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Inha University 1
Chapter 5. MagnetostaticsLecture Note #5A
5.1 The Lorentz Force Law
5.2 The Biot-Savart Law
5.3 The Divergence and Curl of B
5.4 Magnetic Vector Potential
• 움직이는 전하 (= 전류) → 주위에 자기장 형성
- 전기장과 자기장 속에서 있는 전하에 작용하는 힘
- Cyclotron motion
- 자기장 속에서 움직이는 전하에 작용하는 Lorentz force와 전하가 움직이는방향은 수직이므로 자기장이 전하에 한 일은 영.
• 전하가 자기장 속에서 이동할 때 작용하는 힘 = Lorentz Force(즉, 자기장 속에서 전류에 작용하는 힘)
• 전류의 흐름: 연속방정식 (Continuity equation)
dt
dqI
BvQFmag
BvEQF
BldIFmag
dt
dJ
0 dtvBvQldFdW magmag
Inha University 2
5.1 The Lorentz Force Law
from Eqs. (4.30), (4.32), (4.34), (4.22)Electrostatics (정전기학)
+
- 전기장의 공간적인 분포- 전하 간에 작용하는 힘- 전위차
- 전자의 이동 → 전류 → 유도 자기장
-
Electrodynamics (전기동력학)
Magnetostatics (정자기학)
ro
E
0
E
B
0v q
0v q
(일정한 전류가 흐르는 경우)
(자기장이 일정한 경우)
Inha University 3
5.1 The Lorentz Force Law
Q
0v iq
vv iq
iq
Q
For stationary charges For moving charges
“Test” charge
“Test” charge
In this case, what kinds of forces are acting
on the test charge?
When there are moving charges along the wire,
what kinds of forces are acting on the test charge?
Inha University 4
5.1 The Lorentz Force Law
5.1.1 Magnetic Fields 5.1.2 Magnetic Forces
Right-hand rule:
• Thumb (엄지) Direction of Current
(전류 방향)
• The other four fingers (나머지 4손가락)
Direction of Magnetic Field
(자기장 방향)
Q
B v
BvQFmag
When there is a moving charge q with velocity v
inside the magnetic field B, the force acting on the
charge is
F
“Lorentz Force Law”
When a moving charge q is placed in the
presence of both electric and magnetic fields, the
force acting on the charge is
Q
B
v
BvEQF
Fmag
E
(5.1)
(5.2)
Felect
Ftotal
Inha University 5
5.1 The Lorentz Force Law
5.1.2 Magnetic Forces
I
BveFmag
ve
- continued
I
For two wires with the same current
directions
F
I
BveFmag
ve
I
For two wires with the opposite current
directions
F
Inha University 6
5.1 The Lorentz Force Law
: the cyclotron formula
[Example 5.1]
where m = the particle’s mass, and p = mv.
(5.3)
Cyclotron motionThe archetypical motion of a charged particle in a magnetic field is circular, with the
magnetic force providing the centripetal acceleration.
B
Q
v
F
R
pv
R
vmQvBBvQFmag
2
RBQp
This is used to determine the momentum of the charged
particle by measuring the radius of the trajectory.
1) When the charged particle moves in a plan
perpendicular to B.2) When the charged particle moves in a plan
perpendicular to B but with some additional
speed v|| parallel to B
v
v cos
v||=v sin
BParticle path
The charged particle follows
a helical trajectory.Momentum of
the charged particle =
(자기장 속에서 움직이는 전하에 작용하는 힘에 의한 원운동)
Inha University 7
5.1 The Lorentz Force Law
[Example 5.2]
The position of the particle at any time t = (0, y(t), z(t)).
Cycloid MotionWhen a uniform electric field is applied to a positively charged particle in addition
to a magnetic field at right angles, what path will it follow?
zzyymamzyByzBzEQBvEQF ˆ ˆ ˆ ˆ ˆ
Then, we apply the Newton’s 2nd law:
For the particle at rest initially, the electric field accelerates the charge in the z-direction.
