EXERCISE 1: FORCES ACTING ON A CHARGED PARTICLE AND THE EQUATIONS OF MOTION

Imagine that we have a particle of mass m, charge q, and velocity v=(vx,vy,vz) with vz>>vx,vyentering a region of uniform magnetic field B=Bxx^ and uniform electric field E=Eyy^. As shown below, Bx>0 and Ey<0. The length of the field region along the z-axis is L.

Neglecting any other forces (e.g., gravitational forces), show that the Cartesian components of the combined electric and magnetic forces are

(7)Fx=0
(8)Fy=qEy+qvzBx
(9)Fz=qvyBx

resulting in the equations of motion

(10)x¨=0
(11)y¨=qm(Ey+vzBx)
(12)z¨=qmvyBx

where the dot accents indicate differentiation with respect to time. Note that the particle will not experience any transverse acceleration if vz=vpass=Ey/Bx.

(a) Assume that Ey=105 V/m and Bx=2.00×103 T, and that all other field components are zero. Calculate, by hand, the Cartesian components of the acceleration at the instant when a Li+ ion of mass 7 amu and kinetic energy 100 eV enters the field region traveling along the direction u^=(x^+y^+100z^)/10002.

How will these acceleration components compare to those for a doubly ionized nitrogen ion (N+)?

(b) Write a code to perform the calculation in part (a). Note that, as soon as the particle enters the field region, the velocity components will change, and therefore so will the forces. To calculate an accurate trajectory, it is necessary to repeatedly calculate the forces, a task for which the computer is very well suited.

(c) What do you expect the trajectory of this ion to look like as it traverses the field region? Explain your answer.