Transcript III-4
III–4 Application of Magnetic
Fields
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Main Topics
• Applications of Lorentz Force
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Currents are Moving Charges
Moving Charges in El. & Mag.
Specific charge Measurements
The Story of the Electron.
The Mass Spectroscopy.
The Hall Effect.
Accelerators
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Lorenz Force Revisited
• Let us return
to the
Lorentz
force:
F q[ E (v B)]
and deal with its applications.
• Let’s start with the magnetic field only.
First, we show that
F q (v B ) F I ( L B )
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Currents are Moving Charges I
• Let’s have a straight wire with the length L
perpendicular to magnetic field and charge
q, moving with speed v in it.
• Time it takes charge to pass L is: t = L/v
• The current is: I = q/t = qv/L q = IL/v
• Let’s substitute for q into Lorentz equation:
F = qvB = ILvB/v = ILB
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Currents are Moving Charges II
• If we want to know how a certain conductor
in which current flows behaves in magnetic
field, we can imagine that positive charges
are moving in it in the direction of the
current. Usually, we don’t have to care what
polarity the free charge carriers really are.
• We can illustrate it on a conductive rod on
rails.
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Currents are Moving Charges III
• Let’s connect a power source to two rails which
are in a plane perpendicular to the magnetic field.
And let’s lay two rods, one with positive free
charge carriers and the other with negative ones.
• We see that since the charges move in the opposite
directions and the force on the negative one must
be multiplied by –1, both forces have the same
direction and both rods would move in the same
direction . This is a principle of electro motors.
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Moving Charge in Magnetic
Field I
• Let’s shoot a charged particle q, m by speed v
perpendicularly to the field lines of homogeneous
magnetic field of the induction B.
• The magnitude of the force is F = qvB and we can
find its direction since FvB must be a right-turning
system. Caution negative q changes the orientation
of the force!
• Since F is perpendicular to v it will change
permanently only the direction of the movement
and the result is circular motion of the particle.
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Moving Charge in Magnetic
Field II
• The result is similar to planetary motion. The
Lorentz force must act as the central or centripetal
force of the circular movement:
mv2/r = qvB
• Usually r is measured to identify particles:
v 1
r q
m B
• r is proportional to the speed and indirectly
proportional to the specific charge and magnetic
induction.
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Moving Charge in Magnetic
Field III
• This is basis for identification of particles
for instance in bubble chamber in particle
physics.
• We can immediately distinguish polarity.
• If two particles are identical than the one with
larger r has larger speed and energy.
• If speed is the same, the particle with larger
specific charge has smaller r.
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Specific Charge Measurement I
• This principle can be used to measure
specific charge of the electron.
• We get free electrons from hot electrode
(cathode), then we accelerate them forcing
them to path across voltage V, then let them
fly perpendicularly into the magnetic field B
and measure the radius of their path r.
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Specific Charge Measurement II
• From: mv2/r = qvB v = rqB/m
• This we substitute into equation describing
conservation of energy during the
acceleration:
• mv2/2 = qV q/m = 2V/(rB)2
• Quantities on the right can be measured. B
is calculated from the current and geometry
of the magnets, usually Helmholtz coils.
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Specific Charge of Electron I
• Originally J. J. Thompson used
different approach in 1897.
• He used a device now known as a
velocity filter.
• If magnetic field B and electric field E are
applied perpendicularly and in a right
direction, only particles with a particular
velocity v pass the filter.
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Specific Charge of Electron II
• If a particle is to pass the filter the
magnetic and electric forces must
compensate:
qE = qvB v = E/B
• This doesn’t depend neither on the
mass nor on the charge of the particle.
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Specific Charge of Electron III
• So what exactly did Thompson do? He:
• used an electron gun, now known as CRT.
• applied zero fields and marked the undeflected
beam spot.
• applied electric field E and marked the
deflection y.
• applied also magnetic field B and adjusted its
magnitude so the beam was again undeflected.
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Specific Charge of Electron IV
• If a particle with speed v and mass m flies
perpendicularly into electric field of
intensity E, it does parabolic movement and
its deflection after a length L:
y = EqL2/2mv2
• We can substitute for v = E/B and get:
m/q = L2B2/2yE
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Mass Spectroscopy I
• The above principles are also the basis of an
important analytical method mass spectroscopy.
Which works as follows:
• The analyzed sample is ionized or separated e.g. by GC
and ionized.
• Then ions are accelerated and run through a velocity
filter.
• Finally the ion beam goes perpendicularly into
magnetic field and number of ions v.s. radius r is
measured.
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Mass Spectroscopy II
• The number of ions as a function of specific
charge is measured and on its basis the
chemical composition can be, at least in
principle, reconstructed.
• Modern mass spectroscopes usually modify
fields so the r is constant and ions fall into
one aperture of a very sensitive detector.
• But the basic principle is anyway the same.
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The Hall Effect I
• Let’s insert a thin, long and flat plate of material
into uniform magnetic field. The field lines should
be perpendicular to the plane.
• When current flows along the long direction a
voltage across appears.
• Its polarity depends on the polarity of free charge
carriers and its magnitude caries information on
their mobility.
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The Hall Effect II
• The sides of the sample start to charge until
a field is reached which balances the
electric and magnetic forces:
qE = qvdB
• If the short dimension is L the voltage is:
Vh = EL = vdBL
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Accelerators
• Accelerators are built to provide charged
particles of high energy. Combination of
electric field to accelerate and magnetic
field to focus (spiral movement) or confine
the particle beam in particular geometry.
• Cyclotrons
• Synchrotrons
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Cyclotrons I
• Cyclotron is a flat evacuated container which
consists of two semi cylindrical parts (Dees) with
a gap between them. Both parts are connected to
an oscillator which switches polarity at a certain
frequency.
• Particles are accelerated when they pass through
the gap in right time. The mechanism serves as an
frequency selector. Only those of them with
frequency of their circular motion equal to that of
the oscillator will survive.
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Cyclotrons II
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The radius is given by:
r = mv/qB
= v/r = qB/m
f = /2 = qB/2m
f is tuned to particular particles. Their final
energy depends on how many times they
cross the gap. Limits: size Ek ~ r2, relativity
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Homework
• Chapter 28 – 1, 2, 5, 14, 21, 23
• Due this Wednesday July 30
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Things to read
• Repeat chapters 27 and 28,
excluding 28 - 7, 8, 9, 10
• Advance reading 28 – 7, 8, 9, 10
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The vector or cross product I
Let c=a.b
Definition (components)
ci ijk a j bk
The magnitude |c|
c a b sin
Is the surface of a parallelepiped made by a,b.
The vector or cross product II
The vector c is perpendicular to the plane
made by the vectors a and b and they have to
form a right-turning system.
ux
uy
uz
c ax
ay
az
bx
by
bz
ijk = {1 (even permutation), -1 (odd), 0 (eq.)}
^