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References

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  1. A. Sommerfeld, “Über den Wechselstromwiderstand von Spulen.” Ann. d. Physik,  329, p. 609, 1907.
    [CrossRef]
  2. W. Lenz, “Über die Kapazität der Spulen und deren Widerstand und Selbstinduktion bei Wechselstrom.” Ann. d. Physik,  342, 1912, p. 923.
    [CrossRef]
  3. See Sur le calcul des Pertes dans les selfs d’antennes en tubes de cuivre. Note de H. Abraham, L. Bloch, and E. Bloch, Radio telegraphic Militaire, 1919, “.”
  4. See G. Breit, “A Method of Solving Problems on Skin Effect.” Forthcoming publication.

1912 (1)

W. Lenz, “Über die Kapazität der Spulen und deren Widerstand und Selbstinduktion bei Wechselstrom.” Ann. d. Physik,  342, 1912, p. 923.
[CrossRef]

1907 (1)

A. Sommerfeld, “Über den Wechselstromwiderstand von Spulen.” Ann. d. Physik,  329, p. 609, 1907.
[CrossRef]

Abraham, de H.

See Sur le calcul des Pertes dans les selfs d’antennes en tubes de cuivre. Note de H. Abraham, L. Bloch, and E. Bloch, Radio telegraphic Militaire, 1919, “.”

Bloch, E.

See Sur le calcul des Pertes dans les selfs d’antennes en tubes de cuivre. Note de H. Abraham, L. Bloch, and E. Bloch, Radio telegraphic Militaire, 1919, “.”

Bloch, L.

See Sur le calcul des Pertes dans les selfs d’antennes en tubes de cuivre. Note de H. Abraham, L. Bloch, and E. Bloch, Radio telegraphic Militaire, 1919, “.”

Breit, G.

See G. Breit, “A Method of Solving Problems on Skin Effect.” Forthcoming publication.

Lenz, W.

W. Lenz, “Über die Kapazität der Spulen und deren Widerstand und Selbstinduktion bei Wechselstrom.” Ann. d. Physik,  342, 1912, p. 923.
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Über den Wechselstromwiderstand von Spulen.” Ann. d. Physik,  329, p. 609, 1907.
[CrossRef]

Ann. d. Physik (2)

A. Sommerfeld, “Über den Wechselstromwiderstand von Spulen.” Ann. d. Physik,  329, p. 609, 1907.
[CrossRef]

W. Lenz, “Über die Kapazität der Spulen und deren Widerstand und Selbstinduktion bei Wechselstrom.” Ann. d. Physik,  342, 1912, p. 923.
[CrossRef]

Other (2)

See Sur le calcul des Pertes dans les selfs d’antennes en tubes de cuivre. Note de H. Abraham, L. Bloch, and E. Bloch, Radio telegraphic Militaire, 1919, “.”

See G. Breit, “A Method of Solving Problems on Skin Effect.” Forthcoming publication.

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Figures (10)

F. 1
F. 1

Illustration of division of coil.

F. 2
F. 2

Diagram for discussion of relation of current density and magnetic field

F. 3
F. 3

Diagram for discussion of magnetic field at infinity.

F. 4
F. 4

Lines of magnetic force for problem I.

F. 5
F. 5

Lines of magnetic force for problem II.

F. 6
F. 6

Lines of magnetic force for problem III.

F. 7
F. 7

Variation of square of magnetic intensity along circumference of the wire’s cross section according to Sommerfeld’s calculation.

F. 7a
F. 7a

Variation of resistance with pitch.

F. 8
F. 8

Illustration of Method of Inversion used in solution of potential problem I.

F. 9
F. 9

Degenerate case of problem I.

