The motion of a charged particle in an electromagnetic field can be obtained by the Lorentz Equation for the force on a particle in such a field. If the electric field vector is E and the magnetic field vector is B, the force on a particle of mass m that carries a charge q and has a velocity v is given by:
F= qE + qv (cross product) B
Where we assume v<< c (the speed of light)
a.) If there is no electric field and the particle enters the magnetic field in a direction perpendicular to the lines of flux show that the trajectory is a circle with radius:
r = (mv)/(qB)
b.) Choose thr z axis to lie in the direction of B and let the plane containing E and B be the xy plane.
Then: B = Bk, E= Ey j + Ez k
Show that z(t) = zo = zo't + (qEz)/(2m) t2
Where z(0) = zo and z'(0) = zo'
c.) Continue the calculation and find x'(t) and y'(t); show that the time averages of these velocities are:
(x') = Ey/B, (y') = 0
d.) Integrate the velocitiy equations above (c) with the initial conditions:
x(0) = -(Am)/( qB); x'(0) = Ey/B; y(0) = 0y'(0) = A
Note I cannot label vectors but hopefully it is clear from the context which are vectors