Download 26 Magnetism

26. Magnetism: Force & Field
Topics
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The Magnetic Field and Force
The Hall Effect
Motion of Charged Particles
Origin of the Magnetic Field
Laws for Magnetism
Magnetic Dipoles
Magnetism
Introduction
An electric field is a disturbance in space caused
by electric charge. A magnetic field is a
disturbance in space caused by moving electric
charge.
An electric field creates a force on electric charges.
A magnetic field creates a force on moving electric
charges.
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Magnetic Field and Force
It has been found that the
magnetic force depends on
the angle between the velocity
of the electric charge and
the magnetic field
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Magnetic Field and Force
The force on a moving charge can
be written as
F  qvB
where B
represents the
magnetic field
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Magnetic Field and Force
The SI unit of magnetic field is the tesla
(T) = 1 N /(A.m). But often we use a smaller
unit: the gauss (G)
1 G = 10-4 T
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The Hall Effect
The Hall Effect
Consider a magnetic field into the page and a current
flowing from left to right.
Free positive
charges will be
deflected upwards
and free negative
charges
downwards.
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h
The Hall Effect
Eventually, the induced electric force balances the
magnetic force:
qvd B  qE
Hall
Voltage
VH  Eh
1 IB
 vd Bh 
nq
t
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Hall coefficient
h
t is the thickness
Motion of Charged Particles
in a Magnetic Field
Motion of Charged Particles in
a Magnetic Field
The magnetic force on
a point charge
does no work. Why?
The force merely changes
the direction of motion of
the point charge.
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Motion of Charged Particles in
a Magnetic Field
Newton’s 2nd Law
F  ma
2
v
qvB  m
r
So radius of circle is
mv
p
r

qB qB
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Motion of Charged Particles in
a Magnetic Field
Since,
mv
r
qB
the cyclotron period is
2 r 2 m
T

v
qB
Its inverse is the cyclotron frequency
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The Van Allen Belts
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Wikimedia Commons
Origin of the Magnetic Field
The Biot-Savart Law
A point charge produces an electric field.
When the charge moves it produces a
magnetic field, B:
0 qv  rˆ
B
2
4 r
0 is the magnetic
constant:
0  4 107 T  m/A
 4 107 N/A2
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As drawn, the field
is into the page
The Biot-Savart Law
When the expression for B is extended
to a current element, IdL,
we get the Biot-Savart law:
0 I dL  rˆ
dB 
2
4 r
The total field is found by
integration:
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0 I dL  rˆ
B
2

4
r
Biot-Savart Law: Example
The magnetic field due to an infinitely long current
can be computed from the Biot-Savart law:
0 I dL  rˆ 0 I dL sin  ˆ
B

k
2
2


P
4
r
4
r
r x y
2
y
r̂
I
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
dL
0 I ˆ
B
k
4 y
2
x
Biot-Savart Law: Example
Note: if your right-hand thumb points in the
direction of the current, your fingers will curl in the
direction of the resulting
magnetic field
0 I
B
4 y
I
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Laws of Magnetism
Magnetic Flux
Just as we did for electric fields, we
can define a flux for a magnetic
field:
d  B  dA
But there is a profound difference
between the two kinds of flux…
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B
dA
Gauss’s Law for Magnetism
Isolated positive and negative electric
charges exist. However, no one has ever
found an isolated magnetic north or south
pole, that is, no one has ever found a
magnetic monopole
Consequently, for any closed surface the
magnetic flux into the surface is exactly
equal to the flux out of the closed surface
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Gauss’s Law for Magnetism
This yields Gauss’s law for magnetism

B  dA  0
Closed Surface
Unfortunately, however, because this law
does not relate the magnetic field to its
source it is not useful for computing
magnetic fields. But there is a law that is…
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Ampere’s Law
If one sums the dot product B  dr around
a closed loop that encircles a steady current
I then Ampere’s law holds:

B
B  dr  0 I Encircled
Closed Loop
dr
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I
That law can be
used to compute magnetic fields,
given a problem of sufficient symmetry
Ampere’s Law: Example
z
What’s the magnetic field a distance z above an
infinite current sheet of current density l per unit
length in the y direction? From symmetry, the magnetic
field must point in the
y
positive y direction
above the sheet and in
the negative y direction
below the sheet.
x
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Ampere’s Law: Example
z
Ampere’s law states that the line integral of the
magnetic field along any closed loop is equal to 0
times the current it encircles: 
B  dr  0 I Encircled
Closed Loop
y
x
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Draw a rectangular
loop of height
2a in z and length b
in y, symmetrically
placed about the current
sheet.
Ampere’s Law: Example
z
The only contribution to the integral is from the upper
and lower segments of the loop. From symmetry the
magnitude of the magnetic field is constant and the
same on both segments. Therefore,
y
the integral is just 2Bb.
The encircled current is
I = l b. So, Ampere’s
law gives 2Bb = 0 l b and
therefore B = 0 l / 2
x
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Magnetic Force on a Current
Magnetic Force on a Current
Force on each charge:
Force on wire segment:
n = number of charges
per unit volume
q vd  B


