BASICS OF MAGNETISM

1
MAGNETISM

• History of Magnetism
• Bar Magnets
• Magnetic Dipoles
• Magnetic Fields
• Magnetic Forces on Moving Charges and
  Wires
• Electric Motors
• Current Loops and Electromagnets
• Solenoids

• Sources of Magnetism
• Spin Orbital Dipole Moments
• Permanent Magnets
• Earths Magnetic Field
• Magnetic Flux
• Induced Emf and Current
• Generators
• Crossed Fields

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Magnetic Dipoles
Recall that an electric dipole consists of two
equal but opposite charges separated by some
distance, such as in a polar molecule. Every
magnet is a magnetic dipole. A bar magnet is a
simple example. Note how the E field due an
electric dipole is just like the magnetic field
(B field) of a bar magnet. Field lines emanate
from the or N pole and reenter the - or S
pole. Although they look the same, they are
different kinds of fields. E fields affect any
charge in the vicinity, but a B field only
affects moving charges. As with charges, opposite
poles attract and like poles repel.
-

_
Electric dipole and E field
N
S
Magnetic dipole and B field
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Magnetic Monopole Dont Exist
We have studied electric fields to due isolated
or - charges, but as far as we know, magnetic
monopole do not exist, meaning it is impossible
to isolate a N or S pole. The bar magnet on the
left is surrounded by iron filings, which orient
themselves according to the magnetic field they
are in. When we try to separate the two poles by
breaking the magnet, we only succeed in producing
two distinct dipoles (pic on right).
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Force Due to Magnetic Field
The force exerted on a charged particle by a
magnetic field is given by the vector cross
product
F q v ? B
F force (vector) q charge on the particle
(scalar) v velocity of the particle relative to
field (vector) B magnetic field (vector)
Recall that the magnitude of a cross is the
product of the magnitudes of the vectors times
the sine of the angle between them. So, the
magnitude of the magnetic force is given by
F q v B sin?
where ? is angle between q v and B vectors.
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F q v B
sin?
1 N 1 C (m / s) (T)
From the formula for magnetic force we can find a
relationship between the tesla and other SI
units. The sine of an angle has no units, so
1 N
1 N
1 T

C (m / s)
A m
A magnetic field of one tesla is very powerful
magnetic field. Sometimes it may be convenient to
use the gauss, which is equal to 1/10,000 of a
tesla. Earths magnetic field, at the surface,
varies but has the strength of about one gauss.
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Direction of Magnetic Field Force
Near the poles, where the field lines are close
together, the field is very strong (so the field
vector are drawn longer). Anywhere in the field
the mag. field vector is always tangent to the
mag. field line there. The charge in the pic in
moving into the page. Since q is , the q v
vector is also into the page. The - charge is
moving to the right, so the q v vector is to the
left. The mag. force vector is always ? to plane
formed by the q v vector and the B vector.
B
F
The force on the - charge is into the page. If
a charge is motionless relative to the field,
there is no magnetic force on it, but if either a
magnet is moving or a charge is moving, there
could a force on the charge. If a charge moves
parallel to a magnetic field, there is no
magnetic force on it, since sin 0 0.
-

