TELKOM
NIKA
, Vol. 11, No. 10, Octobe
r 2013, pp. 6
017 ~ 6
024
ISSN: 2302-4
046
6017
Re
cei
v
ed Ap
ril 18, 2013; Revi
sed
Jul
y
1
1
, 2013; Acce
pted Jul
y
21,
2013
Optimization and Construction of Single-side Nuclear
Magnetic Resonance Magnet
Ji Yongliang*, He Wei, He
Xiaolong
State Ke
y
L
a
b
o
rator
y
of Po
w
e
r T
r
ansmission Equi
pme
n
t & S
y
stem Secur
i
t
y
an
d Ne
w
T
e
chno
log
y
,
Cho
ngq
in
g Uni
v
ersit
y
, Ch
on
g
q
in
g 40
003
0, Chin
a,
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: ji
yon
g
li
ang
@
g
mail.c
o
m
A
b
st
r
a
ct
Sing
le-si
ded N
M
R devices ca
n oper
ate un
de
r conditi
ons in
a
ccessibl
e to co
nventi
o
n
a
l NM
R w
h
ile
featurin
g porta
bility a
nd the a
b
ility to
an
aly
z
e arbitrary-si
z
e
d obj
ects. In th
is pap
er, a semi-
e
ll
iptic Ha
lb
a
c
h
ma
gn
et arr
a
y
w
a
s desi
g
n
e
d
and
b
u
ilt for
si
ngl
e-sid
e
N
u
cl
ear M
agn
etic
Reso
nanc
e (
N
MR). W
e
pr
es
ent a
n
easy-to-i
mple
me
nt target fi
eld
alg
o
rith
m f
o
r si
n
g
le-s
id
e NMR ma
gnet desi
gn base
d
on
Gra
m
-Sch
mi
d
t
Orthogon
al
me
thod. T
he creat
ing
ma
gn
et
ic field of d
e
sig
n
e
d
magn
et st
ructure coul
d ach
i
eve best flatn
e
s
s
in the reg
i
on
of interestin
g for NMR app
li
cations.
T
he o
p
timi
z
i
n
g
resu
l
t
show
s that the best
mag
n
e
t
structure
can g
ener
ate mag
n
e
t
ic
fields
w
h
ic
h
flatly distrib
u
t
ed i
n
the h
o
ri
zontal
directi
on
and th
e gr
adi
e
n
t
w
a
s distribute
d
in the vertic
al dir
e
ction w
i
th grad
ie
nt of 2mT
/
mm. T
he field stren
g
th and gr
ad
ient
w
e
r
e
me
asur
ed by a
three di
me
nsi
ons Ha
ll pro
be
and a
g
re
ed w
e
ll w
i
th the simul
a
tions.
Ke
y
w
ords
: sin
g
le-si
de
ma
gn
et, Nuclear Ma
gnetic R
e
so
na
nce, curve fittin
g
, opti
m
i
z
a
t
i
on
desi
g
n
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Comp
ared
wi
th the tra
d
itio
nal en
clo
s
e
d
nucl
ear mag
n
e
tic resona
nce (NMR)
equi
pment,
unilateral nu
clea
r mag
n
e
t
ic re
son
a
n
c
e equip
m
ent
sca
n obj
ect
s
from the
surfa
c
e
with
out
encompa
ssin
g them [1, 2]. At
the sam
e
time, its volume is
small
and po
rtable,
so it has b
e
e
n
widely u
s
e
d
i
n
food a
nalysis an
d qu
ality cont
rol, ma
te
rial scie
nce, physi
cal g
e
o
g
rap
h
y, etc.[3
-9].
