The motion of a charged particle in an electromagnetic field can be obtained from 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 charge q and has velocity v is given by:
F = qE +qv X B
If there is no electric field and the particle enters the magnetic field in a direction perpendicular to the lines of teh magnetic flux, show that the trajectory is a circle with radius
r = mv/qB = v/w
where w=gB/m is the cyclotron frequency
Choose the z-axis to lie in direction of B and let the plane containing E and B be the yz-plane.
B = Bk E = E_y j + E_z k
Show that the z-component of the motion is given by:
z(t) = z(0) + (z_dot(0))*t+(qE_z / 2m) * t^2
Continue the calculation and obtain expressions for dx/dt and dy/dt Show that the time average of these velocity components are:
<dx/dt> = E_y/B <dy/dt> = 0
Integrate the velocity equation from (3) and show (with appropriate initial conditions - see attached file) that:
x(t) = -A/w * cos(wt) + E_y/B *t
y(y) = A/w *sin (wt)
Plot the trajectory for the following cases:
A > E_y/B
A = E_y/B
A < E_y/B
The solution is 14 pages long including full explanations and derivation of all equations.