III

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III4 Application of Magnetic Fields
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Main Topics

• Applications of Lorentz Force
• 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
• and deal with its applications.
• Lets start with the magnetic field only. First,
  we show that

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Currents are Moving Charges I

• Lets 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
• Lets 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
  dont 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

• Lets connect a power source to two rails which
  are in a plane perpendicular to the magnetic
  field. And lets 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

• Lets 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
• 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 doesnt 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

• Lets 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

• The radius is given by
• r mv/qB ?
• ? v/r qB/m ?
• f ?/2? qB/2?m
• 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 ca.b
• Definition (components)

The magnitude c
Is the surface of a parallelepiped made by a,b.
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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.
?ijk 1 (even permutation), -1 (odd), 0 (eq.)