Superconducting Magnet Field derivation with Hollow Cylindrical Conductor Current

1. ( )
( )
3
2
3
2
2
0
2 2
2 2
2
2
0
2 2
2
0
2 2
2
1 1
3
21
2
0
2 2
0
3 2 ( )
by ( )( )
Bio
2 ( )
t-Savart Law
4
( )ln ( )l
2 ( )
2
( )
c
c c
c
c
c
z
z
a z z b
z z ba
I r
B
r z
nI r
I nI dz d
a z z b a
z z b z z b
a z
r
z b
dB dzd
z
a
r
r
z
nI r
dzdr
r z
nI
dl xdB I
x
zB
π
μμ
μ
μ
μ
− +
− −
′⇒ =
+
→
⎛ ⎞+ − + +
⎜ ⎟= − + − − −
⎜ ⎟+ − + +⎝
=
⎠
⇒ =
+
+
×=
∫ ∫
( )
( )
2 2
2 2
2 2
1 1
n
c
c
a z z b a
a z z b a
⎧ ⎫⎛ ⎞+ − − +⎪ ⎪⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪+ − − +⎝ ⎠⎩ ⎭
3 3
2 2
3
2
2 2
2 2 2
2
2 2 2
1
2 2
2 2
2 2
tan
sec
( ) (1 tan )
sec
cos sin
(sec )
sinh ( )
ln( )
z r
r
dz d
r z
z
d d
r z
z r
dr z
zr z
z
dr z r r z
r z
θ
θ
θ
θ
θ
θ θ θ θ
θ
−
→
⇒ = =
+ +
= = =
+
=
+
⇒ = + +
+
∫ ∫
∫ ∫
∫
∫
n: turns per area
Magnetic Field Formula
z
zB
ˆIr
cz 2a
1a
b

2. 0
by (0, ) 0 (symmetry) & (0, )
0 ( , ) all ( )
(no source in Maxwell eq.)
1 1
( ) ( ) ( ( ) ) 0
0
Solve ( , ) & ( , )
0
r z
C
z r z r
r z
r z
z
B z B z
B r z B dr I
BB B B
B r z B r z
B
B
rB
r z z r r r
B B
z r
B
r z
θ
θ
θ
μ
θ θ
θ
∇× =
=
= ∀ ∈ =
∂∂ ∂ ∂ ∂∂
⇒ − − + − =
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂
⇒ − ⇒=
∂ ∂
+
∫ i
0
0
( , )
( , ) (0, )
0
( , )1
( , ) (0, )
1 1
( ) 0
( )
z
r
z
r
r
z
z
r r
r
r
B B
rB
r r r z
B
rB
r
B r z
r z dr B z
z
B
B r z
B r z r dr B z
r zz
θ
θ
′∂
′= − +
∂
∇ =
′∂
′ ′⇒
∂ ∂∂
⇒ + + =
∂ ∂ ∂
= +
∂∂
⇒ = −
∂ ∂ ∂
∫
∫
i
0
0
0
( )1
( ) (1)
( )
( ) (2)
z
r
r
z z
r
r
B r
B r r dr
r z
B r
B r dr B
z
′∂
′ ′= − − − − − −
∂
′∂
′= + − − − − −
∂
∫
∫
Magnetic Field (continued)

