0) I did not mean for my previous message to be quite
so terse. I hit the "send" shortcut by accident. Only
by luck was it intelligible at all.
1) Many people (students especially) have a hard time
visualizing a field that is not the gradient of any
potential. The electric field in a betatron ... or
in an ordinary transformer ... is definitely in this
category, definitely not the gradient of any potential.
Here are some diagrams that make this easier to visualize: https://www.av8n.com/physics/non-grady.htm
especially https://www.av8n.com/physics/non-grady.htm#fig-betatron
As always, if you don't want to bother with security
change https: to http:.
2) To analyze the overall behavior of a betatron,
separate the electric contribution from the magnetic
contribution. Also imagine that the magnetic field
is changing /slowly/ relative to the times and sizes
of interest. We start with the non-relativistic limit:
*) A constant magnetic field means the particle moves
in a circle, with a curvature given by its e/m ratio
multiplied by |B|.
*) Gradually increasing |B| means more curvature. So
what we really get is a spiral, not a circle.
*) In the non-relativistic limit, the E field does
not change the shape of the spiral, but just makes
the particle move along the spiral faster and faster.
For a practical betatron, you might want to have a
spatially non-uniform field, so that there is a time-
independent field where the particles are, and a
time-varying field where the particles aren't. That
means the radius of curvature stays the same, but
you can still accelerate the particles, because the
voltage drop around the loop depends on the derivative
of /flux/ ... not the derivative of the local B field.
That's a Maxwell equation you know: For any loop:
voltage = flux dot.
We now lift the speed restriction and consider the
relativistic case. The electric field produces a
force that increases the /momentum/ of the particle
in accordance with the fundamental definition of
force:
F = dp / dτ
where τ is the proper time. In accordance with the
Lorentz force law the force is independent of
velocity to first order; the correction term is
second order. Also, the correction term doesn't
change direction of the force, so it doesn't change
the spatial shape of the orbit; it just changes
how fast the particle moves along the orbit.
Figuring out the effect of the magnetic field is
trickier, because the relativistic correction term
has a more directly noticeable effect. Specifically,
the magnetic force depends on /velocity/ not momentum.
The momentum is increasing without bound, but the
velocity is not proportional to momentum. This is
significant, because it means the force is not as
effective at changing the momentum. The curvature
is less than it would be in the non-relativistic
limit, less by a factor of γ.
As always, less curvature means greater radius
of curvature.
In a practical betatron, you want to compensate
for this by scheduling an increase in the magnetic
field in the region where the particles are. The
schedule is a nonlinear function of time, since
it depends on γ.