Solutions to HW3

 

1)

E_Q = 0, B_Q = 0, i.e. no field components in   Q - direction. Electrodes must be rotationally symmetric with respect to z-axis.

 

Retaining only first order terms in r means: approximation is only valid for small r.

 

R'² << 1 implies approximation is only good for small values of dr/dz, i.e. small slopes of the orbit.

 

V₅=0

V₆= (-1/35) (-1/16) (-1/4)V_o'''''' = -1/2304 V₀^VI

 

2)

g=174mm

focal distance is the same for anti-protons

all potentials doubled: no changes in focal distances

reverse potential on center electrode gives (0, +2kV,0).

  Electrons:       with E_kin=2kV   g=156mm

                        with E_kin=1kV   g=91mm

  Protons:          with E_kin=1kV    reflected

                        with E_kin=2kV   g=5mm

The lens is always positive, i.e. for pos. or neg. potentials on the center electrode and

                                                   for pos. or neg. particles the focal length is positive.