Solving Maxwell's Equations for AC Wire

^E r । dE- ^ । ¹ дЕ ф + dE ^   о r    dr r дф    dz     ,

(1)

1 dE_z дЕ ф _ .to dH_r г дф    dz     c dt ’

(2)

dE_r dE_z _ .to dH ф dz    dr     c dt ’

(3)

Е ф + dE ф 1 dE_r _ .to dH_z r     dr r dф     c dt ’

(4)

H r ₊ dH r . 1 ^dH ф ₊ dH z _ о r    dr r dф    dz     ’

(5)

1 dH z   dH ф _ _I

r dф    dz     ^r ’

(6)

dH_T   dH_z

(7)

—^r--^z _ I dz    dr     ^ ,

^H ф . ^dH ф _ 1 dH r _ i r     dr    r dф     ^z ’

(8)

lr . ^d^ r . 1   ^d^ . ^d^z __ 0

r    dr r  dф dz     ’

(9)

We will search for unknown functions in the following form:

H_r . _ h_r (r)co ,

(10)

^Н Ф . ^_ к ф ^(r) si ,

(11)

H z ._ h z (r)si ,

(12)

E_r . _ e_r (r)si ,

(13)

Е ф ._ е ф (r)co ,

(14)

E z ._ e z (r)co ,

(15)

I_r . _ V_r (r)sic ,

(16)

1 ф . ^_ ! ф ^(r) coc ,

(17)

I z ._ i-z (r)sic ,

(18)

where

co _ cos( окр + /z) cos( cot) ,

(19)

si _ sin( ор + /z) sin( ot) ,

(20)

coc _ cos( ор + /z) sin( ot) ,

(21)

sic _ sin( ар + /z) cos( cot) ,

(22)

a system of equations with one argument and unknown functions

h(r), e(r), i(r) . This system of equations is as follows:

e_r(r)                е ф ⁽г)

_r + ^e r ^{(r)     Ф}_г a x- ^e z ^(r) = 0 ,

(21)

^- 1 ^-e z ^(r) a + ^е ф ^(r) X ^-~^h r = ⁰ ,

(22)

e r ⁽ r)x — e ‘ (r) + ^h ф = 0,

(23)

V' e ф (r)-^ -c + "51 h z = 0,

(24)

^r _r + ^ r ⁽r) + ^Ф _г a + x - h z (r) = 0 ,

(25)

1 h z ⁽ r) a -h , ⁽ r) - i r ⁽ r) = 0 ,

(26)

-h r ⁽ r)x — h z M-i , (r) = 0 ,

(27)

— + h. , (r) + ^ a-i z (r) = 0 .

(28)

^ r ) ^{+ i} r ^(г) + ^Ф _г ^a + X ^-i z ^(r) =0 .

(29)

Next, we introduce into consideration the coefficient k, which connects

the functions h and e :

h_r = ke_r ,

(30)

^h ,      ^ke , .

(31)

h_z = -ke_z.

(32)

We perform the change of variables according to (30-32) in

equations (21-

29) and rewrite them

e_r          б ф

— + ^e r ^- —^{a -xe} z = ⁰ ,

(33)

e_z            "1

^- — ^a + ^e , ^{x -} — ^ke r = ⁰ ,

(34)

-e z + e r X-^t^e , = 0 ,

(35)

~ + e^ - — a + k — e_z = 0 , r     ^ r         c ^z

(36)

k — + ke_r + k — a - kxe_z = 0 ,

(37)

-k^a + ke , x-i r = 0 ,

(38)

ke z - ke r X-i , = 0 ,

(39)

—k “ — ke^ + k ~ a — i_z = 0 , r      ^      r      ^z

(40)

■ ; ^{+ i} r + " 7 ^{a + x - i} z = 0 .

(41)

Note that equations (33) and (40) coincide for

i z = -kxe z

(42)

Note that equations (34) and (38) coincide for

i_T = — ke_r . ^r         c       ^r

(43)

Physics

Note th

at equations (35) and (39) coincide for

1 ф = k ^^- в ф .                                          (44)

Note th

at equations (36) and (40) coincide for

i z = -k^e z .                                     (45)

Из (42,

45) находим:

y-^to

X = —                                    (46)

Conside Finally,

T equation (33-36) e_r , в ф , e

ring (33, 41, 44, 45), we note that equations (33, 41) coincide. equations (33) and (37) coincide.

us, equations (37-41) can be excluded from the system of s and replaced by conditions (42-46). The remaining 4 equations are a system of differential equations with three unknowns _z . Let's rewrite (33-36) taking into account (46) and get:

7 ^{+ e} 7 ^—7«^— X ^e z = ^{0 ,}                       (47)

• — 7 a + в ф / — xke_r = 0 ,                       (48)

• —e z + в г / — k/в ф = 0 ,                          (49)

• 7 + ^в ф ^— 7® + ^k /в z = ^{0 .}                     (50)

Adding

(48, 49), we get:

or

—в z — 7® + / ⁽ 1 — k ⁾ (в ф +в г ) = 0 ,              (51)

^(в ф + ^в г ⁾ = _z ( _1-fc ) ⁽^ + _r ^в Z ⁾ ,                       ⁽⁵²⁾

Adding

1

r

(47, 50), we get:

(в ф +в г ) + — (в ф + в г ) — т(в ф +в г ) — ⁽ 1 — k ⁾ /в z = 0

е_ф = AN r ^a-1 , e_T = AMr ^a-1 ,

(58)

(59)

where M and N are unknowns. Substituting (58, 59) in (48, 49), we

find:

где M и N - неизвестные. Подставляя (58, 59,57в) в (48, 49), находим: — a Ar ^2-a + xAN r ^a-1 — xkAMr ^a-1 = 0 , — aAr ^a-1 + xAMr ^1-a — kxAN r ^a-1 = 0

or

—a + xN — xkM = 0 ,

(60)

• —a + xM — kxN = 0 .

It is seen that N = M . Hence,

• —a + x(1 — k)N = 0 ,

or

a

N = M = ——.

/(1-k)

Thus,

(61)

(60)

(62)

^a

e_r = e₍₀ = A— —— r ^{1 a} .

⁷    V     x(1-k)

Obviously,

(64)

i = e/p,

where p is the resistivity. Moreover, from (42, 46) we find:

(66)

1

.          ^p = ^—5            .

Since the values of ρ are known, we obtain:

(67)

к = — —. XP

(68)

From (43-46, 68) we get: ⁱ z = e z / Р ,

(69)

i-r = -e z / Р ,

(70)

i p = -e z / p .

(71)

[ S/l               FE p H z -E z H pl

S = S p = _n(E x H) = T]\E_zH r - E r H z .

[ S z_|                [E r H p -E p Hj

(3)

or, taking into account the previous formulas,

S r = u(e p h z - e z ^ p )co • si,

(4)

S_p = p(e_zh_r co ² — e_rh_z si ² ),

(5)

S_z = p{e_rh_p si ² — e_ph_r co ² ).

(6)

Substituting here (2.33-2.35, 2.68-2.70), we get:

S_r = u(—ke_pe_z + ke_ze_p)co • si ,

(7)

S_p = T](ke_ze_r co ² + ke_re_zsi ² ) = T]ke_re_z ,

(8)

S_z = T]^—ke_re_psi ² — ke_pe_r co ² ) = —T]ke_re_p .

(9)