup>* ¼ �ð Þ� <sup>1</sup> <sup>10</sup>�10*C*.

In the laboratory coordinate system, the dependence of the electric field strength *Ep <sup>x</sup> t* <sup>0</sup> ¼ 0, *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *z* ð Þ <sup>0</sup> ¼ 0; *t*,*r x*ð Þ , *y* ¼ 0, *z* ¼ 0 , radiated by the beginning of the REB segment *r*<sup>0</sup> *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *z* ð Þ <sup>0</sup> ¼ 0 , on the transverse coordinate *x* was calculated using Eq. (10), (**Figure 1**). The signal radiation time *t* 0 was selected equal to zero *t* <sup>0</sup> ¼ 0. The observation point *r x*ð Þ , *y* ¼ 0, *z* ¼ 0 was selected in the cross section *<sup>z</sup>* <sup>¼</sup> 0 at *<sup>y</sup>* <sup>¼</sup> 0. The observation time *<sup>t</sup>* was determined by the formula *<sup>t</sup>* <sup>¼</sup> j j *<sup>x</sup>*

*c* . The dependence of the potential electric field strength *E<sup>p</sup> <sup>x</sup> t* 0 , *x*<sup>0</sup> ¼ 0, *y* ð <sup>0</sup> ¼ 0, *z*<sup>0</sup> ¼ *vet* 0 ; *t*,*r x*ð ¼ 0*:*3*m*, *y* ¼ 0, *z* ¼ 0ÞÞ, radiated by the beginning of the REB segment *r*<sup>0</sup> *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *z*<sup>0</sup> ¼ *vet* <sup>0</sup> ð Þ, on the signal generation time *t* 0 calculated with the help of Eq. (10), is represented in **Figure 2** where *r x*ð ¼ 0*:*3*m*, *y* ¼ 0, *z* ¼ 0Þ is the observation point coordinates.

The dependence of the magnetic field strength *Hy t* <sup>0</sup> ¼ 0, *x*<sup>0</sup> ¼ 0, *y* ð <sup>0</sup> ¼ 0, *z*<sup>0</sup> ¼ *L*; *t*,*r x*ð , *y* ¼ 0, *z* ¼ 0ÞÞ radiated by the REB segment end *r*<sup>0</sup> *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *z* ð Þ <sup>0</sup> ¼ *L* on the transverse coordinate *x* was calculated using Eq. (22) (**Figure 3**). The signal

#### **Figure 1.**

*The potential electric field strength Ep <sup>x</sup> <sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the REB segment beginning.*

generation time *t*

*REB segment end.*

*Magnetic field strength Hy t*<sup>0</sup>

**Figure 4.**

**Figure 3.**

*z*<sup>0</sup> ¼ *vet*<sup>0</sup>

**69**

*t* was determined by the formula *t* ¼

observation point coordinates.

<sup>0</sup> was selected equal to the zero, *t*

The dependence of the magnetic field strength *Hy t*<sup>0</sup>

The dependence of the electromagnetic energy flux *<sup>i</sup>*

The dependence of the electromagnetic energy flux *<sup>i</sup>*

*<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> � �, on the signal generation time *<sup>t</sup>*<sup>0</sup>

*r x*ð Þ ¼ 0*:*3*m*, *y* ¼ 0, *z* ¼ 0 is the observation point coordinates. The observation time

*Magnetic field strength Hy <sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>L</sup>*;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � � � *radiated by the REB segment end.*

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

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

ffiffiffiffiffiffiffiffiffi *<sup>x</sup>*2þ*L*<sup>2</sup> p

*<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>*;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment end *<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � �, on the signal radiation time *<sup>t</sup>*<sup>0</sup> calculated using Eq.(22), is represented in **Figure 4** where *r x*ð ¼ **0***:***3***m*, *y* ¼ **0**, *z* ¼ **0**Þ is the

*<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment beginning *<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>** � �, on the transverse coordinate *<sup>x</sup>* was calculated with the help of Eqs. (49), (10), (11), (21), and (22) (**Figure 5**). The signal generation time *t*<sup>0</sup> was selected equal to the zero, *<sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**. The *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � is the observation point

coordinates. The observation time *<sup>t</sup>* was determined by the formula *<sup>t</sup>* <sup>¼</sup> j j *<sup>x</sup>*

; *<sup>t</sup>*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment

*<sup>c</sup> :*

<sup>0</sup> ¼ 0 where

, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>*;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � � � *radiated by the*

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

*Sz t*<sup>0</sup> ¼ **0**, *x* ð <sup>0</sup> ¼ **0**,

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

*Sz t*<sup>0</sup>

*c* .

