Physics 122B
Electricity and Magnetism
Lecture 20 (Knight: 32.8-.10)
More Magnetic Effects
Martin Savage
Line Integrals Made Easy
If B is everywhere perpendicular
to the path of integration ds, then:
If B is everywhere parallel to
the path of integration ds, then:
f
f
 B  ds  0
 B  ds  BL
i
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i
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Ampere’s Law
A special case of a line integral is one that runs
in a closed path and returns to where it started,
i.e., a line integral around a closed curve, which, for
a magnetic field, is denoted by:
r r

O Bds

Consider the case of the field at a distance d
 2I
from a long straight wire:
B 0
4 d

r
r
O Bds

= 2  r B = 0 I
This result is:
 independent of the shape of the curve
around the wire;
 independent of where the current passes
through the curve;
 depends only on the amount of current
passing through the integration path.
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
r
r
O Bds

= 0 I
Ampere’s Law
3
Example: The Magnetic Field
Inside a Current-Carrying Wire
A wire of radius R carries current I
uniformly distributed across its cross
section.
Find the magnetic field inside the
wire at a distance r<R from the axis.
I through

I
r2
2
 JA 
r  I 2
 R2
R
r r
 ds 
B  ds  B(2 r )
 BO
Bds
 0 I through
r2
 0 I 2
R
0 I r 2 0 2 Ir
B

2
2 r R
4 R 2
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Symmetry and Long Solenoids
Can B have a radial component inside solenoid ?
y
Original
Solenoid
Rotate 1800
about y axis
Reverse
Current
Radial
B field?
Therefore, radial B field components near center are ruled out by
symmetry. But we can still have B fields in z and q directions.
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The Magnetic Field
of a Solenoid (1)
A solenoid is a helical coil
of wire consisting of multiple
loops, all carrying the same
current.
One can think of the field
of a solenoid by superimposing
the fields from several loops,
as shown in the lower figure. On the axis, the three
fields will add to make a stronger net field, but
outside the loop the fields from loops 1 and 3 will
tend to cancel the field from coil 2.
When the fields from all the loops are
superimposed, the result is that the field inside the
solenoid is strong and roughly parallel to the axis,
while the field outside is very weak. In the limit of
an ideal solenoid the field inside is uniform and
parallel to the axis, while the field outside is zero.
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The Magnetic Field
of a Solenoid (2)
We can use Ampere’s Law to calculate the
field of an ideal long solenoid by choosing the
integration path carefully. We choose a
rectangular LxW loop, with one horizontal side
outside the solenoid and the vertical sides passing
through.
If the loop encloses N wires, then Ithrough = NI.
Therefore, Ampere’s Law says that:

O Bds

 B  ds  B L  B W  B L  B W

3
4
W
2
1
r Br  ds  0 NI
1
2
3
4
The first side in inside and parallel to B, so B1=B.
Sides 2 and 4 are perpendicular to B (no radial
B), so B2=B4=0. Side 3 is outside the solenoid,
so B3=0. Therefore, B = 0NI/L
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If n = N/L is the
number of turns per
unit length, then:
Bsolenoid  0 nI
7
Example: Generating
a Uniform Magnetic Field
We wish to generate a 0.10 T magnetic field near the center of a
10 m long solenoid.
How many turns are needed if the wire can carry a maximum
current of 10 A?
Bsolenoid  0 nI  0 NI / L
Therefore, N 
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BL
(0.10 T)(0.10 m)

 800 turns
7
0 I (4 10 Tm/A)(10 A)
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Solenoids and Bar Magnets
As shown in the figures, the magnetic field
of a solenoid looks very much like that of a
bar magnet.
The north pole of the solenoid can be
identified using yet another right hand rule.
Let the fingers of your right hand curl in the
direction of the solenoid currents. Then your
thumb will be pointing in the direction of the
magnetic field and to the north pole of the
solenoid.
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Example: Magnetic Levitation
A 0.10 T uniform magnetic field is
horizontal, parallel to the floor. A
segment of 1.0 mm copper wire is also
parallel to the floor and perpendicular
to the field. What current through
the wire in what direction will allow
the wire to “float” in the magnetic
field? (rCu=8920 kg/m3)
F  ILB
 mg
 rVg
 r ( r 2 L) g
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I
r r 2 g
B
(8920 kg/m3 ) (0.001 m)2 (9.80 m/s 2 )

(0.10 T)
 0.687 A
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The Force between
Two Parallel Wires
Bwire
0 2 I

4 d
Fparallel wires  I1 LB2
 0 2 I 2 
 I1 L 

4

d


0 2 L I1 I 2

4 d
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Parallel wires carrying current in the
same direction attract each other.
Parallel wires carrying current in
opposite directions repel each other.
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Example: A Current Balance
Two stiff 50 cm long parallel wires
are connected at the ends by metal
springs. Each spring has an unstretched
length of 5.0 cm and a spring constant
of k = 0.020 N/m.
How much current is required to
stretch the springs to a length of 6.0
cm?
Fsp  k y
Fmag
0 2 LI 2

