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)