66
An Introduction to Beam Physics
We observe that the horizontal (x-a) motion is affected mostly by E x and
B y , and the vertical (y-b) motion is affected mostly by E y and B x , and it is a
direct consequence of the Lorentz force law. When a longitudinal component
of the magnetic field B s is present, it acts to mix the horizontal and the
vertical motions through the B s dependent terms in a
and b
, which is even
a linear effect. This phenomenon is readily observed in the spiral motion of
a charged particle moving through a solenoid if any transversal component of
the momentum exists, which is described by a and b in our case.
A careful analysis of the equations of motion reveals that indeed if all the
particle optical coordinates are small, so are their derivatives defined through
the equations of motion; indeed, the system is weakly nonlinear.
An Introduction to Beam Physics
We observe that the horizontal (x-a) motion is affected mostly by E x and
B y , and the vertical (y-b) motion is affected mostly by E y and B x , and it is a
direct consequence of the Lorentz force law. When a longitudinal component
of the magnetic field B s is present, it acts to mix the horizontal and the
vertical motions through the B s dependent terms in a
and b
, which is even
a linear effect. This phenomenon is readily observed in the spiral motion of
a charged particle moving through a solenoid if any transversal component of
the momentum exists, which is described by a and b in our case.
A careful analysis of the equations of motion reveals that indeed if all the
particle optical coordinates are small, so are their derivatives defined through
the equations of motion; indeed, the system is weakly nonlinear.