Equations (138)

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2 I 10 r
curl H = 4 π 10 U
H E F ¯
4 π 10 u E F ¯
4 π 10 u E F ¯ = H E F ¯
u = 10 4 π H
div H = 0 .
4 π 10 I
( H p 1 + H p 2 ) A D = 4 π 10 I
H p 1 + H p 1 = 4 π I 10 h
A D = h ( the pitch of the winding ) .
H p 2 2 π I 10 h = H p 1 + 2 π I 10 h = H p
H p = 2 π I 10 h
H p = + 2 π I 10 h
W ( z ) = log z s = 1 C 2 s s z 2 s z 2 s h 2 s
A 2 s s 1 = 1 , 2 , 2 A 2 s 1 A 2 s + 2 s 1 h 4 s 1 ( 2 s + 2 s 1 1 2 s 1 ) + s 1 , s 2 = 1 , 2 , A 2 s + 2 s 2 2 A 2 s 1 + 2 s 2 2 A 2 s 1 h 4 ( s 1 + s 2 ) ( 2 s + 2 1 1 2 s 1 ) ( 2 s 1 + 2 s 1 1 2 s 2 )
A 2 s = 1 + 1 2 2 s + 1 3 2 s + ; ( m n ) = / m _ / n _ / m n _
C 2 s = A 2 s A 2 s + 2 2 A 2 h 4 ( 2 s + 1 2 ) + h 8 { A 2 s + 4 2 A 4 ( 2 s + 3 4 ) + A 2 s + 2 2 A 4 2 A 2 ( 2 s + 1 2 ) ( 3 2 ) } + h 12 { A 2 s + 6 2 A 6 ( 2 s + 5 6 ) + A 2 s + 4 2 A 6 2 A 2 ( 2 s + 3 4 ) ( 5 2 ) + A 2 s + 2 2 A 6 2 A 4 ( 2 s + 1 2 ) ( 5 4 ) A 2 s + 2 ( 2 A 4 ) 2 2 A 2 ( 2 s + 1 2 ) ( 3 2 ) 2 } + etc .
θ s = 1 2 C 2 s s 1 h 2 s sin 2 s θ
1 s = 1 4 C 2 s h 2 s cos 2 s θ
H θ ( 1 ) = ( 2 I / 10 ) [ 1 s = 1 4 C 2 s h 2 s cos 2 s θ ]
a 2 s + 1 2 s + 1 = b 2 s + 1
b 2 s + 1 b 1 = 2 h ( 2 s + 2 ) [ A 2 s + 2 2 h 2 s 1 = 1 A 2 s 1 + 2 A 2 s + 2 s 1 + 2 ( 2 s + 2 s 1 + 1 2 s 1 ) h 4 s 1 + ( 2 h 2 ) 2 s 1 , s 2 = 1 , 2 , . A 2 s 2 + 2 A 2 s 1 + 2 s 2 + 2 A 2 s + 2 s 1 + 2 ( 2 s + 2 s 1 + 1 2 s 1 ) ( 2 s 1 + 2 s 2 + 1 2 s 2 ) h 4 ( s 1 + s 2 ) ]
h 2 s + 2 b 2 s + 1 b 1 = A 2 s + 2 2 A 4 A 2 s + 4 ( 2 s + 3 2 ) h 6 2 A 6 A 2 s + 6 ( 2 s + 5 4 ) h 10 + 2 2 A 4 A 6 A 2 s + 4 ( 2 s + 3 2 ) ( 5 2 ) h 12 2 A 8 A 2 s + 8 ( 2 s + 7 2 ) h 14 + 2 2 [ A 4 A 8 A 2 s + 6 ( 2 s + 5 4 ) ( 7 8 ) + A 6 A 8 A 2 s + 4 ( 2 s + 3 2 ) ( 7 4 ) ] h 16 [ 2 A 10 A 2 s + 10 ( 2 s + 9 ) + 2 3 A 4 A 2 6 A 2 s + 4 ( 2 s + 3 2 ) ( 5 2 ) 2 ] h 18 + 2 2 [ A 4 A 10 A 2 s + 8 ( 2 s + 7 6 ) ( 9 2 ) + A 8 A 10 A 2 s + 4 ( 2 s + 3 2 ) ( 9 6 ) + A 6 A 10 A 2 s + 6 ( 2 s + 5 4 ) ( 9 4 ) ] h 20