 2k y
4 d
1
   k yd
I  0
 15.5 A
L
 4 
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Forces
on Current Loops
Parallel currents in loops attract.
Opposite currents in loops repel.
Magnetic poles attract or repel
because the moving charges in one
current producing the pole exert an
attractive or repulsive magnetic
force on the moving charges in the
current producing the other pole.
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Torques
on Current Loops
Consider the forces on a current loop
carrying current I that is a square of length
L on a side that is in a uniform magnetic
field B. Its area vector makes an angle q
with B.
Ftop   Fbottom
Ffront   Fback
F  0
  Fd   ILB  L sin q   ( IL2 ) B sin q   B sin q
The magnetic dipole moment is   IL2  IA
  B
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  IA
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An Electric Motor
We can use the torque of a
loop in a magnetic field to make
an electric motor. The current
through the loop passes through
a commutator switch, which
reverses the current as the loop
approaches the equilibrium
position.
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Measuring Current with Torque
The torque on a coil in a uniform field can be
used to measure current. The figure shows a
galvanometer or current meter. The magnetic
field is arranged so that it is always
perpendicular to the coil as the coil pivots on lowfriction bearings.
A spiral spring produces angle-dependent
torque that is opposed by the magnetic field
induced torque. Therefore, flowing current
through the coil produces a rotation and pointer
deflection that is proportional to the current.
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Question
What is the direction of the current in the loop?
(a) Out at the top of the loop and in at the bottom;
(b) Out at the bottom of the loop and in at the top;
(c) Either direction is OK.
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Atomic Magnets
A plausible explanation for the magnetic
properties of materials is the orbital
motion of the atomic electrons. The
figure shows a classical model of an atom
in which a negative electron orbits a
positive nucleus. The electron's motion
is that of a current loop. Consequently,
an orbiting electron acts as a tiny
magnetic dipole, with a north pole and
a south pole.
However, the atoms of most elements contain many electrons. Unlike
the solar system, where all of the planets orbit in the same direction,
electron orbits are arranged to oppose each other: one electron moves
counterclockwise for each electron that moves clockwise. Thus the
magnetic moments of individual orbits tend to cancel each other and the
net magnetic moment is either zero or very small.
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The Electron Spin
The key to understanding atomic magnetism was the
1922 discovery that electrons have an inherent
magnetic moment. Perhaps this shouldn't be surprising.
An electron has a mass, which allows it to interact with
gravitational fields, and a charge, which allows it to
interact with electric fields. There's no reason an
electron shouldn't also interact with magnetic fields,
and to do so, it comes with a built-in magnetic moment.
e
Q=-e
e=9.274x10-24 J/T
An electron's inherent magnetic moment is often called the electron
spin, because in a classical picture, a spinning ball of charge would have a
magnetic moment. This classical picture is not a realistic portrayal of
how the electron really behaves, but its inherent magnetic moment
makes it seem as if the electron were spinning.
An electron also has an intrinsic angular momentum. However, its
magnetic moment is twice as large as a spinning sphere of charge with
that angular momentum should have. This is due to quantum effects.
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Ferromagnetism (1)
It happens that in iron (and other elements nearby in
the periodic table, e.g., Co and Ni) the spins interact
with each other in such a way that atomic magnetic
moments tend to all line up in the same direction.
Materials that behave in this fashion are called
ferromagnetic. The figures show how the spin magnetic
moments are aligned for the atoms making up a
ferromagnetic solid.
In ferromagnetic materials, the individual magnetic
moments add together to create a macroscopic magnetic
dipole. The material has a north and a south magnetic
pole, generates a magnetic field, and aligns parallel to an
external magnetic field. In other words, it is a magnet.
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Ferromagnetism (2)
Although iron is a magnetic material, a typical piece
of iron is not a strong permanent magnet. It turns out,
as shown in the figure on the right, that a piece of iron
is divided into small regions called magnetic domains. A
typical domain size is roughly 0.1 mm. The magnetic
moments of all of the iron atoms within each domain are
perfectly aligned, so that each individual domain is a
strong magnet.
The picture shows a photograph of domains in iron.
Each domain is magnetized in a different direction.
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Induced Magnetic Dipoles
When an unmagnetized ferromagnetic material
is placed in an externally applied magnetic field,
magnetic domains in the material that are
aligned with the field are energetically
favored.
This causes such aligned domains to grow,
and for domains that are nearly aligned to
rotate their magnetic moments to match the
field direction. The net result is that a
magnetic dipole moment is induced in the
material, with a new south pole close to the
north pole of the external magnet.
If, when the field is removed, some fraction
of the magnetic dipole moment remains, the
material has become a permanent magnet.
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Hysteresis*
Some ferromagnetic
materials can be permanently
magnetized, and “remember”
their history of magnetization.
The “hysteresis curve” shows
the response of a ferromagnetic
material to an external applied
field. As the external field is
applied, the material at first
has increased magnetization,
but then reaches a limit at (a)
and saturates. When the
external field drops to zero at
(b), the material retains about
60% of its maximum
magnetization.
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Unmagnetized
Physics 122B - Lecture 120
Partially
magnetized
Saturated
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