Radiation of a uniformly moving charge

Автор: Solomon I. Khmelnik

Журнал: Доклады независимых авторов @dna-izdatelstwo

Рубрика: Physics

Статья в выпуске: 49, 2020 года.

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It is generally accepted that a uniformly moving charge does not radiate. At the same time, Cherenkov-Vavilov radiation is known, which arises in a medium (but not in a vacuum), when a charge moves at a speed exceeding the speed of light in this medium. The energy for such radiation is extracted from the medium under the action of a particle moving in it. Below we will consider the rectilinear and uniform motion of a charge that moves in a vacuum and which has no other source of energy other than its own kinetic energy. It will be shown how from Maxwell's equations it follows that the flow of energy (its radiation) moves with the charge.

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Короткий адрес: https://sciup.org/148311521

IDR: 148311521

Текст научной статьи Radiation of a uniformly moving charge

It is generally accepted that a uniformly moving charge does not radiate. At the same time, it is known that a uniformly moving charge emits in a medium (but not in a vacuum) - this is Cherenkov-Vavilov radiation, which is experimentally observed when a charge moves with a speed (in a medium) exceeding the speed of light in this medium. This radiation is explained as radiation of the medium under the action of a particle moving in it [1].

In [4], the author showed that there is a solution to Maxwell's equations for a charge that moves uniformly in vacuum and does NOT radiate. Below, such a solution will be obtained in a different way and it will be shown that the charge energy flux, i.e. its radiation, moves with the charge.

Here we will use the method for solving Maxwell's equations proposed in [3]. The Maxwell equations in the CGS system of the following form are considered:

rot(E) + ^ = 0,(1)

rot(H) + ^J- LLL = 0,(2)

div(E) = 4лр,(3)

div(H) = 0,(4)

where

H, E is magnetic and electrical strengths, respectively, J is current density, p is the charge density.

We will use cylindrical coordinates г, ф, z , and consider the movement of charge q along the oz axis. We will denote as z₀, z the current, and analyzed coordinates of the point, respectively. The charge v is located at the point z₀ , therefore the charge density is determined through the delta function, i.e.

p = q^s⁽z₀-z⁾ , (5)

J = vp , (6)

where v is the speed of the charge.

To shorten the notation, we will further use the following notation:

co = cos( ар + % • 5 (z₀ — z)) , si = sin( ар + % • 5(z₀ — z)) ,	(7) (8)
where O, % are some constants. Wherein dco „ . „
— = ⁰ if ^z = Z o .	(9)
We represent the unknown functions in the following form:
H_f . = h (r)co ,	(11)
Н_ф .= h< p (r)si ,	(12)
H_z .= h_z (г) si ,	(13)
E_f . = e_T (r)si ,	(14)
Е ф .= е_ф (r)co ,	(15)
E z .= e z (r)co , where	(16)

• E r , Е_ф , E_z is electrical strengths,
• H r , Н ф , H z is magnetic strengths,
• h(r), e(r) are some functions of coordinate r ,

A rigorous proof that Maxwell's equations with electric charges defined by the delta function can be represented in this form is given in [2, Sections 6, 9.6.7].

In the system of cylindrical coordinates, equations (1-4) are:

E_T + dE_T + 1 r dr r	dE ф + dE Z = dф dz	4ттр.	см.	(3)	(17)
1 dE_Z dE ф _ r dф dz	M dH_T , c dt		см.	(1)	(18)
dE_T dE_Z _ м dz dr c	^dH Ф dt ’		см.	(1)	(19)
E ф + dE ф 1 r dr r	dE_T м dH_Z • -— =--, dф c dt		см.	(1)	(20)
H_T dH_T 1	^ + ^ = dф dz	0 ,		(4)	(21)
--+ —--1— • r dr r	^ + ^ = dф dz	0 ,	см.	(4)	(21)
1 dH_Z - dH ф r dф dz	£ dE_T c dt ,		см.	(2)	(22)
dH_T dH_Z _ £ dz dr c	dEg, dt ,		см.	(2)	(23)
H ф + дФф 1 dH_r + 4я . = £ d.E_Zr dr r dф c ^J c dt		■	см.	(2)	(24)

By direct substitution, one can make sure that functions (11-16)

transform the system of equations (17-24) with four arguments r, ф, z,t into a system of equations with one argument r and unknown functions h(r), e(r) . This system of equations is as follows:

e_r (r) , , z x б ф ^Г^ 4я _n

— + 6г(r)-^-oc-X-ez(r)-—p = 0, (25) -1-ez(r^a + х-бф(r) -^ hr = 0,

er (r)X - eZ (r) + ~ hф = 0,

~ф~ + еф (r)-^ .a + ^7hz = 0,

^7^ + hr (r) + а-х-hz (r) = 0,(29)

1hz(r) K~xhv(r)-^er(r) = 0,(30)

hr (rk - hz (r) + — („Ф, (r) = 0,(31)

^+кф(r) + -^ « + «^(Г) + ^ = 0. (32)

We will seek a solution to these equations under the assumption that hr = ker,(33)

кФ keф, hz = —kez. .