a 1 = b 1
a 1 sin θ + a 3 sin 3 θ + a 5 sin 5 θ +
1 = 2 1 b 1 s 1 = 1 ( 2 s 1 1 ) b 2 s 1 1 A 2 s 1 h 2 s 1
b 1 = [ 1 2 + s 1 = 1 ( 2 s 1 1 ) ( b 2 s 1 1 b 1 ) A 2 s 1 h 2 s 1 ] 1
H θ ( 2 ) = H o x ( a 1 sin θ + a 3 sin 3 θ + a 5 sin 5 θ + )
a 2 s + 1 2 s + 1 = β 2 s + 1
h 2 s + 2 β 2 s + 1 2 β 1 = A 2 s + 2 + 2 h 2 s 1 = 1 A 2 s 1 + 2 A 2 s + 2 s 1 + 2 ( 2 s + 2 s 1 + 1 2 s 1 ) h 4 s 1 + ( 2 h 2 ) 2 s 1 , s 2 = 1 , 2 , A 2 s 1 + 2 A 2 s 1 + 2 s 2 + 2 A 2 s + 2 s 1 + 2 ( 2 s + 2 s 1 + 1 2 s 1 ) ( 2 s 1 + 2 s 2 + 1 2 s 2 ) h 4 ( s 1 + s 2 ) +
β 1 = [ 1 2 + 1 ( 2 s 1 ) ( β 2 s 1 / β 1 ) A 2 s h 2 s ] 1
a 1 cos θ a 3 cos 3 θ a 5 cos 5 θ
H θ ( 3 ) = H 0 y ( a 1 cos θ + a 3 cos 3 θ + a 5 cos 5 θ + )
H θ = H θ ( 1 ) + H θ ( 2 ) + H θ ( 3 ) = ( 2 I / 10 ) [ 1 s = 1 4 C 2 s ( cos 2 s θ ) h 2 s ] + H o x s = 1 a 2 s 1 sin ( 2 s 1 ) θ H o y s = 1 a 2 s 1 cos ( 2 s 1 ) θ
u = 10 4 π H θ
0 2 π u d θ = I
σ u 2 d s
0 2 π σ u ( θ ) 2 d θ
R = 0 2 π σ u ( θ ) 2 I 2 d θ
u = I / ( 2 π )
R 0 = σ 2 π
u = 10 4 π H θ ] R R 0 = 1 + 1 2 s = 1 ( 4 C 2 s h 2 s ) 2 + ( 10 H o x 2 I ) 2 a = 1 a 2 s 1 2 2 + ( 10 H o y 2 I ) 2 s = 1 a 2 s 1 2 2
R R 0 = 1 + 8 s = 1 ( C 2 s h 2 s ) 2
A 2 s = ( ) s 1 ( 2 π ) 2 s B 2 s 2 ( 2 s ) !
A 2 = π 2 6 , A 4 = π 4 90 , A 6 = π 6 945 , A 8 = π 8 9450 , etc .
C 2 = 1.190 C 4 = 0.23
R R 0 = 1 + 8 ( ( 1.190 ) 2 16 + ( 0.06 ) 2 256 + ) = 1.71
3.42
a 1 sin θ + a 3 sin 3 θ + a 5 sin 5 θ +
u ( θ ) = sin θ + a 3 a 1 sin 3 θ + a 5 a 1 sin 5 θ +
0 π u ( θ ) d θ = 2 ( 1 + a 3 3 a 1 + a 5 5 a 1 + )
σ 0 π u ( θ ) 2 d θ = π σ 2 ( 1 + ( a 3 a 1 ) 2 + ( a 5 a 1 ) 2 + )
R = π σ 8 1 + ( a 3 a 1 ) 2 + ( a 5 a 1 ) 2 + ( 1 + a 3 3 a 1 + a 5 5 a 1 + ) 2
R 0 = σ 2 π