E
B
Q
Q
v
Fmag
Felec
The velocity of the particle at any time t = (0, ý(t), ź(t)). where
,dt
tdyty
dt
tdztz
Thus,
zyByzB
B
zy
zyx
Bv ˆ ˆ
00
0
ˆˆˆ
,2
2
dt
tydty
2
2
dt
tzdtz
Inha University 8
5.1 The Lorentz Force Law
zzyymzyBzEyzBQ ˆ ˆ ˆ ˆ ˆ
The solution of this equation becomes
For convenience, let
321 sincos CtBEtCtCty
[Example 5.2] - Continued (1)
(5.4)
, ymzBQ zmyBQEQ
m
BQ
: “cyclotron frequency”
, zy Thus, we get
y
B
Ez
(5.5a)
y
B
Ez
dt
yd
2
Differentiating Eq. (5.5a) & using Eq. (5.5b), we get
y
B
Ey 2
(5.5b)
(5.6a)
Eq. (5.6a) Eq. (5.5b) : 412 sincos CtCtCtz (5.6b)
, cossin 21 BEtCtCty
BEtCtC
B
Ey
B
Ez cossin 21
Inha University 9
5.1 The Lorentz Force Law
We let
Since the particle started from rest,
321 sincos CtBEtCtCty
[Example 5.2] - Continued (2)
: The formula for a circle of radius R centered at (0, Rt, R)
at the origin
(5.7a)Thus, finally we get
B
ER
(5.7b)
(5.8)
Eqs. (5.6a) &
(5.6b)
412 sincos CtCtCtz
(5.9)
000 zy 000 zy
BEtCtCty cossin 21
tCtCtz cossin 12
00 31 CCy
00 42 CCz
00 2 BECy
00 1 Cz 01 CB
EC
2
13 CC
24 CC
ttB
Ety
sin
tB
Etz
cos1
Since , 1cossin 22 tt 222RRztRy
The y-coordinate of the center keeps moving with time t with a
constant speed u :
B
ERu
The moving curve is “cycloid”.
(5.10)
Q
v
FmagFelec
0222 R
dt
dRztRy
dt
d
,Rdt
dy 0
dt
dz
022
Rz
dt
dztRyR
dt
dy
2R
Inha University 10
5.1 The Lorentz Force Law
because
If the charge Q is moved to dl = v dt in a magnetic field, the work done is
0 dtvBvQldFdW magmag
Further discussion will be covered in Chapter 8.
“Magnetic forces do no work on the charge ??”
(5.11)
5.1.2 Magnetic Forces - continued
Work done by Lorentz force :
vBv
i
q
B
1F
v
l
Inha University 11
i
q
B
1F
v
l
5.1 The Lorentz Force Law
5.1.3 Currents
Current in a wire : The charge per unit time passing a given point
• The electric current in wire is based on electron
flow. (In some semiconductor materials, both
electrons and holes moves in the opposite directions.)
• The direction of the electron flow is opposite to the
direction of current flow.
dt
dqI
s
CA 1 1
dlBIdlBvdqBvFmag
(5.12)
(5.13)
Unit : Amperes = Coulmb/sec
vdt
dlI
Free electron
AV BV
BA VV
EeFe
Ei
: line chargev : charge flow speed
Magnetic force acting on a segment of current-carrying wire is
vI
( We consider that =q/L is positive charge.)
BldIFmag
Since the current is constant along the wire, BldIFmag
(5.14)
(5.15)
(5.16)
(5.17)
Inha University 12
5.1 The Lorentz Force Law
[Example 5.3]
If I > Ib, the loop rises.
aBIF up
mag
ahBIhFW up
magmag
The magnetic force acting on the wire is
(Solution)
IThis magnetic force should be balanced with the weight :
mgaBIb Ba
mgIb
Then, the work done by the magnetic force over the distance of rise h is
i
q
B
1F
w
l
Let the charge flow speed be w for the current I.
wI
(5.18)
(5.19)
aBIBaBqF up
mag ww Thus,
Inha University 13
5.1 The Lorentz Force Law
[Example 5.3]
During a time interval dt, the charges moves a horizontal distance dl
auBquBF horiz
mag
ahBIdtuaBdtuaBdlFW horiz
maghoriz ww
The speed of the upward rise due to the magnetic
force acting on the charge q is u
(Solution)
The magnetic force acting on the upward motion of the charge
becomes
dtdl w
Then, the work done by the horizontal magnetic force on the
charge over the distance dl is
i
q
B
1F
w
l
- continued (1)
: the same as the work done to flow the charge
along the wire in the magnetic field.