, calculated by Eqs. (49),

#### **Figure 2.**

*The potential electric field strength E<sup>p</sup> <sup>x</sup> t*<sup>0</sup> , *x*<sup>0</sup> ¼ **0**, *y*<sup>0</sup> ¼ **0**, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the REB segment beginning.*

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

#### **Figure 3.**

**8. Numerical results**

*Progress in Relativity*

charge *<sup>Q</sup>* ¼ �ð Þ� <sup>1</sup> <sup>10</sup>�10*C*.

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

observation point coordinates.

*Ep <sup>x</sup> t*

zero *t*

*vet* 0

**Figure 1.**

**Figure 2.**

**68**

*The potential electric field strength Ep*

*The potential electric field strength E<sup>p</sup>*

*the REB segment beginning.*

*<sup>x</sup> t*<sup>0</sup>

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

*REB segment beginning.*

We have considered the filamentary REB of the length *L* ¼ 3 *m*, moving along the *Oz* axis with velocity *ve* ¼ 0*:*94 *c* ( *c* is the speed of light) and having overall

In the laboratory coordinate system, the dependence of the electric field strength

<sup>0</sup> ¼ 0. The observation point *r x*ð Þ , *y* ¼ 0, *z* ¼ 0 was selected in the cross section

0 was selected equal to

, *x*<sup>0</sup> ¼ 0, *y* ð <sup>0</sup> ¼ 0, *z*<sup>0</sup> ¼

0 calculated with the help

<sup>0</sup> ¼ 0, *x*<sup>0</sup> ¼ 0, *y* ð <sup>0</sup> ¼ 0, *z*<sup>0</sup> ¼

*<sup>x</sup> t* 0

*<sup>x</sup> <sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the*

;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by*

*c* .

<sup>0</sup> ¼ 0, *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *z* ð Þ <sup>0</sup> ¼ 0; *t*,*r x*ð Þ , *y* ¼ 0, *z* ¼ 0 , radiated by the beginning of the REB segment *r*<sup>0</sup> *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *z* ð Þ <sup>0</sup> ¼ 0 , on the transverse coordinate *x* was cal-

; *t*,*r x*ð ¼ 0*:*3*m*, *y* ¼ 0, *z* ¼ 0ÞÞ, radiated by the beginning of the REB segment

*<sup>z</sup>* <sup>¼</sup> 0 at *<sup>y</sup>* <sup>¼</sup> 0. The observation time *<sup>t</sup>* was determined by the formula *<sup>t</sup>* <sup>¼</sup> j j *<sup>x</sup>*

of Eq. (10), is represented in **Figure 2** where *r x*ð ¼ 0*:*3*m*, *y* ¼ 0, *z* ¼ 0Þ is the

*L*; *t*,*r x*ð , *y* ¼ 0, *z* ¼ 0ÞÞ radiated by the REB segment end *r*<sup>0</sup> *x*<sup>0</sup> ¼ 0, *y*<sup>0</sup> ¼ 0, *z* ð Þ <sup>0</sup> ¼ *L* on the transverse coordinate *x* was calculated using Eq. (22) (**Figure 3**). The signal

culated using Eq. (10), (**Figure 1**). The signal radiation time *t*

The dependence of the potential electric field strength *E<sup>p</sup>*

<sup>0</sup> ð Þ, on the signal generation time *t*

The dependence of the magnetic field strength *Hy t*

*Magnetic field strength Hy <sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>L</sup>*;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � � � *radiated by the REB segment end.*

#### **Figure 4.**

*Magnetic field strength Hy t*<sup>0</sup> , *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>*;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � � � *radiated by the REB segment end.*

generation time *t* <sup>0</sup> was selected equal to the zero, *t* <sup>0</sup> ¼ 0 where *r x*ð Þ ¼ 0*:*3*m*, *y* ¼ 0, *z* ¼ 0 is the observation point coordinates. The observation time *t* was determined by the formula *t* ¼ ffiffiffiffiffiffiffiffiffi *<sup>x</sup>*2þ*L*<sup>2</sup> p *<sup>c</sup> :*

The dependence of the magnetic field strength *Hy t*<sup>0</sup> , *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, � *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>*;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment end *<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � �, on the signal radiation time *<sup>t</sup>*<sup>0</sup> calculated using Eq.(22), is represented in **Figure 4** where *r x*ð ¼ **0***:***3***m*, *y* ¼ **0**, *z* ¼ **0**Þ is the observation point coordinates.