R R 0 = π 2 4 1 + ( a 3 a 1 ) 2 + ( a 5 a 1 ) 2 + ( 1 + a 3 3 a 1 + a 5 5 a 1 + ) 2
b 3 b 1 = 0.101 a 3 a 1 = 0.304 b 1 b 1 = 0.013 a 5 a 1 = 0.065
R R 0 = π 2 4 1.097 ( 0.889 ) 2 = 3.42 ( The number 0.889 = π 2 16 log 2 )
R = 0 π σ H θ 2 d θ ( 0 π H θ d θ ) 2
R = σ 2 π
R R 0 = 2 π 0 π H θ 2 d θ ( 0 π H θ 2 d θ ) 2
0.378 × 2 π
0.345 × 2 π
0.345 0.378 × 3.73 = 3.41
F x = 2 π a H o y = x M ( x ) = M ( l 1 ) + M ( l 2 )
2 U x 2 + 2 U y 2 = 0
W ( z )
W ( z ) = V ( x , y ) + j U ( x , y )
V x = U y V y = U x
0 2 π ( V ρ ) ρ = 1 d θ = 2 π
2 V x 2 + 2 V y 2 = 0
V e = Ψ ( ρ , θ )
V i = Ψ ( 1 ρ , θ ) = χ ( ρ , θ )
( V e ρ ) ρ = 1 = ( V i ρ ) ρ = 1
( Ψ ( ρ , θ ) ρ ) ρ = 1 = ( χ ( ρ , θ ) ρ ) ρ = 1
( χ ( ρ , θ ) ρ ) ρ = 1 ( Ψ ( ρ , θ ) ρ ) ρ = 1
( V e ρ ) ρ = 1 = ( V i ρ ) ρ = 1
V x = U y V y = U x
2 π j
W ( z ) log z
j θ = cos θ + j sin θ
W ( z ) = log z C 2 s s 1 h 2 s ( z 2 s z 2 s )
V = log ρ s = 1 C 2 s s 1 h 2 s ( ρ 2 s ρ 2 s ) cos 2 s θ
σ = 1 4 π ( V ρ ) ρ = 1 = 1 4 π [ 1 s = 1 4 C 2 s h 2 s cos 2 s θ ]
( ) 2 σ log r d s
1 2 π 2 π [ 1 s = 1 4 C 2 s h 2 s cos 2 s θ ] log ( n h cos θ + cos θ 0 ) 2 + ( sin θ sin θ 0 ) 2 d θ
r n = ( n h + cos θ 0 ) 2 + sin 2 θ 0 tan θ n = sin θ 0 n h + cos θ 0
log r n 2 2 r n cos θ + 1 = log r n s = 1 s 1 r n s cos s θ
log r n + s = 1 s 1 r n 2 s h 2 s C 2 s cos 2 s θ
log | 1 n h | s = 1 C 2 s s 1 h 2 s ( n h ) 2 s
n = n = + log r n | n h | + n = n = + s = 1 s 1 h 2 s C 2 s ( r n 2 s cos 2 s θ n ( n h ) 2 s )
m cos m θ 2 π
σ = 1 4 π [ 1 s = 1 4 h 2 s C 2 s cos 2 s θ ]
s = 1 s 1 h 2 s C 2 s cos 2 s θ
V = s = 1 s 1 h 2 s C 2 s cos 2 s θ 0 + n = n = + log r n | n h | + n = n = + s = 1 s 1 h 2 s C 2 s ( r n 2 s cos 2 s θ n ( n h ) 2 s )
V = 0
z = j θ 0
n = n = + ( 1 z n h + 1 n h ) = 2 s = 1 A 2 s z 2 s 1 h 2 s
A 2 s = 1 + 1 2 2 s + 1 3 2 s +
