mcconnell (kam2342) – homework19 – Turner – (60230)
1
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001
(part 1 of 2) 10.0 points
In an experiment designed to measure the
strength of a uniform magnetic field produced
by a set of coils, electrons are accelerated from
rest through a potential difference of 202 V.
The resulting electron beam travels in a circle
with a radius of 3
.
64 cm.
The charge on an electron is 1
.
60218
×
10
−
19
C and its mass is 9
.
10939
×
10
−
31
kg.
Assuming the magnetic field is perpendic
ular to the beam, find the magnitude of the
magnetic field.
Correct answer: 0
.
00131668 T.
Explanation:
Let :
e
= 1
.
60218
×
10
−
19
C
,
r
= 3
.
64 cm = 0
.
0364 m
,
V
= 202 V
,
and
m
=
m
e
= 9
.
10939
×
10
−
31
kg
.
Since
K
i
= 0 and
K
f
=
1
2
m v
2
,
we have
1
2
m v
2
=

e

V
v
=
radicalBigg
2

e

V
m
e
=
radicalBigg
2 (1
.
60218
×
10
−
19
C) (202 V)
9
.
10939
×
10
−
31
kg
= 8
.
42949
×
10
6
m
/
s
.
From conservation of energy, the increase in
the electrons’ kinetic energy must equal the
change in their potential energy

e

V
:
F
=
e v B
=
m v
2
r
B
=
m v

e

r
=
(9
.
10939
×
10
−
31
kg)
(1
.
60218
×
10
−
19
C)
×
(8
.
42949
×
10
6
m
/
s)
(0
.
0364 m)
=
0
.
00131668 T
.
002
(part 2 of 2) 10.0 points
What is the angular velocity of the electrons?
Correct answer: 2
.
31579
×
10
8
rad
/
s.
Explanation:
For the angular velocity of the electron we
obtain
ω
=
v
r
=
8
.
42949
×
10
6
m
/
s
0
.
0364 m
=
2
.
31579
×
10
8
rad
/
s
.
003
10.0 points
Hint:
Use nonrelativistic mechanics to work
this problem.
A cyclotron is designed to accelerate pro
tons to energies of 6
.
7 MeV using a magnetic
field of 0
.
6 T.
The charge on the proton is 1
.
60218
×
10
−
19
C and its mass is 1
.
67262
×
10
−
27
kg.
What is the required radius of the cy
clotron?
Correct answer: 0
.
623334 m.
Explanation:
Let :
B
= 0
.
6 T
,
q
= 1
.
60218
×
10
−
19
C
,
E
= 6
.
7 MeV
= (6
.
7
×
10
6
eV)
×
(1
.
602
×
10
−
19
J
/
eV)
,
= 1
.
07334
×
10
−
12
J
,
and
m
= 1
.
67262
×
10
−
27
kg
.
The speed of the proton is
v
=
radicalbigg
2
E
m
=
radicalBigg
2 (1
.
07334
×
10
−
12
J)
(1
.
67262
×
10
−
27
kg)
= 3
.
58249
×
10
7
m
/
s
,
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mcconnell (kam2342) – homework19 – Turner – (60230)
2
where
E
is the kinetic energy of the proton.
The magnetic force supplies the centripetal
acceleration, so
q v B
=
m v
2
r
r
=
m v
q B
=
m
q B
radicalbigg
2
E
m
=
√
2
m E
q B
=
radicalbig
2 (1
.
07334
×
10
−
12
J)
(1
.
60218
×
10
−
19
C) (0
.
6 T)
×
radicalBig
(1
.
67262
×
10
−
27
kg)
=
0
.
623334 m
.
004
10.0 points
A mass spectrometer consists of an acceler
ating potential (to give the ion momentum)
and a uniform magnetic field (to momentum
analyze the ion).
The mass spectrometer using a beam of
doubly ionized sodium atoms has the follow
ing settings:
•
the magnetic field is 0
.
0775 T;
•
the charge of an atom is 3
.
20435
×
10
−
19
C;
•
the radius of the orbit is 0
.
0769 m; and
•
the potential difference is 155 V.
Calculate the mass of a sodium atom.
Correct answer: 3
.
67143
×
10
−
26
kg.
Explanation:
Let :
q
= 3
.
20435
×
10
−
19
C
,
r
= 0
.
0769 m
,
V
= 155 V
,
and
B
= 0
.
0775 T
.
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 Spring '10
 Turner
 Electron, Energy, Force, Mass, Magnetic Field, Electric charge

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