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Electron's motion in Electro Magnetic field

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An electron moves in a force field due to a uniform electric field E and a uniform magnetic field B that is at right angles to E. Let E = jE and B = kB. Take the initial position of the electron at the origin with initial velocity vo = ivo in the x direction. Find the resulting motion of the particle. Show that the path of motion is a cycloid.

x = a sin wt + bt
y = a (1 - cos wt)
z = 0

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Start out with writing the Lorenz force on the electron.

F = -e [E + v x B]

Let v = (vx, vy , vz) and F = (Fx, Fy, Fz)

F = -e [E j + (vx,vy,vz) x B k]

(Fx, Fy, Fz) = -e [E j + - vx j + vy i]
Equating the components,

Fx = - e vy B
Fy = -e(E - B vx)
Fz = 0

Apply Newton's second law. F = m a

Fx = m d^2x/dt^2 = - e vy B
Fy = m d^2y/dt^2 = ...

Solution Summary

I have found the path of the electron when it is placed in a mutually perpendicular magnetic and electric fields provided that the electron is in motion. This problem and solution set will be a great practice set for a student in an Electromagnetic Theory course.