Ben Sauerwine
Practice for Qualifying Exams
Thanks to Ryan for his input in this problem.
Problem Source:
CMU February 2006 Qualifying Exam
Consider an azimuthally symmetric magnetic field, such as might be produced by a
dipole magnet with circular pole faces, of the form
( )
z
r
B
B
ˆ
=
v
.
An electron of charge
e orbits in this field at a fixed radius R with momentum
p
v
in the
(
)
θ
,
r
plane.
(a)
What is the frequency of the electron,
0
ω
, assuming the magnetic field is
time independent?
(
)
(
)
(
)
(
)
(
)
R
B
e
m
v
R
R
B
m
eR
v
R
eRB
p
R
B
m
p
e
mR
p
r
mR
p
r
R
v
m
F
r
R
B
m
p
e
B
v
q
F
c
B
π
π
ω
2
2
ˆ
ˆ
ˆ
0
2
2
2
=
=
=
=
=
−
=
−
=
−
=
×
=
v
v
v
v
(b)
Suppose we inductively accelerate this electron by varying the magnetic field
slowly in time in some way.
We want the electron’s orbit to remain fixed at
R.
Derive a condition that relates
( )
r
B
and the average value of the field
within the orbit,
B
, that allows this to happen.
This is the so-called
Betatron condition.
Make a rough sketch of
( )
r
B
vs. r that allows this
acceleration technique to work.
The fact that we are working with the average value of the field inside the orbit seems to
imply that Faraday’s Law would be a good place to start:
( )
( )
B
dt
d
R
l
d
E
dS
n
r
B
R
B
dS
n
r
B
dt
d
l
d
E
C
S
S
C
2
2
ˆ
1
ˆ
π
π
−
=
⋅
⋅
=
⋅
−
=
⋅
∫
∫
∫
∫
v
v
v
v
v
v

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