Now we write ǻPE in terms of q and the potential difference ǻV across the accelerating plates. Finally, we write an equation for ǻKE and
ǻPE based on the conservation of energy, substitute the expression found above, and solve for the unknown mass m of the charged particle.Although we derived the mass equation for a positively charged particle, we could equally well do so for a negatively charged particle (the
charged plates in the mass spectrometer would have to be reversed to accelerate it in the correct direction).Step Reason
6. ǻPE = qǻV change in potential energy
7. ǻKE = –ǻPE conservation of energy
8. substitute equations 5 and 6 into equation 7
9. solve for m
28.16 - Helical particle motion in magnetic fields
In order for a charged particle to move in a circular
path in a uniform magnetic field, it must enter the field
with a perpendicular velocity. But what if its velocity
has a component parallel to the field?
The result is shown to the right: It is called helical
motion. The particle traces out circles that wind
upward (or downward) in a fashion similar to the
motion of a car navigating the “corkscrew” ramps
found in many multistory parking garages. The
particle moves both in circles and up or down at the
same time. You can use the interactive simulation in a
following section to observe helical motion.
To explain why helical motion occurs, we need to decompose the velocity into
components perpendicular and parallel to the magnetic field. The two diagrams in
Concept 1 do this. In the left-hand diagram, the magnetic field is viewed obliquely (but
not fully parallel to the surface of the screen) and you can see both components of the
particle’s velocity.The component vperpendicular is perpendicular to the magnetic field. This component
accounts for the circular motion of the particle. The other component of the velocity,
vparallel, is parallel to the magnetic field. Since there is no magnetic force exerted on a
charge moving parallel to a magnetic field, this velocity component does not change. It
accounts for the constant upward or downward motion of the particle.
The result of the forces exerted on the particle is that the particle moves in circles in a
plane perpendicular to the magnetic field and at a constant speed in a direction parallel
to the magnetic field. The sum of these motions is helical motion.As the particle moves in a helical path, the vertical spacing between the loops of the
helix, known as the pitch, remains constant since the vertical velocity component does
not change.
Helical motion can arise in a nonuniform magnetic field, as well. A magnetic field that is
stronger at its outer edges can cause a particle to become trapped in a magnetic bottle.
The particle moves in a helical fashion, spiraling up and down inside the “bottle.” Physicists construct bottles like this as three-dimensional
systems that can contain charged particles indefinitely.
The Earth creates a magnetic bottle of this type. Its magnetic field is stronger near the poles. Electrons and protons are trapped in the bottle
created by the Earth. They oscillate back and forth over a short distance every few seconds, resulting in what are called the Van Allen radiation
belts.
Such belts, as well as solar flares, are responsible for auroras, the glorious bands of light visible in the sky at high latitudes at certain times of
the year. The auroras result from solar flares that shoot ionized particles, primarily electrons and protons, into the Earth’s atmosphere. These
particles get trapped in the Van Allen belts. As the particles collide with oxygen and nitrogen molecules from the atmosphere, they emit green
and pink light respectively. From a great distance, you may perceive a faint aurora as white light.Aurora Borealis display in the northern sky.Helical motion
Velocity components perpendicular,
parallel to fieldHelical motion results
·Perp: force causes circular motion
·Parallel: force zero, vparallel constant(^520) Copyright 2000-2007 Kinetic Books Co. Chapter 28