The dependence of the electromagnetic energy flux *<sup>i</sup> Sz t*<sup>0</sup> ¼ **0**, *x* ð <sup>0</sup> ¼ **0**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment beginning *<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>** � �, on the transverse coordinate *<sup>x</sup>* was calculated with the help of Eqs. (49), (10), (11), (21), and (22) (**Figure 5**). The signal generation time *t*<sup>0</sup> was selected equal to the zero, *<sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**. The *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � is the observation point coordinates. The observation time *<sup>t</sup>* was determined by the formula *<sup>t</sup>* <sup>¼</sup> j j *<sup>x</sup> c* .

The dependence of the electromagnetic energy flux *<sup>i</sup> Sz t*<sup>0</sup> , *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, � *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ; *<sup>t</sup>*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment *<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> � �, on the signal generation time *<sup>t</sup>*<sup>0</sup> , calculated by Eqs. (49),

#### **Figure 5.**

*The electromagnetic energy flux <sup>i</sup> Sz <sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the REB segment beginning.*

(10), (11), (21), (22), is shown in **Figure 6**. The observation point coordinate is

*<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>L</sup>* � �, on the transverse coordinate *<sup>x</sup>* was calculated with the help of Eqs. (50), (10), (11), (21), (22) (**Figure 7**). The signal radiation time *t*<sup>0</sup> was selected equal to the zero *<sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**. The *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � is the observation point

*Sz t*<sup>0</sup> ¼ **0**, *x* ð <sup>0</sup> ¼ **0**,

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

*Sz t*<sup>0</sup>

, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>*;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � � � *radiated*

ffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>***2**þ*L***<sup>2</sup>** p

, calculated according

*<sup>c</sup> :*

The dependence of the electromagnetic energy flux *<sup>i</sup>*

*Sz t*<sup>0</sup>

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

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

The dependence of the electromagnetic energy flux *<sup>i</sup>*

*<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � �, on the signal radiation time *<sup>t</sup>*<sup>0</sup>

*y* ¼ **0**, *z* ¼ **0**Þ is the observation point coordinates.

moving in vacuum along a linear direction.

dynamic component of the electric field.

*<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>L</sup>*; *<sup>t</sup>*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment end

coordinates. The observation time *t* was determined by the formula *t* ¼

*<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>*; *<sup>t</sup>*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment end

to Eqs. (50), (10), (11), (21), (22), is shown in **Figure 8** where *r x*ð ¼ **0***:***3***m*,

The applicability of relativity in the physics of charged particle beams has been shown from the example of radiation by a filamentary REB segment uniformly

The expressions have been obtained to describe the strengths of the electric and magnetic fields and the electric and electromagnetic energy fluxes in all three zones: near field zone, intermediate, and wave zones. The filamentary REB edges are relativistic point-like sources of electromagnetic energy propagating in the wave zone. The REB edges form a potential component of the electric field strength, which is inversely proportional to the distance from the source point to the observation point. In the wave zone, strength of this field is comparable with that of the

In electrodynamics, in a moving coordinate system, the relative distance between a charged object and an observer does not change. The phenomenon of relativity associated with the field dynamics degenerates to electrostatic processes. In rest, or laboratory, coordinate system, the relative distance is changing with time, the charge density also varies with the time, and as a result, the retardation phenomena came to the scene and the Poisson equation is to be substituted by the

*r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �.

*The electromagnetic energy flux <sup>i</sup>*

*by the REB segment end.*

**Figure 8.**

**9. Conclusions**

wave equation.

