Magnetic Fields

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Magnetic Fields Chapter 26 26.2 The force
exerted by a magnetic field Definition of B
26.3 Motion of a charged particle in a magnetic
field Applications A circulating charged
particle Crossed fields discovery of the
electron The cyclotron and mass spectrometer
26.4 Magnetic force on a current-carrying wire
Last lecture
This lecture
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Magnetic force and field
The definition of B
The sign of q matters!
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Find expression for radius, r
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• CHECKPOINT Here are three situations in which a
  charged particle with velocity v travels through
  a uniform magnetic field B.
• In each situation, what is the direction of the
  magnetic force FB on the particle?
• Left
• Up
• Into page
• Right
• Down
• Out of page

Answers (a) z (out) (b) x (left, negative
particle) (c) 0
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• CHECKPOINT The figure shows the circular paths
  of two particles that travel at the same speed in
  a uniform B, here directed into the page. One
  particle is a proton the other is an electron.
• Which particle follows the smaller circle
• A. p
• B. e
• Does that particle travel
• clockwise or
• anticlockwise?

Answers (a) electron (smaller mass) (b)
clockwise
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Crossed magnetic and electric fields Net force
The forces balance if the speed of the particle
is related to the field strengths by qvB qE
v E/B (velocity selector)
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Measurement of q/m for electron J J Thomson 1897
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Sun-to-aurora TV analogy
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A small part of the sky overhead
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• CHECKPOINT the figure shows four directions for
  the velocity vector v of a positively charged
  particle moving through a uniform E (out of page)
  and uniform B.
• Rank directions A(1), B(2) and C(3) according to
  the magnitude of the net force on the particle,
  greatest first.
• Of all four directions, which might result in a
  net force of zero
• A(1), B(2), C(3) or D(4)?

Answers (a) 2 is largest, then 1 and 3 equal
(v x B 0) (b) 4 could be zero as FE and FB
oppose
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Picture the problem Velocity vector is in the
y-direction. B is in the yz plane Force on proton
must be towards West, ie in negative x-direction
EXAMPLE The magnetic field of the earth has
magnitude 0.6 x 10-4 T and is directed downward
and northward, making an angle of 70 with the
horizontal. A proton is moving horizontally in
the northward direction with speed v 107 m/s.
Calculate the magnetic force on the proton by
expressing v and B in terms of components and
unit vectors, with x-direction East,
y-direction North and z-direction upwards).
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Circular motion of a charged particle in a
magnetic field
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The Cyclotron
It was invented in 1934 to accelerate particles,
such as protons and deuterons, to high kinetic
energies. S is source of charged particles at
centre Potential difference across the gap
between the Dees alternates with the cyclotron
frequency of the particle, which is independent
of the radius of the circle
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Schematic drawing of a cyclotron in cross
section. Dees are housed in a vacuum chamber
(important so there is no scattering from
collisions with air molecules to lose
energy). Dees are in uniform magnetic field
provided by electromagnet. Potential difference
?V maintained in the gap between the dees,
alternating in time with period T, the cyclotron
period of the particle.
Particle gains kinetic energy q ?V across gap
each time it crosses
?V creates electric field in the gap, but no
electric field within the dees, because the metal
dees act as shields.
Key point fosc f qB/2?m is independent of
radius and velocity of particle
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The Cyclotron
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• EXAMPLE A cyclotron for accelerating protons has
  a magnetic field of 1.5 T and a maximum radius of
  0.5 m.
• What is the cyclotron freqency?
• What is the kinetic energy of the protons when
  they emerge?

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26.4 Magnetic force on a current-carrying wire
Wire segment of length L carrying current I. If
the wire is in a magnetic field, there will be a
force on each charge carrier resulting in a force
on the wire.
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• Flexible wire passing between pole faces of a
  magnet.
• no current in wire
• upward current
• downward current

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26.4 Magnetic force on a current-carrying wire
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EXERCISE A wire segment 3 mm long carries a
current of 3 A in the x direction. It lies in a
magnetic field of magnitude 0.02 T that is in the
xy plane and makes an angle of 30 with the x
direction, as shown. What is the magnetic force
exerted on the wire segment?