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It is also possible to fmd a formula for the energy of the particles as they emerge from the cyclotron.
From the dimensions of the machine, we know the radius of the fmal semicircle in which the particle moves,
before it exits the apparatus. The kinetic energy of the particle in its last semicircle is given by
K =+mv2
rqB
=
+ rn(,)'
mv
as r = 9B
--q 2 B 2 r 2
2m
Refer to the Physics Essentials Textbook, Section 8A, for a much more detailed description of the
cyclotron, its operation and its uses.
8.4
Knowledge and Skills Checklist
Do I know that there is no magnetic force on a charged particle that is stationary in a magnetic field or is
moving parallel to the field and can I explain why this should be so?
Do I know the formula for the force on a charged particle moving at an angle 0 to a magnetic field B?
Do I know the formula for the force on a charged particle moving at right angles to a magnetic field?
Do I appreciate that the force on a charged particle moving in a magnetic field is velocity dependent?
Do I know that the magnetic force is at right angles to the plane defined by v and B?
Do I know how to use the right hand palm rule to fmd the direction of the force on a positively charge
particle moving in a magnetic field?
Do I know how to use the right hand palm rule to fmd the direction of the force on a negatively charged
particle moving in a magnetic field?
Can I solve problems using F = qvB sin 0?
Do I know that a particle moving at right angles to a magnetic field will follow a circular path and can I
explain why this is so?
Do I know the formula for the radius of the path, can I derive the formula and solve problems using it?
Can I describe the nature and direction of the magnetic field in a cyclotron?
Can I describe the purpose of the magnetic field in the cyclotron?
Do I know that the period of the circular motion of the particles in the cyclotron is constant (i.e.
independent of the speed of the particles) and can I derive the formula for this period?
Do I know that the kinetic energy of the particles emerging from the cyclotron depends only on the
magnetic field and the radius of the cyclotron and can I derive the formula for this kinetic energy?
Can I solve problems using the formulae for period and kinetic energy?
8.5
Exercises
1
A proton is fired into a magnetic field of strength B = 0.16T. The velocity of the proton is 8.0x10~ms-'
at right angles to the field.
(1)
Find the magnitude of the force on the proton.
"
"
(2)
Find the radius of the path followed by the proton.
(3)
On the diagram at right sketch the path followed by the proton.
Indicate the direction of the magnetic force on the proton at five points
along this path.
w
-
~
-
2
Explain why the proton moves in a circular path.
State the direction of the force on the charged particle in each of the following.
-
-
-
,* +
5
,
+el
<
,
&
(4)
-
*
*
*<
%
-
~
I
, :L
=.,
Motion of Charged Particles in Magnetic Fields
3
A charged particle pl moves in a circular path in a magnetic field, as in the
diagram.
(1)
State whether the particle is positive or negative.
(2)
On the diagram, draw a vector to show the direction of the force at Q.
::
X
p,
/'
9
(3)
On the diagram, draw in the velocity vectors at points P and Q.
(4)
If the particle begins to slow down, draw in the path that it would now follow
.,:
*,<
's'
>:
/
:
X y,
,:< X X
( 5 ) Another particle p2 with the same charge as pl but with ten times its mass moves in exactly the
same path as pl. Find the ratio of the velocities of pl and p2.
( 6 ) Find the ratio of the forces on pl and p2.
4
5
Two charged particles, A and C, enter and move in a magnetic field, as shown.
(1)
Identify the sign of the charge on each particle.
(2)
List the possible differences between the particles and their speeds, in order
to account for the differences in the two paths.
A charged particle of mass m and charge q and moving with speed v enters, at right angles, a magnetic
mv
field of strength B. Prove that the radius r of the circular path followed is given by r = -.
9B
6
Consider beam of alpha particles, moving at a speed v = 5.0x10~ms-'at an angle €I= 80" to a magnetic
field of strength B = 0.09T.
(1)
What is the magnitude of the force that acts on the alpha particles?
(2)
What is the magnitude of the acceleration of the alpha particles?
(3)
Compare this magnetic force with the weight of the alpha particles.
Electrons, in an electron microscope, are accelerated by a potential difference of 50,000V.
(1)
What is the kinetic energy (in Joules) of the electrons as they leave the electron gun?
-
---
(2)
What is the speed of the electrons as they leave the electron gun?
(3)
The electrons now pass through a focussing magnetic field of strength B = 2.OT. They enter the
field at right angles to it. What is the magnitude of the force experienced by the electrons as they
enter this field?
(4)
The path of the electrons in this magnetic field is
depicted in the diagram at right. On this diagram draw
a vector to show the path of the electrons as they leave
the field.
>,< <.3. .\.,
'.
+*' B
x x x x x x x
<:,
/'-,
%
..,
*,<
$;.,
L
,k<
( 5 ) Explain why it is desirable to have the electrons
moving at right angles to the magnetic field.
V
+
<;
;>< <;,
)py
> ( X X X X X X
Mofion of Charged Particles in Magnetic Fields
8
The Dees of a cyclotron have a diameter of 90cm. The magnetic field inside the evacuated Dees is
1.9T. This cyclotron is used to accelerate protons.
(1)
Find the period of the motion of the protons.
(2)
What is the frequency of the alternating potential difference that must be applied to the Dees?
(3)
Determine the energy (in electron volts) of the protons that emerge fiom this cyclotron.
(4)
Consider a cyclotron that is identical to the one above except that its diameter is twice as great.
What effect would this change have on:
the period of the protons?
(a)
"
(b)
9
""
"""
.-.-"."--..---
--
the kinetic energy of the protons?
A cyclotron has a radius r and a magnetic field of strength B. It is used to accelerate particles of mass
m, carrying a charge q.
(1)
2nm
Show that the period T of the motion of the particles in their circular paths is given by T = -.
qB
(2)
q2B2r2
Show that the kinetic energy K of the emerging particles is given by K = -.
2m
10
The accelerating electric field of a cyclotron has a potential difference of 60,000V.
(1)
(a)
Determine the energy gained ( @ Joules) each time that a proton crosses the field between
the two Dees.
...
...........
(b)
find the total energy gained by a proton if it crosses the gap between the Dees 100 times?
(c)
What is the fmal speed of the emerging protons?
........
(2)
Explain why the radius of the circular path followed by the proton within the Dees increases after
each time that the proton crosses the gap between the Dees.
(3)
If the accelerating voltage between the Dees is doubled while all other parameters remained the
same, explain what effect this would have on
(a)
the energy gained by the proton each time it crossed the gap between the Dees;
(b)
the period of the proton in its circular path;
(c)
the number of times that the proton crosses the electric field between the two Dees, before
it emerges from the cyclotron;
(d)
the fiial kinetic energy of the emerging proton.