8.4. Quadrupole focusing. A magnetic quadrupole field can be used as a focusing field for a charged particle beam. The cross section of the pole faces is shown in Fig 8.29. The pole faces are hyperbolas of the form xy-constant. There are two north poles and two south poles, marked on the figure. The dimension of the magnet per- pendicular to the cross section is . The magnetic field in the region 0 zs is B(x, y, z)-b(vi +n) in which b 0; the field is 0 for z < 0 and z > . Particles enter from negative z with velocity vo-ok and are deflected by the force F-ovo × B. (Neglect the small components vx and vy in calculating the force.) FIGURE 8.29 Exercise 4. A quadrupole focusing magnet. The hyperbolic curves are the boundaries of the pole faces. A charged particle beam moves out of the page Chapter 8 Magnetostatics (a) Sketch the B field lines in the xy plane. (b) Explain qualitatively why B produces focusing in the x direction and defocusing in the y direction, assuming the beam particles are positively charged. (c) Write the equations of motion for a beam particle with charge q and mass m, using the approximate force given above. Solve for x as a function of z forz >0, assuming x-x0 and ux = 0 at z = O. Sketch a graph of x (z) 8.4. Quadrupole focusing. A magnetic quadrupole field can be used as a focusing field for a charged particle beam. The cross section of the pole faces is shown in Fig 8.29. The pole faces are hyperbolas of the form xy-constant. There are two north poles and two south poles, marked on the figure. The dimension of the magnet per- pendicular to the cross section is . The magnetic field in the region 0 zs is B(x, y, z)-b(vi +n) in which b 0; the field is 0 for z . Particles enter from negative z with velocity vo-ok and are deflected by the force F-ovo × B. (Neglect the small components vx and vy in calculating the force.) FIGURE 8.29 Exercise 4. A quadrupole focusing magnet. The hyperbolic curves are the boundaries of the pole faces. A charged particle beam moves out of the page Chapter 8 Magnetostatics (a) Sketch the B field lines in the xy plane. (b) Explain qualitatively why B produces focusing in the x direction and defocusing in the y direction, assuming the beam particles are positively charged. (c) Write the equations of motion for a beam particle with charge q and mass m, using the approximate force given above. Solve for x as a function of z forz >0, assuming x-x0 and ux = 0 at z = O. Sketch a graph of x (z)