Mathematical model of charged particles in a magnetic field Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

СИМВОЛ НАУКИ ISSN 2410-700X № 5 / 2018.

УДК 519.635

Douglas A. Suarez R.,

Doctor of Science in Telecommunications Engineering.

E-mail: [email protected]

MATHEMATICAL MODEL OF CHARGED PARTICLES IN A MAGNETIC FIELD

Annotation

The model and equations (or system of equations) that correspond the model is adequate from the point of view of physics that manifests in solutions that eventually keep increasing without bound.

This model is characteristic principally for quantum electrodynamics problems but it can be useful in classical electrodynamics too especially in problems that consider the motion of charged particles and magnetic fields that they generate.

Key words:

mathematical model, magnetic field, dielectric, energy and charged particles.

Existing division of environment in metals, dielectric and etc is quite conventional and depends on values of intensity of electric and magnetic fields, frequency range of electromagnetic radiation and etc, [1].

The environment is considered a metal if

j0(E,B) Ф 0,P(E,B) = 0,M(E,B) = 0 Given conditions signify that mobile free charges on account of their motion countervail exterior electric and magnetic fields, whose intensities exponentially wane from surface inside the model ("skin effect"). Environment is considered dielectric if

j0(E,B) = 0,Р(Е,В) Ф 0,M(E,B) = 0 Environment is called magnetic if

j0(E,B) = 0,P(E,B) = 0,М(Е,В) Ф 0 Also there are environments that called magnetic electric for them

j0(E,B) Ф 0,Р(Е,В) Ф 0,M(E,B) = 0 There are a great number of materials (environments) that are in the middle of mentioned classification or have the characteristics of all «clean» environments. For example, semiconductor materials are in the intermediate position between metals and dielectrics, [2]. In different conditions dielectrics can develop semiconductor characteristics for example when heating, lighting the surface with assigned length of wave, exceeding of intensities of electric and magnetic fields of thresholds and etc. Aside from this for ferromagnetic magnetization depends on the way of process, hysteresis is observed with it the dependence of M(E, B) is not univocal.

M(B)

Dependence of magnetization from induction of magnetic field for ferromagnetic These phenomenons are connected with domain structure of ferromagnetic so complexes of monocrystals

-( '» )-

СИМВОЛ НАУКИ ISSN 2410-700X № 5 / 2018.

(domains) have organization and different from zero resultant magnetic moment even if exterior magnetic field is absent, [8]. Area covered by hysteresis curve characterizes the operation of exterior field to reorientation of summarized magnetic moments of different domains and to marshal them in direction. Indicated operation releases as a heat. The conduct of ferromagnetic in variable exterior electromagnetic fields is interesting and has many practical uses in radio electronic. The analogue of ferromagnetic as to interaction with electric field is ferroelectricity, for it the dependence of P(E) also has hysteresis character, [5].

Definition of concrete dependences j0 (E, B), P(E, B), M(E, B) is a task of microscopic theory and can be fulfilled considering theoretically the models of interaction of electromagnetic radiation with material in micro level or planting experimentally phenomenological dependences, [11]. The complexity of considering theoretically is obvious because only quite simplified models can be fundamentally investigated however the importance of such method is obvious since it gives general consistent patterns for material equations of macroscopic electrodynamics and respectively directions for investigators experimentalists to work on.

Before considering a range of simplified models of interaction of electromagnetic radiation with dielectric that however give the idea of the character of no lineal that are in material equations, we have one more conclusion for three-dimensional density of energy of electromagnetic field that is the result of Poynting's theorem, [7,9]. The difference of this conclusion from the considered before the case of free fields in vacuum is in using received equations of macroscopic electrodynamics and in consecutive accounting of characteristics of environment in phenomenological level that is through material equations.

From Maxwell's equations in environment the following correlations can be received,

1dB

(H,r°t E) = -<H,-—)

As indicated before Then considering (1),

ldD 4n (E,rot H) = (E-—) + — (j0,E) с at с

(H, rot E) - (E, rot H) = div[E, H]

D = E + 4nP(E,B) В = H + 4nM(E, В)

(1)

1 dH 4n dM 1 dE 4n dP 4n

div[E, H] = --(H,—)--(H,—)--(E,—)--(E,—)--(jo, E)

c at c at c at c at c

Vector S = —[E,H] is called after Poynting, [3,4,10,12]. Integrating both parts of equality according to some volume that the environment has and using the divergence theorem to pass from dimensional integral to divergence of vector to integral flow of this vector through surface that restricts the volume of integration, receive,

| divS dV = | (S, n)da

dV

д Г iE2 + H2\ Г дМ Г дР Г (2)

= - I*1 (-+-)dv -1 (И'ИГ) dv-1 (Е-Ш) dv - 1 (J"E)dv

Here dV is surface that restricts the volume of integration, do is element of area of this surface, n is local vector of normal to surface dV in its element da.