v
B
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Motion of a Charge in a Magnetic Field
The ?s represent field lines pointing into the
page. A positively charged particle of mass m
and charge q is shot to the right with speed
v. By the right hand rule the magnetic force on
it is up. Since v is ? to B, F FB q v B.
Because F is ? to v, it has no tangential
component it is entirely centripetal. Thus F
causes a centripetal acceleration. As the
particle turns so do v and F, and if B is
uniform the particle moves in a circle. This is
the basic idea behind a particle accelerator like
Fermilab. Since F is a centripetal force, F
FC m v2 / R. Lets see how speed, mass,
charge,
field strength, and radius of curvature are
related
FB FC
? q v B m v2 / R
m v
? R
q B
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Magnetic Force on a Current Carrying Wire
A section of wire carrying current to the right
is shown in a uniform magnetic field. We can
imagine positive charges moving to right, each
feeling a magnetic force out of the page. This
will cause the wire to bow outwards. Shown on the
right is the view as seen when looking at the N
pole from above. The dots represent a uniform
mag. field coming out of the page. The mag. force
on the wire is proportional to the field
strength, the current, and the length of the
wire.
S
I
I
N
B
Continued
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Magnetic Force on a Wire (cont.)
Current is the flow of positive charge. As a
certain amount of charge, q, moves with speed v
through a wire of length L, the force of this
quantity of charge is
F q v ? B
Over the time period t required for the charge
to traverse the length of the wire, we have
F (q / t ) v t ? B
Since q / t I and v t L, we can write
I
B
F I L ? B
where L is a vector of magnitude L pointing
in the direction of I.
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Electric Motor
I
F

I
d
I
I
B
Current along with a magnetic field can produce
torque. This is the basic idea behind an electric
motor. Above is a wire loop (purple) carrying a
current provided by some power source like a
battery. The current loop is submerged in an
external field. From F I L ? B, the force
vectors in black are perpendicular to their wire
segments. The net force on the loop is zero, but
the net torque about the center is nonzero. The
forces on the left and right wires produce no
torque since the moment arm is zero for each
(they point right at the center). However, the
force F on the top wire (in the background) has
a moment arm d, so it produces a torque F d.
The bottom wire (in the foreground) produces the
same torque. These torques work together to
rotate the loop, converting electrical energy
into mechanical energy.
Continued
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Electric Motor
As the loop turns it eventually reaches a
vertical position (the plane of the loop parallel
to the field). This is when the moment arms of
the forces on the top and bottom wires are the
longest, so this is where the torque is at a max.
90 later the loop will be perpendicular to the
field. Here all moment arms and all torques are
zero. This is the equilibrium point. The angular
momentum of the loop, however, will allow it to
swing right through this position. Now is when
the current must change direction, otherwise the
torques will attempt to bring the loop back to
the equilibrium. This would amount to simple
harmonic motion of the loop, which is not
particularly useful. If the current changes
direction every time the loop reach equilibrium,
the loop will spin around in the same direction
indefinitely. Although a battery only pumps
current in one direction, the change in direction
of current can be accomplished with help of a
commutator, as can be seen with these animations
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Electromagnets Straight Wire
Permanent magnets arent the only things that
produce magnetic fields. Moving charges
themselves produce magnetic fields. We just saw
that a current carrying wire feels a force when
inside an external magnetic field. It also
produces its own mag-netic field. A long straight
wire produces circular field lines centered on
the wire. To find the direction of the field, we
use another right hand rule point your thumb in
the direction of the current the way your
fingers of your right hand wrap is the direction
of the magnetic field. B diminishes with
distance from the wire. The pics at the right

show cross sections of a current
carrying wire.
I into page, B clockwise
I out of page, B counterclockwise
B
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Current Loops and Magnetic Fields
The magnetic field inside a current loop tends to
be strong outside, it tends to be weak. Heres
why Using the right hand rule we see that each
length of wire contributes to a B field into the
page (all lengths reinforcing one another).
Outside the loop, say at P, the field is weak
since the left side of the wire produces a field
out of the page, but the right side produces a
field into the page. Explain why the field is
weak above the top wire. The situation is the
same with a circular loop. The effect is
magnified with multiple turns of wire. Yet
another right hand rule helps with current loops
Wrap your right hand in the direction of the loop
and your thumb points in the direction of B
inside. This is reminiscent of angular momentum
for a spinning body.
I
I
strong field inside loop, directed into page
I
strong field into page
I
P
weak field outside
weak field
I
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Solenoids

• Solenoids are one of the most common
  electromagnets.
• Solenoids consist of a tightly wrapped coil
  of wire, sometimes around an iron core. The
  multiple loops and the iron magnify the effect
  of the single loop electromagnet.
• A solenoid behaves as just like a simple bar
  magnet but only when current is flowing.
• The greater the current and the more turns
  per unit length, the greater the field inside.
• An ideal solenoid has a perfectly uniform
  magnetic field inside and zero field outside.

