Solution in Mathematica

Mathematica's function NDSolve implements various algorithms for solving first and higher order IVP's. Unlike Fortran, it is a symbolic algebra tool so it can convert any higher order IVP's into first order automatically and involve the appropriate solver to obtain an approximate numerical solution. Look at the documentation pages under Help to see typical examples of usage. Just as earlier, in Mathematica it is possible to follow the above physical formulation very closely without worrying much about the numerical aspects we discussed above. Our dependent variable is the position of the particle, r={x[t],y[t],z[t]}, with velocity v={x'[t],y'[t],z'[t]}. Write down the expressions in equations and then calculate the force in and then write the system of equations . Now use NDSolve to solve for the orbit of the charged particle in these simplified cases, and plot the orbit using ParametricPlot3D in each case:

1.

B₀=1, E₀=0, $\delta B=0$, and choose any initial position and velocity. What is the physical situation in this case? What is the solution, as you learned in your EM class? Verify your guesses.

2.

B₀=1, E₀=0, $\delta B=1$, $\omega =0.2$. Same as above.

3.

B₀=1, E₀=1, $\theta =0$, $\delta B=0$, $\delta E=0$. Explain the results.

4.

B₀=1, E₀=1, $\theta =\frac{\pi }{2}$, $\delta B=0$, $\delta E=0$. Explain the results.

5.

Experiment with other combinations that you find interesting.