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Chapter 25
Magnetism
Magnets
Magnets: interact on iron objects / other magnets
1) North pole (N-pole) & south pole (S-pole)
2) Opposite poles attract, like poles repel
3) Magnetic monopoles have not been found
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Magnetic field
Magnets interact each other by magnetic field
Magnets
create
magnetic field
interact on
magnetic field lines
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Currents produce magnetism
H. C. Oersted found in 1820:
An electric current produces a magnetic field.
magnet ←→ magnet
I
magnet ←→ current
current ←→ current
A. M. Ampère:
(molecular currents)
Magnetism is produced by electric currents.
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Nature of magnetism
Magnetism is the interaction of electric currents
or moving charges.
A. Einstein: electromagnetic field in relativity
Magnetic field:
1) Created by / interacts on
currents or moving charges
2) Closed field lines without beginning or end
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Force on a current
Magnetic field exerts a force on a current (wire)
It’s called Ampère force
1) Uniform field:
F  IlB (l  B )
or F  IlB sin 
F  Il  B
I, l
F
B

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Definition of B
Define magnetic field B by using: F  Il  B
SI unit for B : Tesla (T), 1 T = 1 N/A·m =104 G
2) General case: nonuniform B & curved wire
dF  Idl  B
 F   Idl  B
where dl is a differential length of the wire
integral over the current-carrying wire
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F on curved wire
Example1: Uniform magnetic field B . Show that
any curved wire connecting points A and B is
exerted by the same magnetic force.
Solution: Total magnetic force:
F   Idl  B  I (  dl )  B
lab
 Ilab  B
It is equivalent to a straight wire from A to B
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F   Idl  B  Ilab  B
Discussion:
1) It is valid only if B is uniform
2) Total force exerted on a current loop (coil)?
3)
a
R
o
I
Fab  Fab  I  2R  B  sin 45
B
b
Fab  Fao  IRB
.
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Nonuniform field
Example2: I1 and I2 are on the same plane. What
is the force on I2, if the magnetic field created by
I1
I1 is B  k ?
r
Solution: Total force on I2
I1 I 2
k
dr
r
ab
F   I 2 Bdl   a
ab
 kI1 I 2 ln
a
F
I1
a
b
r
dl
dr
I2
direction?
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Interaction of currents
Question: Infinitely long current I1 and circular
current I2 are insulated and on the same plane.
What is the force between them?
I1 I 2
dF  k
 Rd
R sin 
F   sin  dF  kI1 I 2  d
I1
dF

R
I2dl
I2
 2 kI1 I 2
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*Railgun
Weapon for space war in future → railgun
High power ~ 107 J
High velocity ~ 10 Mach
Battery & rail
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Torque on a current loop (1)
Rectangular current loop in a uniform field
Ftotal  FAB  FBC  FCD  FDA  0
  2  FAB
FDA
b
sin   BIab sin   BIS sin 
2
O
B
B
D
FCD
b
A
I
FCD
O(O ')
a

I
FAB
B
D(C )
C
FBC
A( B)
S
FAB
O
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Torque on a current loop (2)
Torque on the loop:
O
  BIS sin 
I
a
Magnetic dipole moment:
  IS
B
b
FAB
FCD

I
S
O
where the direction is defined by right-hand rule
  B

p

Ql

compare with 

   p E
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Magnetic dipoles
  IS
  B
These are valid for any plane current loop
A small circular current → a magnetic dipole
1)  =/2: maximum torque
I

B
2)  =0 or : stable / unstable equilibrium

I

B
3) N loops coil / solenoid:
  NIS
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Magnetic moment of atom
Example3: Show that μ of an electron inside a
hydrogen atom is related to the orbital momentum
L of the electron by μ=eL/(2m).
Solution:

1
 IS  evr
2
v
Q
e
I
2 r
T
S   r2
Orbital (angular) momentum:
eL

2m
r
L  mvr
Classic / quantum model
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μ of rotating charge
Example4: A uniformly charged disk is rotating
about the center axis (σ, R, ω). Determine the
magnetic moment.
Solution: Magnetic moment   IS
Rotating charge: I  Q   Q
T 2
Total magnetic moment:

R
0

1
  2 rdr  r 2   R 4
2
4

dr
r.

R
direction?

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Force on moving charges
Magnetic field exerts a force on a moving charge:
F  qv  B
→ Lorentz force
1) Magnitude: F  qvB sin 
F

B
q
v
2) Direction: right-hand rule, sign of q
If q < 0, F has an opposite direction to v  B
3) Lorentz force doesn’t do work on the charge!
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Motion in a uniform field (1)
Point charge moves in a uniform magnetic field
1) v
B:
F  qv  B  0
Free motion
2) v  B : Uniform circular motion
mv
F  qvB 
R
2
2 R 2 m
mv

R
, T 
v
qB
qB
B
R
F
q
v
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Motion in a uniform field (2)
3) General case:
v  v  v
v
q
v
h
B

v
Free motion + uniform circular motion
Combination: the charge moves in a helix
2 mv cos 
2 m
mv sin 
, h
R
, T 
qB
qB
qB
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Aurora & magnetic confinement
Aurora: Caused by high-energy charges
plasma
Magnetic confinement:
“magnetic mirror”
coil
coil
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Lorentz equation
Example5:A proton moves under both magnetic
and electric field. Determine the components of
the total force on the proton. (All in SI units)
B  0.4i  0.2 j , E  (3i  4 j ) 103 , v  (6i  3 j  5k ) 103
Solution: Total force
F  q( E  v  B)
 e  3i  4 j  103  e(6i  3 j  5k ) 103  (0.4i  0.2 j )
 (6.4i  9.6 j ) 1016 N
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Different regions
Example6:A proton moving in a field-free region
abruptly enters an uniform magnetic field as the
figure. (a) At what angle does it leave ? (b) At
what distance x does it exit from the field?

Solution: Circular motion
(a)   45o
r
(b) r  mv  x  2mv
eB
eB
r
x
v
45o
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Challenging question
Question:A electron is released from rest. If the
field is shown as the figure, how does the electron
move under the field? What is the path?
The path is a cycloid
-
B
E
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The Hall effect
Current-carrying conductor / semiconductor
placed in a magnetic field → Hall voltage
Lorentz force → Hall field → equilibrium
eEH  evB  EH  vB  VH  EH l  vBl
VH  KI H B
B
IH → Hall current
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Applications of Hall effect
1) Distinguish the semiconductors
VC  VD  N  type
VC  VD  P  type
2) Measure magnetic field
Hall sensor / switch
3) Measure the carrier concentration
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