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Solenoids and Bar Magnets
A solenoid produces a magnetic field just like a
simple bar magnet. Since it consists of many
current loops, the resemblance to a bar magnets
field is much better than that of a single
current loop.
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Sources of Magnetism
Charges in motion (as in a current) produce
magnetic fields. This is one source of magnetism.
Another source is the electron itself.
Electrons behave as if they were tiny magnets.
Quantum mechanics is required to explain fully
the magnetic properties of electrons, but it is
helpful to relate these properties back to the
motion of charges. Every electron in an atom
behaves as a magnet in two ways, each having two
magnetic dipole moments Spin magnetic dipole
moment - due to the rotation of an
electron. Orbital magnetic dipole moment - due to
the revolution of an electron about the nucleus.
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Spin Magnetic Dipole Moment
Just as electrons have the intrinsic properties
of mass and charge, they have an intrinsic
property called spin. This means that electrons,
by their very nature, possess these three
attributes. Youre already comfortable with the
notions of charge and mass. To understand spin it
will be helpful to think of an electron as a
rotating sphere or planet. However, this is no
more than a helpful visual tool. Imagine an
electron as a soccer ball smeared with negative
charge rotating about an axis. By the right hand
rule, the angular momentum of the ball due to its
rotation points down. But since its charge is
negative, the spinning ball is like a little
current loop flowing in the direction opposite
its rotation, and the ball becomes an
electromagnet with the N pole up. For an electron
we would say its spin magnetic dipole moment
vector, µs, points up. Because of its spin, an
electron is like a little bar magnet.
µs
-
N
-
-
-
-
-
I
S
-
-
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Orbital Magnetic Dipole Moment
Imagine now a planet that not only rotates but
also revolves around its star. If the planet had
a net charge, its rotation would give it a spin
magnetic dipole moment, and its revolution would
give it an orbital magnetic dipole moment. Charge
in motion once again produces a magnetic field.
Since an electrons charge is negative, its
orbit is like a current loop in the opposite
direction. By the right hand rule, the angular
momentum vector in the pic below would point down
and the orbital magnetic dipole moment, µorb,
points up. An orbiting electron behaves like a
tiny electromagnet with its N pole in the
direction of µorb. Remember, though, that in
reality electrons are not like little planets. In
quantum mechanics, instead of circular orbits we
speak of electrons behaving like waves and we can
only talk of their positions in terms of
probabilities.
µorb
N
S
I
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Permanent Magnets
Each atom in a ferromagnetic material like iron
is like a little magnet, and these magnets are
all aligned in tiny regions called domains. At
high temps these domains can align in the
presence of an external field (like Earths)
leaving a permanent magnet. This happens at the
Mid-Atlantic Ridge beneath the Atlantic Ocean.
Lets melt the iron, and bring in a magnetic field.
Temp
Now, when we let the solid cool down, and take
away the external magnetic field, we have formed
a perma-nent magnet in the same direction as
external field.
Melting point
Domains
Bar Magnet
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Earths Magnetic Field
Earths field looks similar to what wed expect
if there were a giant bar magnet imbedded inside
it, but the dipole axis of this magnet is offset
from the axis of rotation by 11.5. Also, the
south pole of this magnet is near the geographic
north pole, NG. A compass points in the direction
of the magnetic north pole, NM, around which the
field lines reenter Earths surface. (Magnetic
north is actually the south pole of Earths
magnetic dipole.) NM, which is currently located
in Greenland, drifts about over the centuries.
About every million years Earths field reverses
entirely, as we know from the orientations of
magnetic fields near the Mid-Atlantic Ridge. The
field is likely due to the motion of charged
particles in the fluid outer core, and it
protects us from an otherwise deadly solar wind.
11.5
NM
NG
µorb
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Magnetic Flux
Magnetic flux, informally speaking, is a measure
of the amount of magnetic field lines going
through an area. If the field is uniform, flux is
given by
?B B A B A cos?
A
The area vector in the dot product is a vector
that points perpendicular to the surface and has
a magnitude equal to the area of the surface.
Imagine youre trying to orient a window so as
to allow the maximum amount of light to pass
through it. To do this you would, of course,
align A with the light rays. With ? 0, cos?
1, and the number of light rays passing through
the window (the flux) is a max. Note with the
window oriented parallel to the rays, ? 90 and
?B 0 (no light enters the window). The SI
unit for magnetic flux is the tesla-square meter
T m2. This is also know as a weber (Wb).
?
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Changing Magnetic Flux

