p212c28: 1 Chapter 28: Magnetic Field and Magnetic Forces Iron ore found near Magnesia Compass needles align N-S: magnetic Poles North (South) Poles attracted to geographic North (South) Like Poles repel, Opposites Attract No Magnetic Monopoles Magnetic Field Lines = direction of compass deflection. Electric Currents produce deflections in compass direction. =>Unification of Electricity and Magnetism in Maxwell’s Equations.
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P212c28: 1 Chapter 28: Magnetic Field and Magnetic Forces Iron ore found near Magnesia Compass needles align N-S: magnetic Poles North (South) Poles attracted.
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p212c28: 1
Chapter 28: Magnetic Field and Magnetic Forces
Iron ore found near Magnesia
Compass needles align N-S: magnetic Poles
North (South) Poles attracted to geographic North (South)
Like Poles repel, Opposites Attract
No Magnetic Monopoles
Magnetic Field Lines = direction of compass deflection.
Electric Currents produce deflections in compass direction.
=>Unification of Electricity and Magnetism in Maxwell’s Equations.
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Magnetic Fields in analogy with Electric Fields
Electric Field:
– Distribution of charge creates an electric field E(r) in the surrounding space.
– Field exerts a force F=q E(r) on a charge q at r
Magnetic Field:
– Moving charge or current creates a magnetic field B(r) in the surrounding space.
– Field exerts a force F on a charge moving q at r
– (emphasis this chapter is on force law)
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Magnetic Fields and Magnetic Forces
Magnetic Force on a moving charge
– proportional to electric charge
– perpendicular to velocity v
– proportional to speed v (for a given geometry)
– perpendicular to Magnetic Field B
– proportional to field strength B (for a given geometry)
F = q v B
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F = q v B
F = |q| v B sin = |q| v B (v B) v
B
F
+
F = q v B
F = |q| v Bv
B
F
+
v
F = q v B
F = |q| v B
vB
F
+ B
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Units of Magnetic Field Strength:
[B] = [F]/([q][v])
= N/(C m s-1)
= Tesla
Defined in terms of force on standard current
CGS Unit 1 Gauss = 10-4 Tesla
Earth's field strength ~ 1 Gauss
Direction = direction of velocity which generates no force
Electromagnetic Force:
F = q ( E + v B )
= Lorentz Force Law
Magnetic Fields
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Magnetic Field Lines and Magnetic Flux
Magnetic Field Lines
Mapped out with compass
Are not lines of force (F is not parallel to B)
Field Lines never intersect
Magnetic Flux
dB = B . dA
d B dA
B dA
B dA
B
B
0 no magnetic charge! (no monopoles)
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• SI Unit of Flux:
– 1Weber = 1Tesla x 1 m2
– for a small area B = dB /dA
– B = “Magnetic Flux Density”
Flux through an open surface will play an important role
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Motion of Charged Particles in a Magnetic Field
Charged Particle moving perpendicular to the Magnetic Field
– Circular Motion!
– (simulations)
+
+
vF
F
v
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Charged Particle moving perpendicular to a uniform Magnetic Field
+
+
v
vF q vBmv
R
Rmv
q B
v
R
q B
m
| |
| |
| |
2
cyclotron frequency
R
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In a non-uniform field: Magnetic Mirror
Net component of force away from concentration of
field lines.
B
v
F
Magnetic Bottle Van Allen Radiation Belts
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Work done by the Magnetic Field on a free particle:
=> no change in Kinetic Energy!
Motion of a free charged particle in any magnetic field has constant speed.
dW F dx
qv B vdt
0!
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Recall:
Charged Particle moving perpendicular to a uniform Magnetic Field
F q vBmv
R
Rmv
q B
| |
| |
2
+
+
v
vR
Applications of Charged Particle Motion in a Magnetic Field
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Velocity Selector
makes use of crossed E and B to provide opposing forces
No net deflection => forces exactly cancel:
|q| v B=|q| E
v = E/B
E
upwards
F = q v B
downwards
F = qE
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J. J. Thomson’s Measurement of e/m
Electron Gun
and velocity selector:
E
V
2
2
2
2
21
VBE
me
eVmv
BE
v
e/m = 1.76x1011 C/kg
with Millikan’s measurement of e
=> mass of electron
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Example: Using an accelerating Potential of 150 V and a transverse Electric Field of 6x106 N/C. Determine
a) the speed of the electrons,
b) the magnetic field magnitude required for no net deflection
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Mass Spectrometer
E
vE
B
Rmv
qB
mRqB
v
RqB
E
2
One method
velocity selector + circular trajectory
R1
R2
p212c28: 17
Example: Vacuum System Leak Detector uses Helium atoms. Ionized helium atoms (He +) are detected with a mass spectrometer with a magnetic field strength of .1 T. With a velocity selector tuned to 1x105 m/s, where must the detector be placed to detect 4He +
ions?
p212c28: 18
Magnetic Force on a Current Carrying Wire
vd
I
dl
B
Fi
A
F F q v B
Nqv B n volume qv B
nAdlqv B JAdl B
Idl B RHR
i i i
d d
d
( )
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F ILB
A m T
N
sin
. sin
.
50 1 12 45
42 4
Example: A 1-m bar carries 50 A from west to east in a 1.2 T field directed 45° North of East. What is the magnetic force on the bar?
Force will be directed upwards (out of the plane of the page)
I dl
B
p212c28: 20
Torque on a Current Loop (from F = I l x B )Rectangular loop in a magnetic field (directed along z axis) short side length a, long side length b, tilted with short sides at an angle with respect to B, long sides still perpendicular to B.
B
Fa
Fa
Fb
Fb
Forces on short sides cancel: no net force or torque.
Forces on long sides cancel for no net force but there is a net torque.
p212c28: 21
Torque calculation: Side view
Fb = IBb
Fb
moment arm
a/2 sin
= Fb a/2 sin Fb a/2 sin
Iab B sin = I A B sin Magnetic Dipole ~ Electric Dipole
U =
Switch current direction every 1/2 rotation => DC motor
magnetic moment
B
p212c28: 22
Hall Effect
Conductor in a uniform magnetic fieldx
y
z
Bz
Jx
Magnetic force on charge carriers F = q vd B
Fz = qvdB Charge accumulates on edges
Jx
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Equilibrium: Magnetic Force = Electric Force on bulk charge carriers
Bz
Charge accumulates on edges Fz = 0 = qvdBy + q Ez
Ey
vE
B
J nqv nqE
B
nqJ B
E
dz
y
x dz
y
x y
z
Hall EMF V E w
I J tw
nqIB
V t
H z
x
y
H
w
t Jx
p212c28: 24
Negative Charge carriers:
velocity in negative x direction
magnetic force in positive z direction
=> resulting electric field has reversed polarity
Bz
Jx
Ey
p212c28: 25
Example: A ribbon of copper 2.0 mm thick and 1.5 cm wide carries a 75 A current in a .40 T magnetic Field. The resulting Hall emf is .81 V. What is the density of charge carrying electrons?