The magnetic force
The Right-hand rule
Magnetic field is the analogous field to the electric field for magnetism
ultimately electric and magnetic field is just different mirror image of the
same thing.
magnetic field have no ends—there is no magnetic monopole
magnetic goes from north magnetic pole to south magnetic pole
we have not found an isolated north or south pole: they are always found
in pairs.
A moving charge feels a force in a magnetic field
~ B = q~v × B
~
F
Make sure your coordinate system is right-handed, then:
x̂ × ŷ = ẑ ,
Magnetism
ŷ × ẑ = x̂ ,
ẑ × x̂ = ŷ .
Use any even permutation of the equation.
Thumb: first variable.
Index: second variable.
Middle/Rest: third variable.
FB = qvB sin θ
N·s
unit of magnetic field is Tesla: 1 T =
C·m
The force is always perpendicular to both the magnetic field and the
direction of motion.
magnetic force do no work.
just like electric field, magnetic field also have energy
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Let algebra takes care of everything
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Why the right-hand rule?
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Cyclotron motion
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The Hall effect
Electromagnetic (Lorentz) force:
A particle of charge q and velocity v enter a region of uniform magnetic
field B perpendicular to the velocity would under go circular motion.
called cyclotron motion.
mv
cyclotron radius: r =
qB
qB
cyclotron frequency: f = 2πm
~ = qE
~ + q~v × B
~
F
A conductor driven by an electric field is placed in a uniform
perpendicular magnetic field.
Charges is bend toward the edge of the conductor and accumulates there.
The accumulated charge creates a cross electric field and potential
difference.
At equilibrium, the Lorentz force of the cross field cancels the magnetic
force. The corresponding potential difference Hall Potential:
Cyclotron frequency do not depend on radius.
The cyclotron exploit this fact to accelerate charge particles.
Modern particle accelerators are descendants of the cyclotron
In general there can be also be drift in the direction of magnetic field
Magnetic field can “freeze” or “trap” charged particle with potentially very
useful application.
Magnetic mirroring: gradient in magnetic field tends to prevent particle
from entering region of strong magnetic field.
Responsible for Aurora Borealis.
Protects earth from solar wind.
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VH =
What are the directions of current density and total electric field?
The integer and fractional quantum Hall effect.
One Nobel prize each since 1980 (fractional effect unexplained).
Quantum mechanics at macroscopic level—correlated electron effect.
Current standard for measuring the value of e.
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More magnetic force
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What creates magnetic field?
Since magnetic force affects moving charge, an current carrying wire also
experience it:
Uniform current carried by straight wire in uniform magnetic field:
Moving charges or electric current according to the Biot-Savart Law:
~ =
dB
~ = I~l × B
~
F
~ = Id~l × B
~
dF
Since magnetic monopole do not exist1 , dipolar field is the simplest kind
of magnetic field.
Natural magnetic field of stars and planet are not well understood due to
the exceedingly complex dynamics. Earth’s magnetic field:
A current loop constitutes a magnetic dipole:
~
µ
~ =NIA
roughly equal to that of a magnetic dipole with dipole moment
µ = 8.0 × 1022 A · m2 .
due to motion its iron-rich core through not completely understood
mechanism.
periodic reversal about once every million years that last about 10,000
years.
Torque on a magnetic dipole:
~
~τ = µ
~ ×B
with potential energy:
~
U = −~
µ·B
Magnetism
µ0 Id~l × r̂
4π r 2
A current loop not only responds to magnetic field like a magnetic
dipole, but also creates a dipolar magnetic field.
Arbitrarily shaped conductor in non-uniform magnetic field:
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IB
nql
1 as
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far as we know
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Magnetic force between wires
Ampère’s Law
Amère’s Law: The line integral of magnetic field along any closed loop is
proportional to the steady current penetrating an (any) area bounded by
that loop.
I
~ · ~l = µ0 Ienclosed
B
Magnetic field due to a straight wire:
B=
µ0 I
2πr
Magnetic analogue of Gauss’s Law. Magnetic field is not conservative.
Eq. above only applies to steady current that do not change with time.
Allows trivial calculation of magnetic field in few highly symmetric cases.
Direction dictated by the r.h.r.
Line current and cyliner:
The force between two parallel wire:

1
µ0 I
B=
× rr
2π

R2
µ0 I1 I2 l
F=
2πd
Parallel current attract; anti-parallel currents repel.
Sheet current:
B=
outside
inside
1
µ0 J s
2
Solenoid:
B=
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Magnetic material
Ferromagnetism: material that can have permanent magnetic moment
magnetic domain: quantum-mechanical interaction induced region of
aligned magnetic momentum.
Usually transition or rare earth metal with many unpaired electron: Fe,
Co, Nb.
Curie temperature: temperature above which ferromagnetism ceases.
Hysteresis: magnetization of a ferromagnet depends on the history of
magnetization.
A lot of application everywhere and extremely interesting study of
thermodynamic.
Paramagnetism: has intrinsic magnetic moment that become aligned
under magnetic field.
Magnetic moment is induced and temporary.
Attracts weakly to magnet.
Diamagnetism: No intrinsic magnetic moment
Describes most material.
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µ0 nI
0
Magnetism
inside
outside
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