Hi all-
Catching up on some unfinished business, here is my take on
ExB drift: The betatron problem.
e an electric field along the x-axis, oscillating as e=Ecos(ot)
b a magnetic field along the y-axis, oscillating as b=Bcos(ot) and E<<B to
keep things
non-relativistic. (the case where e and b are constant is treated
in Jackson).
an electron, initially at rest with a=the charge/mass ratio.
u,v, and W+w respective x,y,z components the electron velocity vector, W a
constant.
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Eqations of motion:
(1) du/dt = a[e - (W+w)b];
(2)dv/dt =0;
(3) dw/dt =aub
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Solution:
Divide (1) by (3) to get du/dw = [(E/B) -W -w]/u. Set W=E/B (a constant)
to get
udu+wdw = 0, ------> u^{2}+w{2}= a constant. Call the constant V^{2}
(TeX notation)
Then I can write u=Vsin(q) w=Vcos(q) where q is a function of t to be
determined.
The initial condition (electron at rest) requires q(0)= 0 and V=-E/B <<1
(in units where c=1).
Equation (3) now gives us that dq/dt =-aBcost(ot) -------> q
=(a/o)sin(ot) to satisfy q(0)=0.
Then u=-E/Bsin[(a/o)sin(ot)], v=0,vw= E/B{1 - cos[(a/o)sin(ot)]. A
picture of tthe trajectory
in the x-z plane is shown at
The constant term, W=E/B is often referred to as the "ExB drift".
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A similar analysis applies to an electron subjected to a plane
wave (where E=B), but relativistic kinematics must be used.
--
"What did Barrow's lectures contain? Bourbaki writes with some
scorn that in his book in a hundred pages of the text there are about 180
drawings. (Concerning Bourbaki's books it can be said that in a thousand
pages there is not one drawing, and it is not at all clear which is
worse.)"
V. I. Arnol'd in
Huygens & Barrow, Newton & Hooke