In unilate
ral
nucl
ear mag
netic
re
son
a
n
c
e e
quip
m
ent
, the gen
erati
on of its
mai
n
mag
netic fi
eld
depe
nd
s on
the perm
ane
nt magnet. T
he typical m
agnet st
ru
ctu
r
es i
n
cl
ude
U sh
ape m
a
gnet
[10], Halba
c
h array [11, 12], and b
a
rrel-sha
pe
magnet et
c. At the same
time, perma
nent
magnet
s
al
so
have bee
n use
d
to gen
e
r
ate gra
d
ient
m
agn
etic fie
l
d. In the lite
r
ature [10], it
is
pointed o
u
t that singl
e st
rip mag
net h
a
s go
od gr
a
d
ient magn
etic field ch
aracteri
stics in
a
distan
ce al
on
g the magn
etization di
re
ction of t
he upp
er and lo
we
r
surfa
c
e
cent
e
r
s. On the b
a
sis
of supp
osi
n
g
high-permea
b
ility material
surfa
c
e
a
s
a
n
equal
scal
ar mag
netic
potential face
, a
type of mag
net structu
r
e
with ho
rizo
ntally uni
form magn
etic
field whil
e vertically g
r
a
d
i
ent
magneti
c
field wa
s co
nstructed by mea
n
s of se
paration of variable
s
[13].
The currently desig
n of unilateral n
u
cl
ear ma
gneti
c
reso
nan
ce
magnet ne
ed
s extra
gradi
ent coils due to la
ck o
f
gradie
n
t ma
gnetic fiel
d.
Howeve
r, the d
e
sig
n
theo
ry
of gra
d
ient
co
ils
for unilate
ral
nucl
ear m
a
gnetic
re
son
ance is
still not mature. The ma
gnet
desi
gne
d through
method of separation of
variabl
es can g
ene
rate
gra
d
ient
m
agneti
c
field
but its en
cl
ose
d
stru
cture ma
kes it hard to a
pply in
unilate
ral nu
clea
r m
agneti
c
re
son
ance.
In the p
ape
r, we
propo
se
d imp
r
oveme
n
t and
optimi
z
ation
on
the
tradition
al
Halba
c
h
magnet
stru
cture throug
h
the cu
rve fitting me
thod
and o
b
taine
d
a ne
w uni
lateral m
agn
et
stru
cture. Th
e desi
gne
d magnet a
r
ra
y can ge
ner
ate magn
etic field with sound flatne
ss in
hori
z
ontal direction and gradient
in
ve
rti
c
al
di
re
ct
ion i
n
a spe
c
ific a
r
ea. So the
d
e
sig
ned m
a
g
net
stru
cture do
e
s
not n
eed
e
x
tra gra
d
ient
coil
s [
14], wh
ich
simplified
the gra
d
ient
codi
ng
syste
m
desi
gn.
2. Calculatio
n of the Ma
g
n
etic Field
The calculati
on of the m
agneti
c
field
of
perm
anent
magnet a
d
o
p
ts scal
ar m
agneti
c
potential met
hod on th
e b
a
si
s of magn
etic charge
t
heory. In ord
e
r to imp
r
ove
the cal
c
ulati
o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 601
7 –
6024
6018
accuracy, thi
s
pa
pe
r ma
d
e
use of
se
cond o
r
d
e
r
fini
te eleme
n
t al
gorithm [1
5]
and o
b
taine
d
th
e
followin
g
finite element eq
uation,
KR
(1)
,
,
1
,
2
...10
e
ij
i
j
V
KN
N
d
V
i
j
(2)
1,
2
.
.
.
6
e
ir
i
S
Rn
B
N
d
s
i
(3)
whe
r
e
n
rep
r
ese
n
ts the
o
u
tward unit n
o
rmal
ve
ctor on the
surfa
c
e of the m
a
gnet,
B
r
i
s
t
he
resi
dual m
a
g
netism of th
e magn
et,
Ni
is the sh
a
pe functio
n
of second
order tetra
hed
ron
element. After solving the
scalar ma
gne
tic potential
φ
,
the magnetic field distrib
u
tion aro
und
the
spa
c
e of pe
rmanent ma
gn
et can be g
o
t.
