A particle of charge q starts from rest at the origin of coordinates in a region where there is a uniform electric field of strenth E parallel to the x-axis, and a uniform magnetic field B parallel to the z-axis.
Find the equations of motion, and solve them to show that the coordinates of the particle at a time t later will be:
x = (E/B*omega)*(1 - cos(omega*t))
y = - (E/B*omega)*(omega*t - sin(omega*t))
z = 0
where omega = q*B/m. (The path of the circle is a cycloid.)
The parametric equation of a cycloid:
x = constant*(1 - cos(omega*t))
y = constant*(omega*t - sin(omega*t))
The force acting on the particle:
F = q*E + q*vxB
The Attempt at a Solution
I've done some work on this problem and so far the equations of motion that I've got for the particle are as follows:
1) F(x) = q*E + q*v(y)*B -> x[double-dot] = q*E/m + omega*y[dot]
2) F(y) = -q*v(x)*B -> y[double-dot] = -q*B*x[dot]
I've tried integrating these equations once (eg. integrate 2)) and then substituting this into the other equation. This then gave me:
x[double-dot] + omega^2*x = E*B
And this is where I'm stuck. This has the form of a simple harmonic oscillator, except that the r.h.s. isn't zero, so I can't solve it. Also, I'm not even sure if everything that I've done so far is correct.
Any help on this would be very much appreciated!