26. The equation of motion for the proton is
F
=
q
v
×
B = q(v
x
ˆi+ v
y
ˆj + v
z
ˆ
k)
× Bˆi = qB(v
z
ˆj
− v
y
ˆ
k)
=
m
p
a = m
p
dv
x
dt
ˆi+
dv
y
dt
ˆj +
dv
z
dt
ˆ
k
.
Thus,
dv
x
dt
=
0
dv
y
dt
=
ωv
z
dv
z
dt
=
−ωv
y
,
where ω = eB/m
p
. The solution is v
x
= v
0x
, v
y
= v
0y
cos ωt and v
z
=
−v
0y
sin ωt. In summary, we have
v(t) = v
0x
ˆi+ v
0y
cos(ωt)ˆj
− v
0y
(sin ωt)ˆ
k.