Use the Lorentz equations of motion with a time-dependent E-field to model a cyclotron. Use a solver a Runge-Kutta solver for differential equations and create a plot of the trajectory of a Hydrogen ion over 50 orbits. Use a magnetic field of 10 Tesla and an electric field that oscillates between ±1 V/m. (Adopt a consistent unit system, such as the SI unit system.) 2. After 1000 orbits, how fast is the ion going? If the particle were accelerated by a linear electric field of 1 V/m for the same amount of time with no Bz what would the final velocity have been? How do the two velocities compare? 3. Grad students operating cyclotrons had to set the frequency very accurately. How good is the acceleration if the frequency is 0.9ωc? How about 0.999ωc? How accurately does the driving frequency have to match the cyclotron frequency to get ∼ 50% of the velocity of a perfectly-tuned cyclotron after 1000 orbits? Show all code used (mathematica or MATLAB).
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