Inha University 14
I
I
I
5.1 The Lorentz Force Law
5.1.3 Currents
Surface Current Density
The current per unit width ( = current ribbon density) :
vvdl
d
dl
dK
I
daBKdaBdqBFmag v v
(5.22)
where : surface charge density, v : speed of the charge flow
- Magnetic force acting on the current per unit width is
vvda
d
da
dJ
I
v dBvFmag
(5.24)
(5.25)
(5.27)
- continued
vK
(5.23)
The volume current density (= current per unit area) :
where : volume charge density, v : speed of the charge flow
vJ
(5.26)
- Magnetic force acting on the current per unit area is
Inha University 15
5.1 The Lorentz Force Law
2a
I
A
IJ
,ksJ
(5.24)The volume current density (= current per unit area) :
Since the total current in the wire is
[Example 5.4]
(Solution)
(a)
(b)
3
22
3
0
2
0
2
0
kadsskddssksJdaI
aa
Inha University 16
5.1 The Lorentz Force Law
5.1.3 Currents
Current Density vs. Current
(5.28)
From the divergence theorem,
According to charge conservation law
SS
adJdaJI
(5.29)
- continued
: the continuity equation (연속 방정식)
(The minus sign reflects the fact that an outward flow decreases the charge left of the volume v.)
v v
dJadJS
v v v v vvvv
ddt
dd
dt
ddvdJ
vJ
from Eq. (5.26)
dt
dJ
Thus,
dt
dqI
Inha University 17
Chapter 5. Magnetostatics
5.1 The Lorentz Force Law
5.2 The Biot-Savart Law
5.3 The Divergence and Curl of B
5.4 Magnetic Vector Potential
• 일정한 전류가 흐르는 도선 주위의 자기장을 구하는 방법 : Biot-Savart Law
• 일정한 전류가 흐르는 두 도선 사이에 작용하는 힘
P
2
0
2
0ˆ'
4'
ˆI
4 r
rldIdl
r
rrB
d
II
L
Ff
mag 210
2
Inha University 18
5.2 The Biot-Savart Law
Electrostatics (정전기학)
- Stationary charges
Magnetostatics (정자기학)
ro
E
0
t
5.2.1 Steady Currents
+ - Constant electric fields
- Steady currents Constant magnetic fields
0
t
J
: charge density
J : current density (=I/A)
constant
constant J
I
0
t
Q
0 J
: the continuity equation (5.33)
(5.32a)
(5.32b)
Inha University 19
5.2 The Biot-Savart Law
The Biot-Savart Law:
- The magnetic field of a steady line current is given as
2
0
2
0ˆ'
4'
ˆI
4 r
rldIdl
r
rrB
5.2.2 The Magnetic Field of a Steady Current
Unit of the magnetic field (B) :
0 = 4 x 10-7 N/A2 : the permeability of free space
(tesla)mA
N 1T 1
(5.34)
(5.35)
(5.36)
I
r
P
Inha University 20
5.2 The Biot-Savart Law
From the Biot-Savart law,
[Example 5.5]
(5.37)
tan' sl
r
At the point P, the direction of (dl’ x r ) points out of the page.
(Solution)
cos'sin'ˆ' dldlrld
2
0
2
0ˆ'
4'
ˆI
4 r
rldIdl
r
rrB
ds
dl2cos
' dxx
dxxxd2
2
cos
1 sectan
cosrs 2
2
2
cos1
cos1
srsr
1200
22
2
0 sinsin 4
cos 4
coscos
scos
4
2
1
2
1
s
Id
s
Id
sIB
xyyxyx cossincossinsin
Inha University 21
5.2 The Biot-Savart Law
In the case of an infinite wire,
(5.38)
r
Eq. (5.37)
,2
1
s
IB
2
0
[Example 5.5] - Continued (1)
22
: B is inversely proportional to
the distance s from the wire.
