4. [15 points] [3d motion, cylindrical coordinate rz; MATLAB ]. The picture shows a 'quadrupole mass filter'. It consists of four parallel rods, which are charged to induce a special time-dependent electric field in the space between them. Ions are fed into the space between the rods at one end. As this problem will show, only ions with a charge to mass ratio within a range are able to pass through the gap from one end of the rods to the other (z=0 to z=1m) without hitting one of the rods. The ion has a mass m and charge q, and is subjected to an electrostatic force (vector) F=qE due to an imposed time-dependent electric field vector E=E0(1+cos(t))r/d where r is the position in the x-y plane relative to the centerline-axis of the device (direction k in the figure) and E0,,d, are constants that describe the magnitude and geometry of the electric field. (a) Follow the five steps to obtain derive the equations of motion for the ion. [5 points] The final results should be (what coordinate system is this?): r=mdE0q(1+cos(t))r+r22(1+cos(t))r+r2=r2rz=0 where we have introduced 2=mdE0q. What are the units of ? Decide by looking at the 1st equation of motion above. (b) Write a MATLAB code to solve the above pair of second-order differential equations that will give r(t),(t),z(t) with some initial conditions r(0) (initial distance from the center line, r(0)=0,(0)= 0,(0)=0,z(0)=0 (take the point of entry into the electric field as the origin), z(0)=V (some initial value) - i.e. the particle is initially slightly off-center and moving along the k axis with some initial velocity. [5 points] (c) Analyze the motion of a charged particle that enters the rods at z=0 with initial position very slightly off the axis (take the initial position and z-velocity to be r(0)=0.00707m and V=0.01m/s, respectively) with the following parameters: =s10;=20,=1.1/s. Examining the z equations of motion, choose a time range that brings the particle to the end of the device z=1m. Plot the radius versus z (distance along the axis of the device) where "sols" is the name of the array containing your output (r,,z) positions versus time that you use in ode45 (you can use any other name, of course). [3 points] (d) Show that with the same initial conditions and same =s10;=20, the ion will arrive at z=1.0m without a large radial deflection from the center line if 0.711.57 (The results can be shown by trajectories obtained at values of just below and just above the range given). This is how the ion filter works: since depends on the mass of the particle, only ions with a mass in some range can make it to the far end. This can be tuned by changing various parameters and initial velocities. [ 3 points]
