Download Chapter 26 Problem 90 † Given B = 0.1 G Solution Find the radius of

Chapter 26
Problem 90
†
Given
B = 0.1 G
Solution
Find the radius of a 0.1 MeV proton.
Converting the magnetic field from gauss to tesla gives
B = 1.0 × 10−5 T
The energy of the 0.1M eV proton is
−19 J
K = 0.1 × 106 eV 1.602×10
1 eV
−14
= 1.602 × 10
J
The mass of the proton is 1.67 × 10−27 kg. The velocity of the proton is then
K = 21 mv 2
q
q
−14 J)
2K
= 2(1.602×10
v=
m
1.67×10−27 kg
= 4.38 × 106 m/s
The magnetic force provides the centripetal acceleration for the proton as it travels in the circle.
Therefore, from Newton’s 2nd law, the magnitude of the force is
F = ma
v2
r
where v is the component of the initial velocity that is perpendicular to the magnetic field. Solving for
radius gives
qvB = m
r=
mv
qB
(1)
(1.67 × 10−27 kg)(4.38 × 106 m/s)
r=
(1.602 × 10−19 C)(1.0 × 10−5 T )
r = 4.57 × 103 m = 4.57 km
Now find the radius of the 1 M eV proton.
√
Since this particle has 10 times the energy of the first, the velocity will be 10 times as large as the
velocity of the √
0.1 M eV proton. Since the velocity is in the numerator when calculating the radius, the
radius will be 10 times as large as the original proton. Therefore, the radius of the 1 M eV proton is
√
√
r1 M eV = 10r0.1 M eV = 10(4.57 km)
r1 M eV = 14.4 km
√
Likewise the 10 M eV proton will have 10 times the radius of the 1 M eV proton or 10 times the radius
of the 0.1 M eV proton. Therefore, the radius of the high energy proton is
r10 M eV = 45.7 km
†
Problem from Essential University Physics, Wolfson
The period it takes the proton to make one circle can also be calculated from the velocity of the proton.
Use the velocity for the 0.1 M eV proton and the fact that the distance travelled by the proton is the
circumference of a circle. Time is then
t=
2πr
2π(4.57 × 103 m)
d
=
=
v
v
4.38 × 106 m/s
t = 6.56 × 10−3 s = 6.56 ms
Since velocity and radius of the circle are directly proportional to each other when a particle is moving in
a magnetic field, the time for the 1 M eV and 10 M eV protons will also be 6.56 ms.