In [3, Chapter 2] it is shown that in this case the energy flux densities satisfy the energy conservation law .

We change the variables according to (33-35) in equations (25-32) and rewrite them:

e_r . . e - 4я 7 ^{+ e} r —7“ ^{— /e} z ~~P = ⁰ ,	(36)
e z MW r ^{a +} ^/е ф _c ^ke r ⁰ ,	(37)
MW ^e z ⁺ ^/e r ^k _c ^е ф ⁰ ,	(38)
— + e^ — — a — k — e_z = 0 , r ^ф r c ^z	(39)
к ^ + ke_r — k^-a — k/e_z = 0 ,	(40)
—к у a + kxе_ф — ^? e_r = 0 ,	(41)
ke z — kxe r + ^ё ф = 0 ,	(42)
—k^^- — ke^ + k^a + ^^ e_z + 4^ 1 = 0 . r ^ф r c c	(43)

This system of equations takes on different forms for different values of z . Let's consider the solution of these systems of equations.

3. Solution at z = z₀

Taking into account (9), we have:

7 + e r — r “ + /e z — 4яр = 0 ,	(44)
e_z MW i ^— ~^a— Х ^е ф 7 ^ke r = ⁰ ,	(45)
—e z -k^e ф = 0 ,	(46)
+ — + е_ф — — a — k — e_z = 0 , r ^ф r c^	(47)
k^ + ke_r — k^-a + k/e_z = 0 ,	(48)

-k^ct-—e_r = 0 , г с

ke_z + ^£ - e < = 0 ,

—k — — кё_ф + k — ct " e_zг V г с

Note that equations (44) and (51) coincide for

—J _ —k — р, с £

£—)

кс ^_ ^

It follows from (52) and (6) that

4я ^^ к ^4я р

к _ —-. с

It follows from (53) and (55) that -

^х ^{_ —} •

+ 4^ j _ 0 . с

(49)

(50)

(51)

(52)

(53)

(54)
(55)
(56)

4. Solution at Z Ф Z_o

In this case, we have:

е_т . вф

~ ⁺ ^e г ^— ~ ^{а —} X ^e z ^{_ 0} ,

e_z цы , „

^— 7 ^{а +} % ^е< Р ^— ~ ^ке г ^{_ 0} , ^e z ⁺ ^ si ^e r ^к с ^е < ⁰ ^,
^вф + е_ф — —а — k — e_z _ 0 , г < г с ^z
k — + ke_T — k — а — k/e_z _ г - г ^v ^z

e z £ш

^k г ^а ⁺ ^k X ^e
^e г ⁰ ,

^ke z ^^ХЯ г ""I ~^e(P ⁰ ,

—k — — ke») + k — а " e_zг < г с

Note that equations (58) and (62) coincide for

£Ш _ /хшк кс ^_ с •

0 ,

_ 0 .

(57)
(58)
(59)

(60)

(61)

(62)

(63)
(64)
(65)

Under the same condition, equations (60) and (64) coincide, as well as equations (59, 63). Equations (57) and (61) coincide under condition (55). Finally, equations (64) and (68) coincide. Thus, equations (62, 64, 63,

61) can be excluded from the system of equations and replaced by conditions (65, 55). The remaining 4 equations (57-60) are a system of differential equations with 3 unknowns.

The solution to this overdetermined system of equations exists for ez = 0.(66)

Moreover, from (57, 60) we find ep eT.

From (57, 66, 67) we find:

— + eT —- c = 0
5. Energy flows

or ep = eT = Ara"\(69)

where A is the amplitude of the strength.

The frequency in this solution is not fixed. Consequently, the charge can radiate with any frequency.

In [3, Chapter 2] it is shown that under conditions of the form (7, 8, 33, 34, 35), the energy flux densities are determined by the formulas

ST = 0,(70)

Sp T]kerez,

Sz = Tkerep,(72)

T = с/4k,(73)

i .e. there is no radial energy flux, and the energy flux densities at a given radius around the circumference and along the trajectory do not depend on time and other coordinates. Since the functions e p and e_T are defined by (69), the energy flux is limited in space. This means that the flow of energy moves with the charge, but is not radiated.
6. Conclusions

So, the obtained solution describes the motion of the charge Q in the medium (ц, e) with a certain speed V . In this case, the flow of energy is not emitted, but moves with the charge in a certain limited area.

With this strengths

HT. = hT (r)co,(11)

Нф .= hp (r)si,(12)

ET. = eT (r)si,(14)

	Е ф	.= б ф (r)co ,	(15)
where
	co	= cos( ap + Z" 5(z₀ -z)} ,	(7)
	si =	= sin( ap + z • 5(z₀ — z)) ,	(8)
	*^Ф	= e_r = Ar^a-1,	(69)
	^h r	= ke_r ,	(33)
	Е ф	= ^—к б ф ,	(34)
	к =	vs ^-- ,	(55)
	Z ^z	= — —. v	(56)

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