magnetic potential gradient -...
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7.C. Magnetic Potential and
Magnetic Potential Gradient
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rrr
drr
mdr
r
mdrHU
2
0
2
0
1
44
◈ Magnetic potential and Unit
r
m
r
mr
00 4
1
4
r
mU
04
1
2
04
1
r
mH
rHU ][][]/[
][][
AmmA
mHU
Magnetic Potential
• The unit for the magnetic potential is the [J/Wb] or [A].
Magnetic potential
at point P.
Magnetic field
intensity at point P.
• Work of necessity for unit magnetic pole (+1Wb) move from position ∞ to position P (r m).
→ We define the magnetic potential U at point P in a magnetic field H.
“Magnetic potential U at point P (distance r m) in
a magnetic fields by point magnetic pole +m Wb”
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▷ Definition of Electric Potential (Fig. 3-4)
Work of necessity for unit point charge(1C) move from position ∞ to
position P
≡ We define the electric potential V at point P in an electric field.
VdlEdlqEdlFWPPP
Electric Potential (Chapter 3)
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◈ Magnetic Potential Difference
BA
BAAB drHdrHUUU
A
B
B
A
B
AdrHdrHdrHdrH
◈ Magnetic Potential Gradient
BA UUdUxdHdw
gradUUdx
dUH
• The work done required to move from B to A for magnetic pole +1[Wb] in the magnetic field H.
• Magnetic potential gradient is equal to the energy change if the unit magnetic pole +1[Wb] displacement from A to B in the direction of the magnetic field.
Magnetic Potential Difference
Intensity of the magnetic field is equal to the
magnetic potential gradient.
(−) sign : direction of magnetic potential decreases
as the direction of the magnetic field.
Magnetic Potential Difference & Gradient
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▷ Potential Difference
abab VqW 0
The work done by the electric
force for positive charge qo
paths from a to b
21 rr
baab drEdrEVVV
2
1
2
1
r
r
r
rdrEdrEdrE
2
1
2
1
2
1
1
4
1
44 0
2
0
2
0
r
r
r
r
r
r r
Qdr
r
Qdr
r
Q
210
11
4 rr
Q
Potential difference between position a and b
Fig. 3-6
Electric Potential Difference (Chapter 3)
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▷ Potential Gradient (Fig. 3-11)
Position vector of E and displacement
vector dl in rectangular coordinate space
kEjEiEE zyxˆˆˆ
kdzjdyidxld ˆˆˆ
)ˆˆˆ()ˆˆˆ( kdzjdyidxkEjEiEldEdV zyx
dzz
Vdy
y
Vdx
x
VdzEdyEdxE zyx
)(
VVz
kx
jx
idl
dV
ˆˆˆ
gradVVE
dl
dVE potential
gradient
Electric Potential Gradient (Chapter 3)
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7.D. Magnetic Dipole and
Magnetic Shell
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Magnetic Dipole
22
200210 cos4
cos
4cos
2
1
cos2
1
4
11
4 lr
lm
lr
lr
m
rr
mU
2
0
cos
4 r
lmU
where, r≫l 이므로 0cos
2
2
2
lr
2
04
cos
r
MU
where, M=ml “magnetic dipole moment”
◈ Magnetic potential due to a magnetic dipole
cos2
1
lrr
cos2
2
lrr
with
2104
1
r
m
r
mU
A magnetic dipole is a pair of minute magnetic pole
with equal magnitude and opposite sign (±m[Wb])
separated by a distance l[m].
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E(r’) = (1/4o) (q/r’2) r’ ^
r
q
r
qVVrV
00 4
1
4
1)(
rr
rrq
rr
q
00 4
11
4
2
0
2
0 4
coscos
4)(
r
M
r
dqrV
Where, cosdrr
"" ntdipolemomeqdM
2rrr
Electric Dipole (Chapter 3)
▷ Potential due to a dipole
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◈ Magnetic potential of magnetic dipole → Magnetic Field
2
04
cos
r
MU
ˆˆ HrHH r
]/[4
cos21
4
cos3
0
2
0
mAr
M
rr
M
r
UH r
]/[4
sincos
4
13
0
3
0
mAr
M
r
MU
rH
ˆsinˆcos2
4ˆˆ
3
0
rr
MHrHH r
2
3
0
22
3
0
22 cos314
sincos44
r
M
r
MHHH r
“Intensity of magnetic field at point P”
Magnetic Dipole
The magnetic field at point P is
represented by the sum of magnetic
field Hr in the direction of r and Hθ
in the direction of θ. ̂ˆ r
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Magnetic Shell
◈ Magnetic potential of magnetic shell
2
0
2
0 4
cos
4
cos
r
M
r
mlU
with, M=ml
(cf) magnetic potential
of magnetic dipole :
d
M
r
dS
r
dSdU
0
2
0
2
0 4
cos
44
cos
04
MU
Q. 7.3
Both side of an extremely thin plate with thickness δ [m] are distributed to magnetic
charge density ±σ [Wb/m2] each, we call this configuration a magnetic shell.
where, Magnetic charge of the minute area dS is σdS (cf. m of magnetic dipole)
Thickness of thin plate is δ (cf. l of magnetic dipole)
Intensity of magnetic shell is M =σδ (cf. M=ml of magnetic dipole moment)
Solid angle to create a point P on the area ds is dω
Magnetic potential U at point P for the magnetic shell
area S is proportional to solid angle ω.
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Fig. 3-16 ▷ Electric Double Layer
• Two charged extremely thin plate of
magnitude σ but of opposite sign, we
call this configuration an electric
double layer.
• Magnitude of the electric double layer
is defined as the m=σδ
• If dV is the electric potential at point P by differential surface dS, dS part
of the charge ±(σdS) can be seen as an electric dipole. ±q = ±(σdS)
d
r
dS
r
dS
r
dqrdV
0
2
0
2
0
2
0 4
cos
4
cos
4
cos
4)(
Where, dS forming solid angle from point P
2
cos
r
dSd
04
)(m
rV
Electric Double Layer (Chapter 3)
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Magnetic potential difference of magnetic shell between two points P, Q
QPPQ UUU
21
0
2
0
1
0 444
MMM
00
224
MMU
where, the size of the solid angle when approaching infinity ω1=2π, ω2=2π
Magnetic Shell
◈ Magnetic potential difference of magnetic shell
Magnetic potential difference of magnetic shell when both sides approach infinitely
where, There is a solid angle ω by the polarity.
Solid angle ω1 created by + magnetic charge side is the positive(+),
Solid angle ω2 created by − magnetic charge side is the negative(−)
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• Electric potential
at point P 1
04
mVP
Fig. 3-17
• Electric potentila
at point Q 2
04
mVQ
Where, a solid angle 221
• Potential difference between two points
21
04
mVVV QPPQ
00
44
mmVPQ
▷ Potential difference of electric double layer : VPQ
Electric Double Layer (Chapter 3)