Show that 0 = E+v×B can be solved to give
Hint: Use vector identities for the triple cross product.
2 SINGLE PARTICLE MOTION
(a) Starting from the Lorentz equation of motion for a single particle,
ma = q(E+v×B),
and assuming that the only components of the fields are: (i) radial for the electric field, E = Er rˆ, and (ii) axial for the magnetic field, B = Bz zˆ. Show that Eq. 2.1 can be recast as
where Ωi = qB/m and θ = ArcTan(y/x).
(b) Use the following boundary conditions: x(0) = xo, y(0) = 0, x˙(0) = 0, and y˙(0) = −vo = −Er /Bz , where xo = 0.15 m, Er = 14300 V/m, and Bz = 0.2 T. Plot, using the software of your choice (e.g. Mathematica, Matlab, etc.), the trajectory of a singly ionized helium atom for two full revolutions around the z-axis.
(c)Repeat part (b), but using a singly ionized argon atom instead.