So the vector Poynting gives density of integral flow of energy of electromagnetic field in unity of time through surface that restricts the volume of environment where this field is. The first addend on the right corresponds to change in time of energy in electromagnetic field. The sum of squares of intensity of electric and magnetic fields can be compared with full energy of mechanical oscillator through the principle of electromagnetic analogue as the sum of kinesthetic and potential energy also with the energy of oscillating circuit that consists of energy of compensator (electric field) and induction coil (magnetic field).

The second and the third addends on the right (2) correspond to capacity (operation in the unity of time) of

{ ■ }

СИМВОЛ НАУКИ ISSN 2410-700X № 5 / 2018.

magnetic field on formation of magnetization environment and electric field on creation (division) induction charges. The forth addend defines joule in heat that releases in a unity of time in case of conductance of environment. In vacuum,

d f (E2 + H2\

- (3)

Г д Г (E2 + H2\

J<w=- Ttlb+rr

dV 4 7

That corresponds to before received expression for tensor of energy impulse of free magnetic field. If the dependence of magnetization and polarization from magnetic and electric fields is lineal as in (4), and the conductance of environment is absence, [3,4,10,12].

D = eE В =ßH

(4)

г д [ (ее2 + ^Н2\

J (S,n)d<T = — -J (-+-)dV (5)

dV

So if there is lineal dependence of polarization and magnetization from electric and magnetic fields and there is no conductance the conduct of field in environment is like the conduct of field in vacuum from the point of view of balance and energy.

It is worth noting that at interaction of tagged electric field with parts of environment come dimensional and temporary delays in circulation of field connected with relax processes in microscopic level. These facts are called dimensional and temporary dispersion. So even in case of lineal dependence of polarization and magnetization of environment from electric and magnetic fields, the expression of the form (4) acquires operator aspect and these operators that define material correlations for environment can be represented in class of integral operators, roll of functions of field with nucleus that define dielectric and magnetic penetrability of environment including the facts of dimensional and temporary dispersion,

D(x, t) = £E = J £(x-f,t- t)E(1 T)dfdr = (e * E) (6)

B(x, t)=flE = J p(x-f,t- t)H(1 T)dfdr = (/u * H) (7)

Here and further ( *) is operation of roll. Analogous to conduct environment in lineal approach.

J 0(X,t) = TE = I <T(V - ® ^ - i- i-\rli.rli- f sr 4. 1/ 1 (8)

j0(x, t) = ÔE = J a(x — — t)E(%, r)d%dr = (a * E)

Here T is name of conductance in lineal case if there is not dimensional temporary dispersion is value that inverse active electric resistance of conduct environment.

Note that in case of dimensional no isotropic environment nucleuses of (6) are tensors of correlated dimensions. Also it is worth noting that dielectric penetrability can be considered as complex value, its actual part characterizes redistribution of energy in process of extension of electromagnetic wave in environment on account of effects of dispersion and the supposed part defines leakage of energy of electromagnetic wave. Using the causality principle (electromagnetic signal cannot wind faster than the speed of light) general analytic characteristics of nucleus of operators of dielectric and magnetic penetrability can be investigated. To begin with let us imagine Fourier image of complex dielectric penetrability as,

£ = 1 + 4na

or

(£ * ) = 1 + 4n(a * )

The value a is called polarization. Dielectrics are characterized by damping of electromagnetic field in time if exterior sources of field are disabled. It means that the nucleus of operator a is limited and aims for zero when t ^ <x. For investigation of analytic characteristics of dielectric penetrability let us consider the representation of nucleus

СИМВОЛ НАУКИ ISSN 2410-700X № 5 / 2018.

as integral Fourier.

o

Introduction of polarization and extraction in (9) of addend «unity» let us avoid considering singular representation like delta function that precisely is a Fourier image of unity. Also note that the inferior limit of integration 0 is chosen and not <x as a consequence of causality principle. Indeed in (6) the condition t < t, has to be fulfilled that signifies that induction field in material (environment) is defined by values of fields in all previous moments of time. The expression a can be equal to zero at negative times and then to extend integration on time in all actual axis.

As far as in frequency area (at transformation of Fourier) roll pass to sum, from (6),

D(k, u) = e(k, u)E(k, u)

B(k,a) = p(k,a)H(k,a) (10)

Let us go back to dissipation of energy of electromagnetic wave in dielectric at lineal connection of intensity of fields and inductions in correlations (9) and (10). The presentation of intensities and inductions as the integral Fourier (further we consider only temporary frequency presentation, considering the field that makes little changes in three dimensional areas to simplify the computation).