n = n = + ( 1 z n h ) 2 s = 2 [ A 2 s h 2 s + ( 2 s + 1 2 s 1 ) A 2 s + 2 h ( 2 s + 2 ) z 2 + ( 2 s + 3 2 s 1 ) A 2 s + 4 h ( 2 s + 4 ) z 4 + ]
n = n = + ( r n 2 s cos 2 s θ n ( n h ) 2 s ) = 2 p = 1 ( 2 s + 2 p 1 2 s 1 ) A 2 s + 2 p h 2 s 2 p cos 2 p θ 0
n = n = + s = 1 s 1 h 2 s C 2 s ( r n 2 s cos 2 s θ n ( n h ) 2 s ) = s = 1 2 s 1 h 2 s C 2 s p = 1 ( 2 s + 2 p 1 2 s 1 ) A 2 s + 2 p cos 2 p θ 0 h 2 s + 2 p
n = n = + log ( n h + cos θ 0 ) 2 + sin 2 θ 0 n 2 h 2
s = 1 s 1 h 2 s A 2 s cos 2 s θ 0
C 2 s = A 2 s s 1 = 1 2 C 2 s 1 A 2 s + 2 s 1 ( 2 s + 2 s 1 1 2 s 2 ) h 4 s 1
log ( h 2 ) s = 1 s 1 2 2 s C 2 s cos 2 s θ
lim h = C 2 s 1 C 2 s 2
lim h = C 2 s = A 2 s
h 4 ( C 2 s A 2 s ) = 2 C 2 A 2 s + 2 ( 2 s + 1 2 ) 2 C 4 A 2 s + 4 ( 2 s + 3 4 ) h 4 +
lim h = h 4 ( C 2 s A 2 s ) = 2 A 2 A 2 s + 2 ( 2 s + 1 2 )
2 A 2 A 2 s + 2 ( 2 s + 1 2 )
C 2 s = A 2 s s 1 = 1 2 A 2 s 1 A 2 s + 2 s 1 ( 2 s + 2 s 1 1 2 s 1 ) h 4 s 1 + s 1 , s 2 = 1 2 A 2 s 1 2 A 2 s 1 + 2 s 2 2 A 2 s + 2 s 1 ( 2 s 2 + 2 s 1 1 2 s 2 ) ( 2 s 1 + 2 s 2 1 2 s 2 ) h 4 ( s 1 + s 2 ) +
C 2 s = A 2 s A 2 s + 2 2 A 2 ( 2 s + 1 2 ) h 4 + [ A 2 s + 4 2 A 4 ( 2 s + 3 4 ) + A 2 s + 2 2 A 4 2 A 2 ( 2 s + 1 2 ) ( 3 2 ) ] h 8 + [ A 2 s + 6 2 A 6 ( 2 s + 5 6 ) + A 2 s + 4 2 A 6 2 A 2 ( 2 s + 3 4 ) ( 5 2 ) + A 2 s + 2 2 A 6 2 A 4 ( 2 s + 1 2 ) ( 5 4 ) A 2 s + 2 ( 2 A 4 ) 2 2 A 2 ( 2 s + 1 2 ) ( 3 2 ) 2 ] h 12 +
C 2 s = C 2 s
I m W ( z )
I m ( z W ( z ) ) = 0
I m ( z W ( z ) ) = 0
σ = 1 4 π s = 1 a 2 s 1 sin ( 2 s 1 ) θ
sin θ 0 = s = 1 a 2 s 1 2 ( 2 s 1 ) sin ( 2 s 1 ) θ 0 n = n = + s = 1 a 2 s 1 2 ( 2 s 1 ) r n 2 s + 1 sin ( 2 s 1 ) θ
n = n = + ( z n h ) 2 p + 1 = 2 s = 1 A 2 s + 2 p 2 z 2 s 1 ( 2 s + 2 p 3 2 p 2 ) h 2 s 2 p + 2
sin θ 0 = s = 1 a 2 s 1 sin ( 2 s 1 ) θ 0 2 ( 2 s 1 ) p , s = 1 a 2 s 1 A 2 s + 2 p 2 2 s 1 h 2 s 2 p + 2 ( 2 s + 2 p 3 2 s 2 ) sin ( 2 s 1 ) θ 0
