magnetic field
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1
EMLAB
Magnetic field
2
EMLAB
qI
Generation of magnetic field
• A charged particle in motion generates magnetic field nearby.
• In the same way, currents generate mag-netic field nearby.
3
EMLAB
Magnetic field due to currents or magnet
B due to magnetic moment of electron
4
EMLAB
Forces on charges due to magnetic field (Lorentz force)
B
B
BvF q
B
Electron beams are deflected by Lorentz force
Horizontal and vertical deflection yoke control the path of electron beams.
5
EMLAB
Force and Torque on a closed circuit
BvF q
CId BrF
FrT
Current loops in a magnetic field experi-ence torque, and are rotated until the plane of loops are perpendicular to the applied B.
F
FB
F
Angular acceleration is proportional to the applied torque.
Torque is proportional to the product of radius and force.
If the sum of torques due to A and B has nonzero value, the seesaw is rotated.
6
EMLAB
1r
Relative permeability
Magnetic flux density
]H/m[104 70
Magnetic field and magnetic flux density
HHMHB rm 000 )1()(
• Arrows represent magnetic field due to orbiting elec-trons.
• The orbits of electrons are aligned due to external magnetic field.
extH
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EMLAB
(a) Hard disk tracks. (b) Sketch of qualitative shapes of hysteresis curves required for the head and track magnetic materials.
The magnetic head aerodynamically flies over the disk sur-face at a distance above it of only about 1mm while following the surface profile. In the figure, the surface profile is shown as ideally flat, which in practice is not the case.
Hard disk application
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EMLAB
Electromagnetic forces on a charge
1) Electric force
EF q
2) Magnetic force
BvF q
F
E
F
B
v
(Lorentz force)
)(total BvEF q
(Coulomb force)
9
EMLAB
Prediction of magnetic field : Biot-Savart law
24
ˆ
R
Idd
Rs
H
rrR r
r
sId
Direction of H-field
Current segment
The magnetic field can be predicted by Biot-Savart’s law with known current distribution.
10
EMLAB
Ampere’s law
path closed
IdsH
• Ampere law facilitates calculation of mangetic field like the Gauss law for electric field..
• Unlike Gauss’ law, Ampere’s law is related to line integrals.
• Ampere’s law is discovered experimentally and states that a line integral over a closed path is equal to a current flowing through the closed loop.
• In the left figure, line integrals of H along path a and b is equal to I because the paths enclose cur-rent I completely. But the integral along path c is not equal to I because it does not encloses com-pletely the current I.
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EMLAB
SCdId aJsH
From the integral form, we will derive the differential form of Ampere’s law.
JH
n
CndrH
C
drH
Line integrals from these adjacent currents add up to zero.
nC
Line integrals over a closed path is equal to the sum of line integrals over infinitesimally small loops.
Differential form of Ampere’s law
nn S nC
dd aJrH
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EMLAB
Example- Coaxial cable
I
I
a b cI
I
002
)4(
2ˆˆ
1
ˆˆ
ˆ2
)3(2
ˆˆ2
)2(2
ˆˆ
ˆ
2ˆ
)1(
22
22
22
22
0
2
0
2
0
0
2
0
2
2
2
0
2
0
2
0
H
H
zJzJ
zJ
H
H
zJaJ
Hs
rH
cr
bc
rc
r
IH
Ibc
br
dddd
ddrH
crbr
IHIrH
braa
rIH
Ia
rddd
rHrddH
ar
a r
b
outin
r
in
r
in
S
C
zJ ˆa
I2in
zJ ˆ)bc(
I22out
H
•The direction of magnetic fields can be found from right hand rule.
• The currents flowing through the inner conductor and outer sheath should have the same magnitude with different polar-ity to minimize the magnetic flux leakage
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EMLAB
Example : Surface current
nK
x
xH
s
ˆ20z
2
Kˆ
0z2
Kˆ
KL)L)(H(LHdH x
C
x
•The direction of magnetic field con be conjectured from the right hand rule.
14
EMLAB
Example : Solenoid
)0H(
ˆK
KL)L(HLHdH
out
out
C
in
zH
s
•The direction of magnetic field con be conjectured from the right hand rule.
• If the length of the solenoid becomes in-finite, H field outside becomes 0.
d
I
15
EMLAB
Example : Torus
2
NIKHNId aC
rH
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EMLAB
NS
]V[dt
dV
Electromotive force (emf)
• (-) sign explains the emf is induced across the terminals of the coil in such a way that hinders the change of the magnetic flux nearby.
1. A time-varying flux linking a stationary circuit.
2. A constant magnetic flux with a moving circuit
3. Combination of the above two cases
Situations when EMF is generated
Faraday’s law
1) Faraday experiment
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EMLAB
+V-
SC
ddt
d
dt
ddV aBrE
C
B
t
dt
dSS
B
E
aB
aE
(1) A time-varying flux linking a stationary circuit.