3. Design of
Unilate
ral Magne
t Struc
t
ure
Due to the g
e
neratio
n of u
n
iform ma
gn
etic fi
eld in its cylindri
c
al
ca
vity, Halba
c
h
magnet
is ha
s b
een
paid g
r
eat at
tention to. Figure
1(
a)
sh
ows Halba
c
h
magnet
con
s
tru
c
ted by
16
magneti
c
ba
rs with the same sh
ape
and re
sid
ual
magnetism [11]. Howeve
r, its stru
cture is
encl
o
sed, wh
ich ma
ke
s it
difficult to be appli
ed in u
n
ilateral n
u
cl
ear ma
gnetic reso
nan
ce.
For
this rea
s
o
n
, the belo
w
9 m
agneti
c
bars i
s
rem
a
ine
d
(Figure 1(b
)).
(a)
z
y
o
(b)
Figure 1. (a)Halb
a
ch mag
net array. (b)
Single-side m
agnet array.
The
m
agn
et size
i
s
3 cm ×
3
cm × 10 cm,
Br
is 1.2
8
T. Th
e a
r
ro
w i
s
th
e di
re
ction of
bar
magnet
mag
n
e
tization. In
Y
O
Z pl
ane, Fi
g
u
re
3
sho
w
s t
he Z
-
compo
n
ent of m
agn
e
t
ic flux de
nsity
in regio
n
of interestin
g (RO
I
) in a squa
re
area of 5
c
m l
ength (Fi
gure
2).
From Fig
u
re 2, it can be
seen that the
m
agneti
c
fiel
d pre
s
e
n
ts g
r
adient ap
pro
x
imation
cha
nge
s alon
g Z axis and
equip
o
tential line sa
gs to
the neg
ative directio
n of Z axis. The rea
s
on
for this is du
e
to the
clo
s
e
r
distan
ce
from
point 1
an
d
point 3 to
the
magn
et, wh
e
r
e the
mag
n
e
t
ic
field is st
ron
g
while the fu
rt
her di
stan
ce f
r
om p
o
int 2 to the mag
net
, where the
magneti
c
fiel
d is
wea
k
. T
herefore,
we
can
dedu
ce
that i
f
move th
e
m
agnet
s in
Fig
u
re
1(b) on
the b
o
th
side
s to
the cente
r
of
Y and move
down the ma
gnet, the ma
gnet equi
pote
n
tial line in ROI area
will sag
more
sh
arply,
sho
w
n
as Fi
gure
4. Fo
r th
is spe
c
if
ic
move, please refer to Figure 3. If performing
the reverse o
peratio
n on the magn
et in Figure 1
(
b
)
, the magnet e
quipote
n
tial line in ROI area
will sag towards the upper
side
of Z axis (see in Fi
gure 5).
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
i
zation and Con
s
tru
c
tion of Single-side
Nu
clea
r Magneti
c
Re
sonan
ce … (Ji
Yongliang
)
6019
Y(
m)
Z(
m
)
Bz
(
T
)
-0
.
0
2
-0
.
0
1
0
0.
0
1
0.
0
2
-0
.
0
2
5
-0
.
0
2
-0
.
0
1
5
-0
.
0
1
-0
.
0
0
5
0
0.
0
0
5
0.
01
0.
0
1
5
0.
02
0.
0
2
5
0.
0
3
5
0.
0
4
0.
0
4
5
0.
0
5
0.
0
5
5
0.
0
6
0.
0
6
5
0.
0
7
0.
0
7
5
0.
0
8
0.
0
8
5
3
1
2
Figure 2. The Z-compo
n
e
n
t of magnetic flux density in ROI.
Figure 3. Moving operatio
n of magnet a
rray.
Y(
m
)
Z(
m
)
Bz
(
T
)
-0.
0
2
-0.
0
1
0
0.
01
0.
02
-
0
.
025
-0.
0
2
-
0
.
015
-0.
0
1
-
0
.
005
0
0
.
005
0.
01
0
.
015
0.
02
0
.
025
0.
03
0.
04
0.
05
0.
06
0.
07
0.
08
0.
09
0.