3. 0
(0)
0
(1) 0
2
(2) 20
2
3
(3) 30
3
0
0
Set (zero order)
( )1
( ) (1)
( )
( ) (2)
1
(1)
(2)
(1)
2
1
4
1
)
16
(2
z
r
r
z z
z z
z
r
z
z
z
r
r
r
B r
B r r dr
r z
B r
B r dr B
z
B B
B
B r
z
B
B r
z
B
B r
z
B
′∂
′ ′= − − − −
∂
′∂
′= + − − −
∂
=
⎧
⎪⎪
⎨
⎪
⎪⎩
⇒
⇒
∂
= −
∂
∂
= −
∂
∂
⇒
∂
⇒ =
∫
∫
4
(4) 40
4
5
(5) 50
5
6
(6) 60
6
1
64
1
384
1
23
(1
0
(2)
4
)
z
z
z
r
z
z
B
r
z
B
B r
z
B
B r
z
∂
=
∂
∂
= −
∂
∂
= −
∂
⇒
⇒
2 11
(2 1) 2 1 (2 1)0
2 1 2 1
0
2
(2 ) 2 (2 )0
2 2 2
0
( 1)
for
2 ( 1)! !
( 1)
for
2 ( !)
nn
n n nz
r r rn n
n
nn
n n nz
z z rn n
n
B
B r B B
n n z
B
B r B B
n z
++
+ + +
+ +
=
=
∞
∞
∂−
= =
+ ∂
∂−
= =
∂
∑
∑
0
0
1
( ( ) ) (the component)
& 0 (by symmetry)
&
The componen
(0, ) 0 (by symme
t of vecter potential
1
( , ) [ ( , )] (0, )
1
( ) [t y) (r
θ
θ θ
θ
θ
θ
θ
θ
θ
θ
θ
≡
∇× =
′ ′ ′= +
′
∂∂
⇒ − =
∂ ∂
∂
= ⇒
=
∂
= ⇒
∫
∫
r
z
r
r
z
r
z
A
rA B
A A
A B
A r z r B r z dr A z
r
A r
z
r r
A
r B
r
A z
2 4 6
3 5 70 0 0
0 2 4 6
2
(2 1) 2 1 (2 1)0
2 1 2
0
)]
1 1 1 1
2 16 384 18432
( 1)
for
2 ( 1)! !
θ
θ θ θ
+ + +
+
=
∞
′ ′
∂ ∂ ∂
= − + −
∂ ∂ ∂
∂−
= =
+ ∂
∑
z z z
z
nn
n n nz
n n
n
r dr
B B B
A B r r r r
z z z
B
A r A A
n n z
Magnetic Field (continued)

4. 0
2 2
2 3 4
2 2
0 0 0
2 2
2 3
2 2
0 0 0
Approximating
1 1
( ) (0) ( ) (1)
Th
2! 3!
ink about Taylor's series of ( ) a
1 1
t
( ) (0)
2! 3
0
!
n
z
n
x x x
x x x
B
z
df d f d f
f x f x x x x O x
dx dx dx
df d f d f
f x f x x x x
dx d
f x x
x dx
= = =
= = =
∂
∂
= + + + +
− = − + − +
=
4
2 2 3 2
2 3 4
2 2
0 0 0
2 2 3 2
2 3 4
2 2
0 0 0
( ) (2)
2 2
(2 ) (0) 2 16 (2 ) (3)
2! 3!
2 2
( 2 ) (0) 2 16 ( 2 ) (4)
2! 3!
(1)
x x x
x x x
O x
df d f d f
f x f x x x x O x
dx dx dx
df d f d f
f x f x x x x O x
dx dx dx
d
= = =
= = =
⎧
⎪
⎪
⎪
⎪
⎪⎪
⎨
⎪
⎪
−
= + + + +
−
⎪
⎪
⎪
⎪
= −
⎩
− + +
⇒
−
( )
( )
[ ] [ ]
( )
2
2
3
2
3
(2 ) 8 ( ) 8 ( ) ( 2 )
0
12
1
(error: )
2!
1
(1) (2) (error: )
( ) (0)
0 (zero order)
( ) ( )
0 (1st orde
3!
8 (1) (2) (3) (4)
r)
2
df f x f x f x f x
dx x
f f x f
dx x
df f x f x
d f
x
dx
d f
x
dx xx d
+ − − − −
=
+ ⇒
× + + +
−
=
− −
⇒
=
4
4
4
1
(error: ) (3rd order
4!
)
d f
x
dx
( )
( )
2
2 2
2
2 2
( ) 2 (0) ( )
Simiarly 0 (1st order)
(2 ) 16 ( ) 30 (0) 16 ( ) ( 2 )
& 0 (3rd order)
12
− + −
=
− + − + − − −
=
d f f x f f x
dx x
d f f x f x f f x f x
dx x
Magnetic Field (continued)

5. The magnetic field profile in axis direction at the coil current 1 A
in NTHU HF lab.
0 30 60 90 120 150
z (cm)
0
100
200
300
400
500
Bz(Gauss)
Main
Long
ShortTrim3
Trim2
Trim1
Coil current 1 A
Magnetic Field Profile