
396 Microwave Engineering
−
− = +
e
U B U B U
d
dU
z r r z r
( )
φ
φ
(11.33b)
− =
e
m
U B U B
dU
dt
r r
z
( )
φ φ
(11.33c)
As shown in Figure 11.8a, it may be assumed that B = B
0
a
z
and B
r
= B
ϕ
= 0.
Further, an electron (–e) is assumed to enter the eld at t = 0 at r = a,
ϕ
= 0 and
z = 0 with an initial velocity U = U
r0
a
r
. As U
z
= dz/dt = 0 and the particle starts
at z = 0, z remains 0 for all times. Also, U
ϕ
= 0 at t = 0, d
ϕ
/dt =
ω
and U
ϕ
= r
ω
.
Thus, Equation 11.33 may be written as
−
= −
e
B r
d r
r
0
2
2
2
ω ω
(11.34a)
e
m
B
dr
dt r
d
dt
r
0
2
1
= ( )ω
(11.34b)
dU
z
= 0
(11.34c)
These equations can be solved by following the procedure used in the case
of the rectangul