magnetic potential gradient -...
TRANSCRIPT
7.C. Magnetic Potential and
Magnetic Potential Gradient
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”
▷ 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)
◈ 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
▷ 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)
▷ 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)
7.D. Magnetic Dipole and
Magnetic Shell
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].
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
◈ 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
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.
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)
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(−)
• 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)