= j rf/.^SD-itet

E(t) = J Е(ш)е- dw

— œ

+ œ

D(t) = J D(w)e—iMtdw

—œ

+ œ +œ

9D(t) = -i J = —i J œe(œ)È(œ)e—iœtdœ

dt

— œ — œ

let us integrate according to time the change of energy of field on account of operation of electric potency according to induction of charges in dielectrics,

+ œ + œ

j (E,^)dt = -i j Ms(M)(È(M),È(M1))e—i(M+Mi)tdMdM1dt —œ —œ

As transformation of Fourier delta function is one then

+ œ

j e—i("+"i)tdt = s(M + Ml) —œ

including conditions of corporeality of electric field

Ê*(œ) = Ê(-w)

(sign * signifies operation of complex interface) and definition of delta function with dislocated argument,

+ œ

j f(M1)S(œ + œ1)dœ1=f(-œ)

—œ

receive

+ œ +œ

j (E,—)dt = -i j ms(m)IE(m)I2dtà

—œ —œ

Also results that for corporeality D(t) accomplishment of the condition is needed,

ë(-œ) = ë*(œ)

or

Re£(-œ) = Reè(^)

i - y

Im£(-m) = -Im£(w)

Then conclusive (including analogous transformations for contribution in modification of energy from magnetic field).

So we see that the modification (wane, leakage) of energy of electromagnetic field really is defined by supposed parts of dielectric and magnetic penetrability of dielectric environment. Also from positivity of density of energy of electromagnetic field comes that at m > 0 has to accomplish,

Strictly speaking adduced affirmations are right only for a field that extends in environment close to the state of thermodynamic balance. If the environment is for some reason in a state far from thermodynamic balance supposed parts of dielectric and magnetic penetrability can be negative that leads to increment of energy of electromagnetic wave if it extends in such environment. The action of generators of radiation (lasers and masers) is based on this principle.

If for any range of frequency for this dielectric environment the conditions are fulfilled,

then such range often is called area of transparency of this environment for electromagnetic radiation. The environment can have several areas of transparency. The presence of crossing areas of transparency of different environments is a point of interest. So water and air are transparent for optic radiation at the same time in so called «blue and green» range that opens aspects of creation of laser connection for submarines through satellite systems.

In case of radiation with frequency inside the area of transparency electromagnetic energy wanes slowly so we can introduce the conception of average value of density of energy of the field analogous to the case of fields in vacuum or quasi stationary fields in dielectric environment.

1. Batigin V.V. Toptiguin. IN PROBLEMS OF ELECTRODYNAMICS AND THE SPECIAL THEORY OF RELATIVITY. Editorial USSR-Moscow. 1995, p. 98-124.

2. Bredov M. Rumyantsev A. Toptiguin I. CLASSICAL ELECTRODYNAMICS. Editorial MIR-Moscow. 1985, p. 198-230.

3. Budak B.M., Samarskii AA, Tikhonov A.N. PROBLEMS OF MATHEMATICAL PHYSICS. Editors MIR-Moscow, 1984, vol. I, p. 11-32.

4. Godunov S.K. EQUATIONS OF MATHEMATICAL PHYSICS. Editorial USSR-Moscow, 1994, 66-89, p. 108-127.

5. Grigoryev V.I. QUANTUM FIELD THEORY. Origin and development. Editorial USSR-Moscow. 2015, p. 41-97.

6. Gurevich Y., Rodriguez F. TRANSPORT PHENOMENA IN SEMICONDUCTORS. Editorial Fondo de Cultura Económica. Mexico City, 2000. c. 251-284.

7. Kulyakov A. Rumyantsev A. INTRODUCTION TO THE PHYSICS OF NONLINEAR PROCESSES. Editorial MIR-Moscow. 1990, p. 127-180.

8. Landau L., Lifshitz E. ABBREVIATED COURSE OF PHYSICAL THEORY. Editorial MIR-Moscow. 1982, Vol. I, p. 209-266.

9. Landau L., Azhizer A., Lifshits E. COURSE OF GENERAL PHYSICS. Editorial MIR-Moscow. 1979, c. 56-74.

10. Smirnov M.M. PROBLEMS OF THE EQUATIONS OF MATHEMATICAL PHYSICS. Editors MIR-Moscow, 1991, p. 34-44.

11. Sokolov AA. Ternov IM, Zhukovsky V.C. Borisov A.V. QUANTUM ELECTRODYNAMICS. Editorial MIR-Moscow. 1991, c. 151-218.

12. Tikhonov A.N., Samarskiy A.A. EQUATIONS OF MATHEMATICAL PHYSICS. Editorial MIR-Moscow. 1980, p. 29-147.

Im ё(ш) > 0 1тД(<у) > 0

Reè(^) » Imè(^) Refi(w) » Imfi(w)

Literature:

© Suárez D A., 2018.

í » }