Physics 122B Electricity and Magnetism

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Physics 122B Electricity and Magnetism
Lecture 20 (Knight 32.8-.10) More Magnetic
Effects

• Martin Savage

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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
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Amperes 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
Consider the case of the field at a distance
d from a long straight wire
2 p r B m0 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.

m0 I
Amperes Law
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Example The Magnetic FieldInside 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 rltR from the axis.
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Symmetry and Long Solenoids
Can B have a radial component inside solenoid ?
y
Rotate 1800 about y axis
Original Solenoid
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 Fieldof 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 fieldof a solenoid by
superimposingthe 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 Fieldof a Solenoid (2)
We can use Amperes 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, Amperes
Law says that
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W
2
1
If n N/L is the number of turns per unit
length, then
The first side in inside and parallel to B, so
B1B. Sides 2 and 4 are perpendicular to B (no
radial B), so B2B40. Side 3 is outside the
solenoid, so B30. Therefore, B m0NI/L
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Example Generatinga 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?
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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? (rCu8920
kg/m3)
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The Force betweenTwo Parallel Wires
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?
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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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Torqueson 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.
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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
low-friction 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?
• Out at the top of the loop and in at the bottom
• Out at the bottom of the loop and in at the top
• 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. Thefigure shows a
classical model of an atomin which a negative
electron orbits apositive nucleus. The
electron's motionis that of a current loop.
Consequently,an orbiting electron acts as a
tinymagnetic dipole, with a north pole anda
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
me
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.
Q-e
me-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.
Partially magnetized
Saturated
Unmagnetized