a 2 p 1 2 p 1 + s = 1 , 2 , 2 a 2 s 1 2 s 1 A 2 s + 2 p 2 ( 2 s + 2 p 3 2 s 2 ) h 2 s 2 p + 2 = 0
1 = a 1 2 s = 1 a 2 s 1 2 s 1 A 2 s h 2 s ( 2 s 1 2 s 2 )
b 2 s 1 = a 2 s 1 2 s 1
b 2 s 1 + s 1 = 1 2 A 2 s 1 + 2 s 2 ( 2 s + 2 s 1 3 2 s 1 2 ) h 2 s 2 s 1 + 2 b 2 s 1 1
1 + b 1 2 + s 1 = 1 A 2 s 1 h 2 s 1 ( 2 s 1 1 2 s 1 2 ) b 2 s 1 1 = 0
b 2 s + 1 b 1 = 2 A 2 s + 2 h 2 s 2 s 1 = 1 b 2 s 1 + 1 b 1 2 A 2 s + 2 s 1 + 2 ( 2 s + 2 s 1 + 1 2 s 1 ) h ( 2 s + 2 s 1 + 2 )
b 2 s + 1 b 1
b 2 s + 1 b 1 = 2 A 2 s + 2 h ( 2 s + 2 ) + s 1 = 1 2 2 A 2 s 1 + 2 A 2 s + 2 s 1 + 2 ( 2 s + 2 s 1 + 1 2 s 1 ) h ( 2 s + 4 s 1 + 4 ) s 1 , s 2 = 1 2 3 A 2 s 2 + 2 A 2 s 1 + 2 s 2 + 2 A 2 s + 2 s 1 + 2 ( 2 s + 2 s 1 1 2 s 1 ) ( 2 s 1 + 2 s 2 + 1 2 s 2 ) h ( 2 s + 4 s 1 + 4 s 2 + 6 ) +
Im ( z W ( z ) )
σ = 1 4 π ( a 1 cos θ + a 3 cos 3 θ + )
1 2 π 0 2 π log ( n h cos θ + cos θ 0 ) 2 + ( sin θ sin θ 0 ) 2 s = 1 a 2 s 1 cos ( 2 s 1 ) θ d θ
cos θ = s = 1 a 2 s 1 2 ( 2 s 1 ) cos ( 2 s 1 ) θ 0 n = n = + s = 1 a 2 s 1 2 ( 2 s 1 ) r n 2 s + 1 cos ( 2 s 1 ) θ n
n = n = + r n 2 p + 1 cos ( 2 p 1 ) θ n = 2 s = 1 A 2 s + 2 p 2 cos ( 2 s 1 ) θ ( 2 s + 2 p 3 2 p 2 ) h 2 s 2 p + 2
cos θ 0 = s = 1 a 2 s 1 2 ( 2 s 1 ) cos ( 2 s 1 ) θ 0 + s , p = 1 a 2 s 1 2 s 1 A 2 s + 2 p 2 ( 2 s + 2 p 3 2 s 2 ) h 2 s 2 p + 2 cos ( 2 p 1 ) θ 0
{ a 2 p 1 2 p 1 = s = 1 2 a 2 s 1 ( 2 s 1 ) 1 A 2 s + 2 p 2 ( 2 s + 2 p 3 2 s 2 ) h 2 s 2 p + 2 if p > 1 anh 1 = a 1 2 + s = 1 a 2 s 1 ( 2 s 1 ) 1 A 2 s ( 2 s 1 2 s 2 ) h 2 s
β 2 s 1 = a 2 s 1 / ( 2 s 1 ) β 2 s 1 = s 1 = 1 2 A 2 s 1 + 2 s 2 ( 2 s + 2 s 1 3 2 s 1 1 ) h 2 s 1 2 s + 2 β 2 s 1 1 1 = β 1 2 + s 1 = 1 A 2 s 1 ( 2 s 1 1 2 s 1 2 ) h 2 s 1 β 2 s 1 1
h 2 s + 2 2 β 2 s + 1 β 1 = A 2 s + 2 + 2 A 4 A 2 s + 4 ( 2 s + 3 2 ) h 6 + 2 A 6 A 2 s + 6 ( 2 s + 5 4 ) h 10 + 2 2 A 4 A 6 A 2 s + 4 ( 2 s + 3 2 ) ( 5 2 ) h 12 +