• A changing magnetic flux in a wire loop induces
  an electric current.
• The induced current is always in a direction
  that opposes the change in flux.

These facts were discovered by Michael Faraday
and represent a key connection between
electricity and magnetism. One simple example of
this is a magnet moving in and out of a wire
loop. As a bar magnet approaches a wire loop
along a line perpendicular to the loop, more and
more field lines poke through the loop and the
flux increases. To oppose this change in flux a
current is induced in the direction
shown. Note that the induced current produces its
own magnetic field pointing to the right. Also
note that there is no battery in the loop! This
current will only exist when the flux inside the
loop changes. When the magnet is withdrawn the
flux decreases and current is induced in the
other direction. There is no current when the
magnet is still.
v
I
Java script
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Induced emfs and Currents
The current induced in a loop come not from a
battery but from a changing magnetic flux. We can
think of the loop containing an imaginary battery
that gets turned on whenever flux in the loop
changes. The strength of this battery is called
the emf (electromotive force) its symbol is a
script E E, and its measured in volts. The
induced current is given by
I E / R
where R is the internal resistance in the loop.
E itself depends on the rate at which the flux
inside the loop is changing. If the flux is
changing at a constant rate,
This Faradays law. The negative sign here
indicates the emf opposes the change in flux.
E - ??B / ?t
The greater the change in flux the greater, the
greater the induced emf, and greater the induced
current.
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Electric Generators
In a motor we have seen that a current loop in an
external magnetic field produces a torque on the
loop. In a generator well see that a torque on a
current loop inside a magnetic field produces a
current. In summary Motor Current
Magnetic field ? Torque Generator Torque
Magnetic field ? Current
Turbines in a power plant are usually rotated
either by a waterfall or by steam created heat
produced from nuclear power or the burning of
coal. As the turbines rotate, current loops turn
through a magnetic field to generate electricity.
This process converts mechanical energy into
electrical energy.
The simplest form of an electric generator is
called an alternating current (AC) generator. The
current produced by an AC generator switches
directions every time the wire inside of it is
rotated through a half turn. In the United
States, generator generally have a frequency of
60 Hz, which means the current switches direction
120 times every second! A graph of the current
output from an AC generator produces a sinusoidal
curve due to the periodic nature of the
generators rotation.
Continued
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Electric Generator (cont.)
B
Iinduced
As a turbine turns (due to some power source like
coal) a current loop (purple) is rotated inside a
magnetic field. The field is static but as the
loop turns as the number of field lines poking
through it changes. Thus we have a changing flux
and a corresponding induced emf and current. The
pic shows a loop just after it was horizontal
(perpendicular to the field). The flux is
decreasing since the loop is becoming more
vertical. Since fewer field lines are entering
the loop, the induced current is in a direction
to produce more field lines downward. Just prior
to this, as the loop was approaching horizontal,
the number of field lines inside it was
increasing, so the current was in the other
direction to oppose this change. The current
changes direction twice with each turn--whenever
the loop is horizontal. The result here is AC,
but (direct current) DC motors exist as well in
which current only flows in one direction.