1
Figure 4. The
moved magn
et array an
d its magn
etic field in ROI.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 601
7 –
6024
6020
Therefore, i
n
evitably there
is a
conditio
n
und
er which the m
agn
etic e
quipote
n
tial line i
n
ROI h
a
s b
e
st flatness
an
d g
r
adi
ent a
pproxim
atio
n
dist
ribut
io
n
cha
r
a
c
t
e
ri
st
ic
s.
Thi
s
kin
d
of
magnet st
ru
cture an
d mag
netic field is
what t
he unil
a
teral nu
cle
a
r magnetic
re
sonan
ce n
eed
s.
Y(
m
)
Z(
m
)
Bz
(
T
)
-0.
0
2
-0.
0
1
0
0.
01
0.
02
-0.
025
-0.
0
2
-0.
015
-0.
0
1
-0.
005
0
0.
00
5
0.
01
0.
01
5
0.
02
0.
02
5
0.
06
0.
08
0.
1
0.
12
0.
14
0.
16
0.
18
0.
2
0.
22
0.
24
Figure 5. The reverse o
p
e
r
ated ma
gnet
array an
d its magneti
c
field in ROI.
4. Optimizati
on of Magn
e
t
Struc
t
ure
In terms
of different magnet structures,
the
elliptic
curve can be
used to sim
u
l
a
te. Pu
t
the magnet e
qual arc len
g
th on the ellipt
i
c cu
rve and
cha
nge the
magnet st
ru
cture by adju
s
ti
ng
the length of elliptic semi-major axi
s
.
The elliptic e
quation in YO
Z plane is
sh
own a
s
22
22
1
yz
ab
(4)
whe
r
e
a, b
repre
s
e
n
t the length
s
of elliptic se
mi-m
aj
or
axis a
nd semi-min
or axi
s
, re
spe
c
tivel
y
.
To facilitate the optimization of
magnet
stru
cture, the followin
g
con
s
traint
conditi
on is ad
ded.
2
ab
r
r
r
(5)
r
rep
r
e
s
ent
s the radiu
s
of
the circl
e
where the
c
ent
er of Halb
ach magnet is l
o
cate
d, thus
the
magnet
stru
cture i
s
determ
i
ned by
a
. When
a < b, a
= b = r
a
nd
a >
b
, three m
agnet st
ru
ctu
r
es
in above secti
on ca
n be re
a
lized respe
c
tively.
In order to
descri
b
e
the
flatness of
equi
pot
enti
a
l line,
ch
o
o
se
21
eq
u
a
l-inte
rval
segm
ents i
n
the ROI alon
g
Z axis. In ea
ch
segm
en
t,
sele
ct 21
equ
al-inte
r
val poi
nts. In this
way,
441 gri
d
point
s are fo
rmed.
Then we
cal
c
ulate
d
t
he magneti
c
field of Z-com
pon
ent of each p
o
int
in ea
ch
se
g
m
ent respe
c
tively. The fla
t
ness of
the
equip
o
tential diagram
in
this are
a
can
be
approximatel
y repre
s
e
n
ted
by the following formul
a.
1
1
n
i
i
Std
s
td
n
(6)
2
1
1
()
m
iz
i
j
i
z
j
st
d
B
B
m
(7)
whe
r
e
B
zi
j
is
the mag
neti
c
field of Z
-
comp
one
nt o
f
point
j
on
segm
ents
i
,
B
iz
is the me
an
magneti
c
fiel
d of Z-co
mpo
nent of
all
fie
l
d point
s o
n
segm
ent
i,
st
d
i
is the
stan
dard
varia
n
ce of
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
i
zation and Con
s
tru
c
tion of Single-side
Nu
clea
r Magneti
c
Re
sonan
ce … (Ji
Yongliang
)
6021
magneti
c
fiel
d of Z-comp
onent of all
field points
o
n
seg
m
ent
i
,
Std
is the mean
stand
a
r
d
varian
ce of al
l segm
ents. S
o
mean
stand
ard vari
an
ce
Std
approxim
ately represe
n
ts the flatne
ss
of the eq
uipo
tential line. T
he
smalle
r th
e value i
s
, th
e cl
ose
r
th
e
value of the
magneti
c
fiel
d Z
axis com
pon
ent on all se
g
m
ents an
d the flatter the equipote
n
tial line is.
