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Particle in a magnetic field

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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

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Solution Summary

The solution is 14 pages long including full explanations and derivation of all equations.