I guess for question 2, wiki should answer most of your queries.
I shall post the more relevant parts here... The diagram in your
question isn't very self-explanatory...
Diagram of cyclotron operation from Lawrence's 1934 patent.
The electrodes shown above would be in the vacuum chamber, which
is flat, in a narrow gap between the two poles of a large
magnet.
In the cyclotron, a high-frequency alternating voltage applied
across the "D" electrodes (also called "dees") alternately attracts
and repels charged particles. The particles, injected near the
center of the magnetic field, accelerate only
when passing through the gap between the electrodes. The
perpendicular magnetic field (passing vertically through the "D"
electrodes), combined with the increasing energy of the particles
forces the particles to travel in a spiral path.
With no change in energy the charged particles in a magnetic
field will follow a circular path. In the Cyclotron, energy is
applied to the particles as they cross the gap between the dees and
so they are accelerated (at the typical sub-relativistic speeds
used) and will increase in mass as they approach the speed of
light. Either of these effects (increased velocity or increased
mass) will increase the radius of the circle and so the path will
be a spiral.
(The particles move in a spiral, because a current of electrons or ions, flowing perpendicular to a
magnetic field, experiences a perpendicular force. The charged particles move freely in a vacuum, so
the particles follow a spiral path.)
The radius will increase until the particles hit a target at the
perimeter of the vacuum chamber. Various materials may be used for
the target, and the collisions will create secondary particles
which may be guided outside of the cyclotron and into instruments
for analysis. The results will enable the calculation of various
properties, such as the mean spacing between atoms and the creation
of various collision products. Subsequent chemical and particle
analysis of the target material may give insight into nuclear
transmutation of the elements used in the target.
The centripetal force is provided by the transverse magnetic
field B, and the force on a particle travelling in a
magnetic field (which causes it to be angularly displaced, i.e
spiral) is equal to Bqv. So,
(Where m is the mass of the particle, q is its charge, v is its
velocity and r is the radius of its path.)
The speed at which the particles enter the cyclotron due to a
potential difference, V.
Therefore,
v/r is equal to angular velocity, ω, so
And since the angular frequency is
Therefore,
This shows that for a particle of constant mass, the frequency
does not depend upon the radius of the particle's orbit. As the
beam spirals out, its frequency does not decrease, and it must
continue to accelerate, as it is travelling more distance in the
same time. As particles approach the speed of light, they acquire
additional mass, requiring modifications to the frequency, or the
magnetic field during the acceleration. This is accomplished in the
synchrocyclotron.
The relativistic cyclotron frequency is
,
where fc is
the classical frequency, given above, of a charged particle with
velocity v circling in a magnetic
field.
The rest mass of an electron is 511 keV, so the frequency
correction is 1% for a magnetic vacuum tube with a 5.11 kV direct
current accelerating voltage. The proton mass is nearly two
thousand times the electron mass, so the 1% correction energy is
about 9 MeV, which is sufficient to induce nuclear reactions.
An alternative to the synchrocyclotron is the isochronous
cyclotron, which has a magnetic field that increases with radius,
rather than with time. The de-focusing effect of this radial field
gradient is compensated by ridges on the magnet faces which vary
the field azimuthally as well. This allows particles to be
accelerated continuously, on every period of the radio frequency,
rather than in bursts as in most other accelerator types. This
principle that alternating field gradients have a net focusing
effect is called strong focusing. It was obscurely known
theoretically long before it was put into practice.
