Introduction to Electromagnetism Static Magnetic Fields
(6-10, 6-11)
Yoonchan Jeong
School of Electrical Engineering, Seoul National University Tel: +82 (0)2 880 1623, Fax: +82 (0)2 873 9953
Email: [email protected]
2
Boundary Conditions for Magnetostatic Fields
Boundary conditions:
I d
d S
C ⋅ = ⋅ =
→ ∫ H l ∫ J s
s
n H H J
a × − =
→ 2 ( 1 2 )
← For linear media Two basic equations for magnetostatics:
J H B
=
×
∇
=
⋅
∇ 0
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
Recall:
) T
2 (
1 n B n
B =
n
n H
H 1 2 2
1 µ
µ =
w J
d s
abcda ⋅ = ⋅ ∆ + ⋅ − ∆ = ∆
→ ∫ H l H 1 w H 2 ( w )
) A/m
2 (
1 t H t J s
H − =
→
← J s is non-zero only when an interface with
an ideal perfect conductor is assumed.
Inductance and Inductors
Magnetic flux:
Inductance (Self inductance):
) Wb
∫ ⋅ (
=
Φ S B ds
∫ ⋅
= Φ
→
1
1 1
11 S
L B ds → B 1 ∝ I 1 → Φ 11 L ∝ I 1
1 11 I
= L
) H (
1 11
11 I
L
Φ L
=
→ ( H )
1 11
11 dI
L d
Φ L
=
→ Mutual inductance:
∫ ⋅
= Φ
→
2
2 1
12 S
L B ds
1 12 I
= L
) H (
1 12
12 I
L
Φ L
=
→ ( H )
1 12
12 dI
L d
Φ L
=
→
21 L
L
L =
= Φ ? Question:
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
Total magnetic flux linkage
Entire surface enclosed by I 1 Any surface
By I 1
Linked to S 2
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
4
Mutual Inductance
Mutual magnetic flux linkage:
∫ ⋅
= Φ
→
2
2 1
12 S
L B ds = L 12 I 1
12
21 L
L =
→ Similarly:
∫ ⋅
Φ =
=
→
2
2 1
1 1
12 12
1
S L
I d
L I B s = ∫ 2 ∇ × 1 ⋅ 2
1
) 1 (
S d
I A s
∫ ⋅
=
2
2 1
1
1
C d
I A l ← 1 = 4 0 1 ∫ C 1 R 1 d
I l
A π
µ
∫ ∫ ⋅
=
2 1
2 1
0
4 C C R
d d l l π
µ
∫ ∫ ⋅
=
→
1 2
2 1
0
21 4 C C R
d
L d l l
π µ Consequently:
← Neumann formula
Entire surface
enclosed by I 2
Example 6-14
For a toroid:
∫ ⋅
Φ =
=
→ S
L
I d
L I 1 B s
φ φ B a B =
→
π rB φ C ⋅ d = 2
→ ∫ B l
r NI π µ
φ 2 a 0
B =
→
∫ ⋅
= S Nhdr
r NI
I ( )
2
1 0
φ
φ π
µ a
a
r dr h
N b
∫ a
= 1
2
2 0 π µ
) H ( 2 ln
2 0
a b h
N π
= µ
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
∫
∫ C B ⋅ d l = µ 0 S J ⋅ d s
0 NI µ
=
Entire surface
enclosed by I
6
Example 6-18
For a solenoid with two windings:
1 12 I
= L
∫ ⋅
= Φ
→
2
2 1
12 S
L B ds
2 2 1
1
1 I N a
l
N π
µ ⋅
=
) H
2 (
2 1 1
12 N N a
L = l µ π
→
2 21 I
= L
∫ ⋅
= Φ
→
1
1 2
21 S
L B ds
2 1
2 1 2
2
2 a
l N l l I
N π
µ
⋅
=
) H
2 (
2 1 1
21 N N a
L = l µ π
→
Alternatively:
21
12 L
L =
→
As expected:
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
1 1 1
1 l N I
B = µ
←
Entire surface
enclosed by I 2