The semi
-ma
j
or
axi
s
wa
s cha
nge
d
fro
m
0.7
r
to 2
r
with step of 0.05
r.
Cal
c
ul
ate
mea
n
stand
ard va
riance
Std
of
magneti
c
field of Z-com
p
onent
in ROI
in each ma
gnet stru
ctu
r
e,
respe
c
tively.
The drawn scatter diag
ram
is as follo
ws.
0.
06
0.
08
0.
1
0.
12
0.
1
4
0.
16
0.
18
0.
2
0
0.
5
1
1.
5
2
2.
5
3
x 1
0
-3
a(
m
)
Std
(
T
)
Figure 6. The
scatte
r diag
ram of
Std
about
a
It can b
e
fou
nd that the
r
e
exists th
e lo
we
st point
ab
out mea
n
sta
ndard vari
an
ce
Std
in
scatter di
ag
ram Fig
u
re
6.
In order to fi
nd out th
e lo
cation
of the
lowe
st, this
p
aper carried
out
fitting on the
data points.
The gen
eral
polynomial
fo
r fitting will result in the fitting equation
to
present pathologi
cal
characteri
stic
s, which will
obtai
n
large calcul
ation error [16]. So this paper
made u
s
e of Gram
-Schmi
dt orthogo
nali
z
ation
meth
o
d
to make
curve fitting through.
Orthog
onal p
o
lynomial can
be obtaine
d by following e
quation.
1
0
()
,
0
,
1
.
.
.
k
k
kj
k
j
j
P
xP
x
k
m
(8)
0
2
0
()
0,
1
.
.
.
1
,
0,
1
.
.
.
()
n
k
ji
i
jk
n
ji
i
xP
x
j
k
km
Px
(9)
Fitting coeffici
ent is
0
2
0
()
,0
,
1
.
.
.
()
n
ij
i
i
j
n
ji
i
fP
x
j
m
Px
(10
)
Fitting equati
on is
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TELKOM
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Vol. 11, No
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r 2013 : 601
7 –
6024
6022
0
()
m
jj
j
yP
x
(11
)
5. Results a
nd Discu
ssi
ons
Grad
ually improve the de
gree of fitting poly
nomial, we can find out the fitting
equatio
n
satisfying
the
req
u
irement
s of
sq
ua
re
e
rro
r. In
thi
s
p
aper,
the
req
u
irem
ent of fi
tting sq
uare
error
is le
ss th
an
10-8. T
hen
o
b
tained fittin
g
equ
ati
on a
nd the fitting
cu
rve are shown a
s
bel
ow
(
F
ig
ur
e
7)
76
54
3
2
242676.4
213909
78124.2
15294.
2
1730.7
113.55
4.
06
0.
0838
Std
a
a
aa
a
aa
(12
)
0.
0
6
0.
0
8
0.
1
0.
12
0.
14
0.
1
6
0.
1
8
0.
2
0
0.
5
1
1.
5
2
2.
5
3
x 1
0
-3
a(m
)
Std
(
T
)
Figure 7. The
fitting curve of
Std
about
a
Y(
m
)
Z(
m
)
Bz
(
T
)
-0.
0
25
-0.
0
2
-0
.
0
1
5
-0
.
0
1
-0
.
0
05
0
0.
0
0
5
0.
01
0.
0
1
5
0.
0
2
0.
0
2
5
-0
.
0
2
5
-0.
0
2
-0
.
0
1
5
-0.
0
1
-0
.
0
0
5
0
0.
0
0
5
0.
0
1
0.
0
1
5
0.
0
2
0.
0
2
5
0.
0
7
0.
0
8
0.
0
9
0.
1
0.
1
1
0.
1
2
0.
1
3
0.
1
4
0.
1
5
0.
1
6
Figure 8. The simulate
d magneti
c
field
distrib
u
tion of
optimized magnet
a
rray
.