F  q vd  B nAL
  nq vd A L  B
 IL  B
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Magnetic Force on a Current
Note the direction
of the force on
the wire
For a current element
IdL the force is
dF  IdL  B
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Magnetic Force
Between Conductors
Since the force on a current-carrying
wire in a magnetic field is
dF  Idl  B
two parallel wires,
with currents I1 and I2 exert
a magnetic force on each
other. The force on wire 2 is:
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d
0 I1 I 2 dl2
dF2 
2 d
Magnetic Dipoles
Magnetic Moment
A current loop experiences no net force
in a uniform magnetic field. But it does
experience a
F
torque
B
F
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The force is
F = IaB
Magnetic Moment
Magnitude of torque
  bF sin  bIaB sin 
  IAB sin 
where A = ab
For a loop with N turns, the
torque is   NIAB sin 
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Magnetic Moment
It is useful to define a new vector
quantity called the magnetic dipole
moment
  NIAn̂
then we can write the torque as
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ˆ    B
Example: Tilting a Loop
ˆ    B
  IAnˆ  I R nˆ
2
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Example: Tilting a Loop
ˆ    B
  IAnˆ  I R nˆ
2
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Magnetic Moment
The magnetic torque that causes the
dipole to rotate does work and tends to
decrease the potential energy of the
magnetic dipole
If we agree to set the potential energy to zero
at 90o then the potential energy is given by
U    B
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

B
Magnetization
Magnetization
Atoms have magnetic dipole moments due to
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orbital motion of the electrons

magnetic moment of the electron
When the magnetic
moments align we
say that the material
is magnetized.
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Types of Materials
Materials exhibit three types of magnetism:
 paramagnetic
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
diamagnetic

ferromagnetic
Paramagnetism
Paramagnetic materials
 have permanent magnetic moments
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moments randomly oriented at normal
temperatures
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adds a small additional field to applied
magnetic field
Paramagnetism
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Small effect (changes B by only 0.01%)
Example materials
 Oxygen, aluminum, tungsten, platinum
Diamagnetism
Diamagnetic materials
 no permanent magnetic moments
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
magnetic moments induced by applied
magnetic field B

applied field creates magnetic moments
opposed to the field
Diamagnetism
Common to all materials.
Applied B field induces a magnetic field
opposite the applied field, thereby weakening
the overall magnetic field
But the effect is very small:
Bm ≈ -10-4 Bapp
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Diamagnetism
Example materials
 high temperature superconductors
 copper
 silver
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Ferromagnetism
Ferromagnetic materials
 have permanent magnetic moments
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align at normal temperatures when an
external field is applied and strongly
enhances applied magnetic field
Ferromagnetism
Ferromagnetic materials
(e.g. Fe, Ni, Co, alloys)
have domains of randomly
aligned magnetization
(due to strong interaction
of magnetic moments of neighboring
atoms)
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Ferromagnetism
Applying a magnetic field causes domains
aligned with the applied field to grow at
the expense of others that shrink
Saturation magnetization is reached
when the aligned domains
have replaced all others
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Ferromagnetism
In ferromagnets, some magnetization
will remain after the applied
field is reduced to zero,
yielding permanent magnets
Such materials exhibit
hysteresis
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Summary
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Magnetic Force
 Perpendicular to velocity and field
 Does no work
 Changes direction of motion of charged
particle
Motion of Point Charge
 Helical path about field
Summary
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Magnetic Dipole Moment
 A current loop experiences no net magnetic
force in a uniform field
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But it does experience a torque
Summary
The magnetism of materials is due to the
magnetic dipole moments of atoms, which
arise from:
 the orbital motion of electrons
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and the intrinsic magnetic moment of each
electron
Summary
Three classes of materials
 Diamagnetic
M = –const • Bext,
small effect (10-4)
 Paramagnetic
M = +const • Bext
small effect (10-2)
 Ferromagnetic
M ≠ const • Bext
large effect (1000)
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