Transcript Lecture 13:B fields

Magnetic Fields
Chapter 26
Last lecture
26.2 The force exerted by a magnetic field
Definition of B
This lecture
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
Magnetic force and field
The definition of B
F  qv  B
The sign of q matters!
Find
expression
for radius, r
Charged particle moving in a plane
perpendicular to a uniform magnetic
field (into page).
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?
Answers: (a) +z (out)
A. Left
(b) –x (left, negative particle)
B. Up
C. Into page
(c) 0
D. Right
E. Down
F. Out of page
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.
p
e
(a) Which particle follows the smaller circle
A.
p
Answers: (a) electron (smaller mass)
B.
e
(b) Does that particle travel
A. clockwise or
B. anticlockwise?
(b) clockwise
Crossed magnetic
and electric fields
Net force:
F  qE  qv  B
The forces balance if
the speed of the
particle is related to
the field strengths by
qvB = qE
v = E/B (velocity selector)
Measurement of q/m
for electron
J J Thomson 1897
EXERCISE: Find an
expression for q/m
Sun-to-aurora TV analogy
A small part of the sky overhead
9
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.
(a) Rank directions A(1), B(2) and C(3)
according to the magnitude of the net force
on the particle, greatest first.
(b) Of all four directions, which might result in
a net force of zero:
Answers:
A(1), B(2), C(3) or D(4)?
(a) 2 is largest, then 1 and 3 equal (v x B = 0)
(b) 4 could be zero as FE and FB oppose
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).
Picture the problem:
Velocity vector is in the ydirection.
B is in the yz plane
Force on proton must be
towards West, ie in negative
x-direction
Circular motion
of a charged
particle in a
magnetic field
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
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
Key point: fosc= f = qB/2m
is independent of radius
and velocity of particle
V creates electric field in the gap, but no
electric field within the dees, because the metal
dees act as shields.
The Cyclotron
EXAMPLE: A cyclotron for
accelerating protons has a
magnetic field of 1.5 T and a
maximum radius of 0.5 m.
(a) What is the cyclotron freqency?
(b) What is the kinetic energy of the
protons when they emerge?
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.
Flexible wire passing
between pole faces of
a magnet.
(a) no current in wire
(b) upward current
(c) downward current
26.4 Magnetic force on a current-carrying wire
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?