Magnetic Fields

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Magnetic Fields
Recall how a charged plastic rod produces an
electric field E at points in space around it.
Likewise, a magnetic produces magnetic field B
at all point around it.
In an electromagnetic, a wire coil is wound
around an iron core and an controlled electric
current is sent through the coil. The strength of
field is determined from the size of the current.
Sorting scrap metal is very common use of
electromagnets.
The most familiar type of magnets are the
permanent magnets, like the refrigerator-door
type which do not need external currents to have
a magnetic field.
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Magnetic Fields
Electric charges sets up an electric field that
effects other charges, so one would expect that a
magnetic charge sets up a magnetic field that
effects other magnetic charges. However,
magnetic charges (sometimes called magnetic
monopoles) only exist in some theories, and their
existence has yet to be confirm.

• Magnetic fields are set up in two different ways
• Moving electric charges (currents)
• (2) Intrinsic spin of elementary charged
  particles such as protons

When a charged particle moves through a magnetic
field, a force due to the field acts on the
particle. We will discuss the relationship
between the field and this force.
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The Definition of B
The electric field E can be determine at a point
by introducing a stationary test charge and then
we measure the force on the charge.
Because magnetic monopoles do not exist, we must
define the B field in a different way.
One way is to measure the magnetic force FB
acting on a moving test charge.
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The Definition of B
We may explore the properties of the magnetic
field by directing a beam of charged particles
into a region where magnetic field exist.
By investigating the paths of charged particle,
we can deduce the magnetic force acting on them.
Our finding should the reveal the following
(1) The magnetic force is always perpendicular to
the particles velocity.
(2) At every point, there exist one particular
direction of the velocity for which the force
FB0. We define a magnetic field B to be a vector
quantity that is directed along this zero-force
direction.
(3) The magnetic force is proportional to the
particles charge q, both in magnitude and sign.
(4) The magnitude of the force FB is directly
proportional to the velocity vectors
perpendicular component to the magnetic field
vector. FB ? v?
v sin? where ? is the angle between the
direction of v and the direction of B
(5) The magnetic force direction is
perpendicular to the plane containing both the
velocity vector and magnetic field vector
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The Definition of B
These observations can be simply expressed
defining the magntitude of B field through the
relation
B
We can summarize these results with the following
vector equation
FB q v x B
The force FB on the particle is equal to the
charge q times the cross product of its velocity
v and the magnetic field B.
Recall that the cross-product of an arbitrary
vectors a and b, written as a x b, produces a
third vector c whose magnitude is
c a b sin ?
where ? is the angle between the vectors a and b.
Using this relationship for the cross-product we
can write
FB q v B sin ?
where ? is the angle between the velocity and
magnetic field.
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Magnetic Fields
FB q v B sin ?
This equation tells us that magnitude of the
force is zero, if q 0 or the particle is
stationary. It also tells us that the force is
zero when velocity of the particle is aligned
with the direction of the magnetic field.
The right hand rule allows us to determine the
direction of cross-products and therefore the
direction of FB given the direction of v and B
The right hand rule (in which v is swept into B
through the angle between them) gives the
direction of v x B as the direction of the thumb
and if q is positive then also the direction of
FB, however is q is negative the direction is
180o of of that of v x B.
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Magnetic Force Directions
z
F0
-x (corrected)
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Units of Magnetic Fields
Recalling the definition of the B-field presented
earlier
We can derive the MKS unit for B that follows is
newton per coulomb-meter/second which is defined
as a Tesla
A non-SI unit for B which is commonly used is the
gauss, where
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Magnetic Field Lines
Magnetic field can be represented with field
lines as with electric fields. The direction of
the tangent to a magnetic field line at any point
gives the direction of B at that point. The
density of lines represent the magnitude of B.
This figure shows magnetic field lines which
represents the magnetic field from a bar
magnetic. The lines pass through through the bar
and they all form closed loops. As the field
lines suggest, the magnetic is the strongest near
the ends. The end of the magnetic in which the
field line emerge is referred as the north pole
of the magnet and the other end where fields
enter the magnet is called the south pole.
Regardless of the shape of the magnets, if two
were place near to each other, we would find.
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Crossed Fields Discovery of the Electron
An electric force can be used to balance a
magnetic force on a charge, but the magnitude and
direction of the electric field depends on the
velocity of the charge
E -v x B The
balancing the forces requires that the electric
field is perpendicular to both the magnetic field
and velocity. When the two fields are
perpendicular to each other they are considered
to be crossed fields as require for balancing the
magnetic force. An important special case occurs
if the velocity of charge is perpendicular to
magnetic field. The magnitude of the balancing
electric field becomes E vB
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Crossed Fields Discovery of the Electron
Crossed electric and magnetic fields can be used
to measure the mass to charge ratio as with
Thomson modified cathode ray tube. A beam of
electrons, emitted from a hot filament, is formed
by accelerating the electrons which pass through
a slit. They enter region of cross electric and
magnetic fields. The beam hits a fluorescent
sceen and produces a spotwhere it hits.
Thomsons apparatus for measuring the ratio of
mass to charge of the electron. An electric field
and magnetic magnitudes could be controlled. The
E field is downward and B-field is into the plane
of the figure.
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Crossed Fields Discovery of the Electron
The steps of the measurement are
1) The fields are turned off and the position of
the spot is recorded.
2) Only the electric field E is turned on and the
deflection y of the beam at the edge of the
deflecting plates is computed from the
deflection of the spot on the screen. It is given
by
where L is the length of the plates and v is the
speed of the electrons
The magnetic field is now adjusted until the spot
returns to its original position meaning the
magnetic and electric forces cancel so the speed
of the electron is vE/B. Substitute the
velocity in the above equation and solve for m/q.
The result is
All the quantitites on the right can be
determined experimentally
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Crossed Fields Hall Effect
The Hall effect can be used to obtain the sign
and concentration of charges in a current through
a flat metal sample. A magnetic field,
perpendicular to the current, pushes charges to
one side of the sample. These displaced charges
create a transverse electric field perpendicular
to the current and as charges continue to collect
on one side the electric force increases. When
the electric force balances the magnetic force,
additional charge collection is stopped and the
system is in a steady state. Since an imbalance
of charges is now present in the transverse
direction, a transverse voltage must also be
present.
Electric Field across the strip times the width
d Voltage
Balance of Forces
where vd is the drift speed of the current, n is
number particles per volume, A cross-sectional
area of sample
Making substitution for drift velocity , we have
n d B i/A V