ˆ 2
0
s
IB
(5.39)
: The direction of B follows the right-hand rule.
Inha University 22
5.2 The Biot-Savart Law
The magnetic field at the wire (2) due to the wire (1) is
(5.40)
Eq. (5.17)
Calculation of the electric force acting on two long and parallel wires
the force is repulsive.
The force per unit length becomes
The Lorentz force law acting on the wire (2) due to the wire (1) is
ydld
II
d
IldI
BldIFmag
ˆ 2
ˆ2
210
210
22
122
x
y
z
ˆ 2
10
d
IB
d
II
L
Ff
mag 210
2
The magnitude of the force :2
102
102
2
2L
d
IIdl
d
IIF
The direction of the force : toward the wire (1) “attractive force”
i) When the directions of the two currents are parallel,
ii) When the directions of the two currents are antiparallel,
Inha University 23
5.2 The Biot-Savart Law
[Example 5.6]
• The magnetic field dB attributed to the segment dl’ points
as shown in the figure.
Since
P
z
I
R• As we integrate dl’ around the loop, dB sweeps out a cone.
Bd
horizontaldB
verticaldB
The horizontal components cancel, and the vertical components combine,
to give the following relation:
Rr
Idl
r
I
r
dlIzB
2
cos
4'
cos
4cos
'
4 2
0
2
0
2
0
r
Eq. (5.34)
2
0ˆ'
4 r
rldIrB
,and cos 22 zRrr
R
2322
2
0
2 zR
RIzB
(5.41)
Find the magnetic field a distance z above the center of a circular loop of radius R, which
carries a steady current I.
Inha University 24
5.2 The Biot-Savart Law
For surface and volume current densities, the Biot-Savart law becomes
(5.34)
- continued
All are related to the “charge flow”
(A point charge cannot provide a steady current.)
I5.2.2 The Magnetic Field of a Steady Current
2
0
2
0ˆ'
4'
ˆI
4 r
rldIdl
r
rrB
'ˆ'K
4 2
0 dar
rrrB
For a current flowing in a wire, the Biot-Savart law is
The current per unit width (= current ribbon density) :
vvdl
d
dl
dK
I
(5.22)
'ˆ'J
4 2
0 dvr
rrrB
vvda
d
da
dJ
I
(5.25)
The volume current density (= current per unit area):
(5.13)vdt
ldI
The line current :
vI
vK
vJ
When there are a collection of many source currents,
the total magnetic field can be obtained by superposition principle.
Inha University 25
5.2 The Biot-Savart Law
The magnetic force acting on a current-flowing wire due to the magnetic field B is
(Hint)
aSS
aI
aS
aI
sS
aIFFFFFFFtotal
222
22
0
2
0
2
0314321
BldIFmag
Eq. (5.17)
(1)
(2)
(3)
(4)
ˆ 2
0
s
IB
For side (1), the magnetic field is constant:
Eq. (5.39)
(1)
I
I
I
I(2)
(3)
(4)
S
aS
Idl
S
IIF
2 2
2
001
,
2
01
S
IB
(upward)
For side (2),
S
aSIds
s
IIdl
s
IIdlBIF
aS
S
aS
S
aS
Sln
2 2 2
2000
22
(toward right-hand side)
For side (3), the magnetic field is constant : aS
IB
2
03
a
aS
Idl
aS
IIF
2 2
2
003
(downward)
For side (4), 2
2000
44 ln 2 2 2
FS
aSIds
s
IIdl
s
IIdlBIF
S
aS
S
aS
S
aS
(toward left-hand side)
(upward)
Problem 5.10
Find the force on a square loop placed as shown in Figure near an infinite straight
wire. Both the loop and the wire carry a steady current I.
(opposite-direction to F(2))
Inha University 26
Next Class
Chapter 5. Magnetostatics
5.1 The Lorentz Force Law
5.2 The Biot-Savart Law
5.3 The Divergence and Curl of B
5.4 Magnetic Vector Potential
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