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TELKOM
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ISSN:
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046
Optim
i
zation and Con
s
tru
c
tion of Single-side
Nu
clea
r Magneti
c
Re
sonan
ce … (Ji
Yongliang
)
6023
Take
n th
e d
e
rivative of fi
tting Equatio
n (12) with
resp
ect to
variable
a
, the
value of
variable
a
can be o
b
tai
ned corre
s
p
ondin
g
to the lowe
st po
sition in fitting cu
rving.
The
equip
o
tential
of the corre
s
p
ondin
g
magn
etic field Z-co
mpone
nt
B
z
is sh
own as b
e
low (Figu
r
e
8).
Acco
rdi
ng to
the above d
e
sig
ned m
a
g
net
paramete
r
, the experi
m
ental mag
n
e
t array
wa
s co
nstructed (se
e
Figu
re 9(
a)). A motor-driven th
ree
-
dime
ns
i
o
nal magn
etic
field measuri
ng
platform (se
e
Figure
9(b)) wa
s
used to
measure the
magneti
c
fiel
d in ROI a
r
e
a
gene
rate
d by
magnet a
rray
.
The measured re
sults
we
re
sh
own as
Figure 10(a)
and Figu
re 1
0
(b
).
Figure 9. (a)T
he prototype
of optimiz
ed
magnet a
rray
.
(b)Th
e
moto
r-d
riven
three-dimen
s
ional mag
neti
c
field mea
s
u
r
ing platfo
rm
Z(
m
)
Y(
m
)
Bz(
T
)
-
0
.
025
-0
.
0
2
-
0
.
015
-0
.
0
1
-
0
.
005
0
0.
005
0.
01
0.
015
0.
02
0.
0
2
5
-0
.
0
2
5
-0
.
0
2
-0
.
0
1
5
-0
.
0
1
-0
.
0
0
5
0
0.
0
0
5
0.
01
0.
0
1
5
0.
02
0.
0
2
5
0.
0
7
0.
0
8
0.
0
9
0.
1
0.
1
1
0.
1
2
0.
1
3
0.
1
4
0.
1
5
0.
1
6
0.
1
7
Figure 10. (a
) The mea
s
ured magn
etic field
distri
butio
n of optimize
d
magnet a
r
ray.
(b) T
he value
of simulated
and mea
s
u
r
e
d
magneti
c
flux
Bz
Acco
rdi
ng to the simulation and mea
s
urem
e
n
t re
sults, it can
be seen th
at both
simulatio
n
cal
c
ulatio
n valu
e
and
me
asured valu
e
of
m
agneti
c
field
Z axis compo
nent h
a
ve g
o
o
d
flatness and
magnet g
r
adi
ent along Z a
x
is (2mT/mm
)
, simultaneo
u
s
ly.
At the same
time, carry o
u
t simulatio
n
cal
c
ulatio
n o
n
Z plan
e ma
gnet field at
differen
t
height.
We fo
und th
at at th
e heig
h
t of Z
= 1
7
mm,
within the
box of
(-2mm, 2m
m
)
(-2mm, 2m
m),
the magneti
c
field is relativ
e
ly uniform a
nd unifo
rmity is 130
0 ppm
and field stre
ngth is 0.07
3
9
T.
6. Conclusio
n
On the basi
s
of the improvement and optimizat
io
n of single
-
si
de
(unilateral)
Halb
ach
magnet
stru
ct
ure, thi
s
pa
p
e
r o
b
taine
d
a
nd con
s
tru
c
te
d a n
e
w
unila
teral
Halb
ach
stru
ctu
r
e a
r
ray
according
to
the
optimization
re
sults.
The
resu
lts of the
si
mu
lation
cal
c
ula
t
ion an
d
act
ual
measurement
of the magnetic field bo
th indi
cated t
hat the magnet
has bett
e
r flatness a
n
d
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Vol. 11, No
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7 –
6024
6024
gradi
ent (2
m
T
/mm) in a
r
e
a
of 50 mm
× 50 mm. The
uniformity in
ROI area of
4 mm × 4 m
m
is
1300 p
p
m an
d its field stre
ngth is 0.07
3
9
T.
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