**71**

#### **Figure 6.**

*The electromagnetic energy flux <sup>i</sup> Sz t*<sup>0</sup> , *x*<sup>0</sup> ¼ **0**, *y*<sup>0</sup> ¼ **0**, *z*<sup>0</sup> ¼ *vet*<sup>0</sup> ;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the REB segment beginning.*

#### **Figure 7.**

*The electromagnetic energy flux <sup>i</sup> Sz <sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>L</sup>*;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the REB segment end.*

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

#### **Figure 8.**

**Figure 5.**

**Figure 6.**

**Figure 7.**

**70**

*segment end.*

*The electromagnetic energy flux <sup>i</sup>*

*The electromagnetic energy flux <sup>i</sup>*

*REB segment beginning.*

*Sz t*<sup>0</sup>

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

*segment beginning.*

*Progress in Relativity*

*The electromagnetic energy flux <sup>i</sup>*

*Sz <sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the REB*

;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the*

*Sz <sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>L</sup>*;*t*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** *radiated by the REB*

*The electromagnetic energy flux <sup>i</sup> Sz t*<sup>0</sup> , *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>*;*t*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � � � *radiated by the REB segment end.*

(10), (11), (21), (22), is shown in **Figure 6**. The observation point coordinate is *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �.

The dependence of the electromagnetic energy flux *<sup>i</sup> Sz t*<sup>0</sup> ¼ **0**, *x* ð <sup>0</sup> ¼ **0**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>L</sup>*; *<sup>t</sup>*, *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment end *<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>L</sup>* � �, on the transverse coordinate *<sup>x</sup>* was calculated with the help of Eqs. (50), (10), (11), (21), (22) (**Figure 7**). The signal radiation time *t*<sup>0</sup> was selected equal to the zero *<sup>t</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**. The *r x*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � � is the observation point p

coordinates. The observation time *t* was determined by the formula *t* ¼ ffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>***2**þ*L***<sup>2</sup>** *<sup>c</sup> :*

The dependence of the electromagnetic energy flux *<sup>i</sup> Sz t*<sup>0</sup> , *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, � *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>*; *<sup>t</sup>*, *r x* <sup>¼</sup> **<sup>0</sup>***:***3***m*, *<sup>y</sup>* <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>* <sup>¼</sup> **<sup>0</sup>** � �Þ, radiated by the REB segment end *<sup>r</sup>*<sup>0</sup> *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *vet*<sup>0</sup> <sup>þ</sup> *<sup>L</sup>* � �, on the signal radiation time *<sup>t</sup>*<sup>0</sup> , calculated according to Eqs. (50), (10), (11), (21), (22), is shown in **Figure 8** where *r x*ð ¼ **0***:***3***m*, *y* ¼ **0**, *z* ¼ **0**Þ is the observation point coordinates.

#### **9. Conclusions**

The applicability of relativity in the physics of charged particle beams has been shown from the example of radiation by a filamentary REB segment uniformly moving in vacuum along a linear direction.

In electrodynamics, in a moving coordinate system, the relative distance between a charged object and an observer does not change. The phenomenon of relativity associated with the field dynamics degenerates to electrostatic processes. In rest, or laboratory, coordinate system, the relative distance is changing with time, the charge density also varies with the time, and as a result, the retardation phenomena came to the scene and the Poisson equation is to be substituted by the wave equation.

The expressions have been obtained to describe the strengths of the electric and magnetic fields and the electric and electromagnetic energy fluxes in all three zones: near field zone, intermediate, and wave zones. The filamentary REB edges are relativistic point-like sources of electromagnetic energy propagating in the wave zone. The REB edges form a potential component of the electric field strength, which is inversely proportional to the distance from the source point to the observation point. In the wave zone, strength of this field is comparable with that of the dynamic component of the electric field.

The dynamic component of the electric field strength and the axially symmetric magnetic field form both a constant flux into the given solid angle, i.e. electromagnetic radiation, and a flux per time unit directed along the normal to the conical surface of the above 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 conical surface. The fluxes crossing the above conical surface are independent of 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.

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Relativistic point-like sources create in the wave zone the vortex components of the magnetic field. The REB edges radiate hybrid electromagnetic waves, comprising of potential and vortex electric fields, as well as a vortex magnetic field. The electric and magnetic field strengths radiated by the REB segment edges have opposite signs. In the wave zone, the radiated electromagnetic field fluxes are compound of the electromagnetic energy fluxes, produced by both the REB segment beginning and its end, as well as of their interference components. In the intermediate zone, the electrical energy flux takes place due to the electric potential field and the displacement current. The REB segment, between the beam edges, having a constant charge density, produces a quasi-static electromagnetic field in the near zone.
