Lectures 14-15 (Ch. 28) Sources of Magnetic Field
description
Transcript of Lectures 14-15 (Ch. 28) Sources of Magnetic Field
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Lectures 14-15 (Ch. 28)Sources of Magnetic Field
1. B of a moving charge2. Bio-Savarat law3. Superposition principle4. Force between two currents5. Flux of B6. Amper’s Law7. Bohr’s magneton8. 4 types of magnetic materials
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B of a moving charge
200
4 r
rvqB
0204
1r
r
qE
A
Tm70 104
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Compare electric and magnetic forces
BvqFEqF me
,
0204
1r
r
qE
200
4 r
rvqB
11
2
2
002
v
c
vF
F
m
e
2
00
1c
s
mc 8103
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Biot & Savart law (B of a segment of a current)
dlr
rIBd
vqnAIqnAdldQ
r
rvdQBd
200
200
4
,
4
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2/322
20
2/3220
2/322
0
2/3220
200
)(2)(4
2
)(4
)(4
cos;4
Rx
IR
Rx
RIR
Rx
dlIRdBBB
Rx
IRdldB
dBdBr
dlrIBd
xx
x
x
B of a loop of a current
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Electromagnet (magnetic coil or solenoid) N loops of the current)
2/3220
2/322
20
)(2)(2 axax
NIaB
Most of magnets used in industry are electromagnets.
Attraction is due to magnetization of iron items.(see the end of the previous lecture)
For closely spaced loops (each loop at the same distance to observation point) of the same radius:
In the center of the coil (x=0)
a
NIB
20
N times stronger than from 1 loop
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B of a stright wire
2/12222/322
2/1220
2/3220
22
222
20
200
)(
2
)(
)(2)(4
sin,
sin
4;
4
axx
a
yx
dy
axx
Ia
yx
dyIxB
yx
xyxr
dyr
IdBdl
r
rIBd
a
a
a
a
Long wire: a>>x
x
IB
20
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Oersted’s experiments and Oersted’s RH rule
Hans Christian Ørsted (Oersted) (1777 – 1851)
1820: current produces B
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Superposition principle
21 BBB
d
IBx
00
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Examples. Find B at point P.
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A force between two currents
r
IlIlBIFII 2'''' 0
'
r
II
l
FII
2
'
'0'
Al
rFI
IIm
N
l
Fmr
12
',102,1
0
7
NB: currents in the same direction attract each other.Currents in the opposite directions repeal each other.It’s different from charges of the same or opposite signs!
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Example. Magnetic bottle again (see also lecture 12)
Now we understand also the structure of magnetic field between as a result of superposition of two fields produced by two coils. NB: rotation direction of a positive charge is opposite to the current in the coils. Repulsive interaction of opposite currents results in trapping the charged particles in the bottle.
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Example. Find the force acting on each piece of a rectangular conductor and the net force acting on
the rectangular conductor.
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Flux of B
dABdABAdBAAA cos
0enclq
AdE
Gauss’s law for E:
Gauss’s law for B: 0 AdB
Magnetic lines are closed lines.There are no magnetic carges.
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dlBldBlineline cos
ldBCirculation of B
Example: Find a circulation of B produced by a current in the long straight wire for suggested
integration passes and circulation directions.
r
IB
20
0)(2 2
2
1
10 r
r
r
rIldB
IldB
circle
0
Ir
rIldB
circle
00
2
2
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Irdr
IdlBldB
lineline
0
2
0
0
2cos
Arbitrary shape of the closed line
0 ldBcircle
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Amper’s law
encl
line
IldB 0
Example. Find a circulation.
Amper’s law allows one to find B for symmetric configurations of current.
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Conducting cylinder
r
IB
IrB
RrR
IBRr
R
IrB
R
IJrJrB
Rr
2
2
.22
:
2
,2
.1
0
0
0
20
22
0
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Coaxial cable
0,.4
)1(2
)(
)(2
.32
.22
.1
22
220
22
2200
0
20
Bcrbc
br
r
IB
bc
IJ
brJIrB
crbr
IB
braR
IrB
ar
Self-shielding
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Example. An infinite current sheet. There are n conductors per unite length.Each of them carries a current, I. Find B.
22 0
0
InBInllB
B
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Example. Prove that if in the absence of currents B is unidirectional it has to be uniform.
2121 0)( BBBBlldB
B1
B2
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Solenoid (N loops)
L
NnInB
InLINBLldBb
a
,
'''
0
00
L
L’
In the long solenoid (L>>a) B at the exit =(1/2) B in the center
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Toroidal solenoid (toroid)
0,.32
2
2
.2
0,.1
00
0
0
Bbr
Ina
INBbaif
r
INB
INrB
bra
Bar
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Example. Find a) the net force on the loop and b)the flux of B through the front surface of the loop.
I2I
a
bd
L
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Bohr’s magneton
m
e
Jshh
Lm
e
Lg
m
eL
rm
mrevrI
r
ev
T
e
dt
dQI
LB
LL
L
2
10626.6,2
,...3,2,,0,2
22
2
34
22
Planck’s constant
Bohr’s magneton
e m
e
SgS SSBS
,2
,
Different symmetries of e destributions
S-state(L=0,μL=0) 1e:S≠0,μs ≠ 02e:Stotal=0, μs=0
P-state:L= μL ≠ 0
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4 types of magnetic materials1.Paramagnetics:μi≠0, 00 0
Bwhen
VM i
i
Under the action of B0 alignment of μi struggles with chaotic thermal motion, resulting in Curie’s law
T
BCM 0
Inside the material:
Magnetic susceptibility
Relative magnetic permeability
...,,,,,,,
111
,1010~
,1
43
0
0
0
00
OUPlAlNaEuSmCe
K
B
M
KB
BMBB
m
m
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2. Ferromagneticsμi≠0, M ≠0 in domains
,...,,,
11
,1010~)(
),(1)(
63
0
00
00
0
steelCoNiFe
K
B
BM
BBKB
B
m
m
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Electromagnets typically contain a ferromagnetic core
nIB 00
In the absence of an iron core:
0
000
m
mm
K
BnInIKBKB
In the presence of an iron core:
Magnetic permeability
Narrow hysteresis loop,Often use superconducting wires
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Both paramagnetics and ferromagnetics are attracted to the magnet
NS B
1. Alignment (randomly oriented μ or M become parallel to B, i.e. material becomes magnetized )
2. Attraction(opposite poles attract each other)
S NB
S N
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NS B0
1. Induced M is anti- parallel to B0
2. Repulsion(similar poles repel each other)
S NB0
N S
DiamagneticsIn the absence of magnetic field μi=0,M=0In the presence of magnetic field M is induced in the direction opposite to external magnetic field (consequence of Lens’s low, see next lecture)These materials a repelled by magnets, though this repulsion is very week.
,...,,,,
11
0,1010|~|
),(1)(
64
00
0
HgPbAuCCu
K
BBKB
B
m
m
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Superconductors (R=0)
Perfect diamagnetics:Km=0, No magnetic field inside ofsuperconductors
000 MBB
1
Meisner’s effect