Time varying
E
18
EMLAB
(2) A constant magnetic flux with a moving circuit
Bdvdt
dyBd
dt
dV
ByddS
emf
aB
(1) A phenomena observed by a stationary person
Direction of induced current
Due to the motion of a conducting bar, the charges in it moves in the (+y) direction. The moving charges experience Lorentz force such that
BvEEF
xzyBvF
mq
BqBqq ˆˆˆ
1. Effectively, the motion of bar gener-ates electric field which has the strength of (υ x B)
2. emf = Ed = υBd
20
EMLAB
(3) Combination of the two
rBvaB
rE ddt
demfS
)(
t
B
BvE )(
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EMLAB
Example : AC generator
tBabdS
cos aB
tBabdt
demf sin
A simple AC generator
dt
dd
dt
dd
tddemf
SSSC
aBaB
aErE )(
n̂ B
Observer’s coordinate frame is rotating with the loop.
22
EMLAB
Example : Eddy current
B υ
BυE
B
υ
BυE
Relative velocity of the copper tube to the magnet.
Falling magnet inside a copper tube
Insulator tube Conductor tube Conductor tube
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EMLAB
Inductance
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EMLAB
Two important laws on magnetic field
Current generates magnetic field (Biot-Savart Law)
inducedV
Time-varying magnetic field generates induced electric field that opposes the variation. (Faraday’s law)
Current
Current
B-field Top view
Electric field
B-field
dt
dVind
i
25
EMLAB
Current B i
l
NB
length
lengthl
Ni
l
iNSBSadB
lengthS
Magnetic flux :
Magnetic flux
B in a solenoid with N turn coil
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EMLAB
Concept of inductance
Current ,B
iN 00 ,Magnetic flux : i
NL
dt
diL
dt
dNL
Ф is the magnetic flux due to the coil wound N times.
Ф0 is magnetic flux due to the single turn coil.
Self inductance is proportional to the square of winding N.
The change of magnetic flux intensity due to changing current generates electro-motive force. The proportionality constant between the emf and current is called a inductance.
SNL ,2
S
27
EMLAB
Mutual Inductance
dt
idM
dt
dN 1
2122
(1) When the secondary circuit is open
dt
idL
dt
dN 1
1111
1
11111 i
NL
1
21221 i
NM
1121 :: NN
The current flowing through the primary circuit generates magnetic flux, which influences the secondary circuit. Due to the magnetic flux, a repulsive voltage is induced on the secondary circuit.
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EMLAB
Work to move a current loop in a magnetic field
Idt
dt
dIdttIVW
tt )())((
rr
IBA
If we want to move a current loop with I flowing in a region with a magnetic flux density B, energy should be supplied from an external source.
The voltage induced in the current loop hinders the current flow, which should be canceled by an external source.
S
)(ti
)(tVR
29
EMLAB
B
The energy is equal to assemble circuits with current Ii.
Magnetic energy : Mutual interaction
Ii
Ij
iIWi
N
n
n
iintotal
n
iin
ii
IW
IW
IW
IW
2
1
1
1
1n
2
133
122
1
1 1
11-N
322
211
)(
)(
)(
N
n
N
niintotal
NN
N
ii
N
ii
IW
IW
IW
IW
N
n
N
nii
intotal IW1 12
1
•Energy needed to as-semble I1, I2~IN in a
free space.
•Energy needed to dis-integrate I1, I2,~,In.
Magnetic material
S
daB
N
i
N
jjiij
N
i
N
jjitotal IILIW
1 11 1 2
1
2
1(Including self energy)
30
EMLAB
Magnetic energy
S
)(ti
RtitS )()(
)(tLi
20
2
0
2
00
2
1)(
2
1)]([
2
1
)()()()(
00
00
LItiLdtdt
tidL
dtdt
tdiLtidttVtiW
tt
tt
R
(Initially, this circuit has a zero current flowing. Then , the current increases to I.)
(To support current i(t), the current source should provide additional voltage which cancels induced voltage by Faraday’s law.)
dt
tdiL
dt
dtVt R
)()()(
)(tVR
Self energy : The energy needed for the circuit to have a current I flow in spite of the re-pelling electromotive force from Faraday’s law.
31
EMLAB
)()(2)()(2
1
)()(2)()(2
1
)()()(
)()()(
)()()()(
121112221
211
0
21222
211
0
212
2121
1
0
2211
1
1
1
tItIMtILtIL
dttitiMtiLtiLdt
d
dttidt
tdiM
dt
tdiLti
dt
tdiM
dt
tdiL
dttittitW
t
t
t
Magnetic energy : two coil system
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