University of Surrey, Department of Physics
2nd Year Electromagnetism Laboratory
Paths of Charged Particles in Electric and Magnetic Fields — 2 weeks
(Set 2)
1. Introduction
The force acting on a (non-relativistic) particle with charge q and velocity v in any
combination of electric and magnetic fields can be described by the equation
F  mr  q(E  v  B)
[1]
In this practical, you will use this equation to investigate the paths of electrons in
electric and magnetic fields and write a program to calculate the trajectory of particles
in a simple particle accelerator, called a cyclotron.
2. Aims of the Practical
The practical skills learnt will be:
 connecting up of apparatus and making measurements of voltage;
 making measurements involving parallax errors and investigating other possible
causes of systematic error;
 programming in FORTRAN and use of the xvgr graph-plotting program.
3. The e/m Apparatus
Two separate sets of apparatus are available for this part of the practical. Both allow
the same measurement of the specific charge (e/m) for the electron, but achieve the
measurement in slightly different ways. This is the lab script for Set 2 of the apparatus,
made by Pasco. You should aim to take approximately one laboratory period (4 hours)
to perform the measurements in this section. Pay careful attention to your error
estimates.
In this experiment, you will examine the effects of electric and magnetic fields on a
high speed electron beam and, by noting the deflections produced by these fields,
calculate the specific charge of the electron by two different methods.
The PASCO Model SE-9638 e/m Apparatus, shown in Figure 1, provides a simple
method for measuring e/m, the charge to mass ratio of the electron. The method is
similar to that used by J.J. Thomson in 1897. A beam of electrons is accelerated
through a known potential, so the velocity of the electrons is known. A pair of
Helmholtz coils produces a uniform and measurable magnetic field at right angles to the
electron beam. This magnetic field deflects the electron beam in a circular path. By
measuring the accelerating potential V, the current to the Helholtz coils I and the radius
of the circular path of the electron beam r, e/m is easily calculated.
The e/m apparatus also has deflection plates that can be used to demonstrate the
effect of an electric field on the electron beam. This can be used as a confirmation of
the negative charge of the electron, and also to demonstrate how an oscilloscope works.
A unique feature of this e/m tube is that the socket rotates, allowing the electron
beam to be oriented at any angle (from 0–90 degrees) with respect to the magnetic field
from the Helmholtz coils. You can therefore rotate the tube and examine the vector
nature of the magnetic forces on moving charged particles. Other experiments are also
possible with the e/m tube. For example, you can use a small permanent magnet instead
of the Helmholtz coils to investigate the effect of a magnetic field on the electron beam.
Equipment

The e/m tube (see Figure 1) is filled with helium at a pressure of 10-2 mm Hg, and
contains an electron gun and deflection plates. The electron beam leaves a visible
trail in the tube, because some of the electrons collide with helium atoms, which are
excited and then radiate visible light.

The electron gun is shown in Figure 2. The heater heats the cathode, which emits
electrons. The electrons are accelerated by a potential applied between the cathode
and the anode. The grid is held positive with respect to the cathode and negative
with respect to the anode. It helps to focus the electron beam.
CAUTION: The voltage to the heater of the electron gun should NEVER
exceed 6.3 volts. Higher voltages will burn out the filament and destroy the e/m
tube.

The Helmholtz Coils are two coils of wire whose separation is equal to the coil
radius. This provides a highly uniform magnetic field. The Helmholtz coils of the
e/m apparatus have a radius and separation of 15 cm. Each coil has 130 turns. The
magnetic field B produced by the coils is proportional to the current through the
coils I. The constant of proportionality is 7.80  10-4 T A1.

The control panel of the elm apparatus is straightforward. It is illustrated in Figure
3 and all connections are labelled. The hook-ups and operation are explained in the
next section.

A cloth hood can be placed over the top of the apparatus so the experiment can be
performed in a lighted room.

A mirrored scale is attached to the back of the rear Helmholtz coil. It is illuminated
by lights that light automatically when the heater of the electron gun is powered. By
lining the electron beam up with its image in the mirrored scale, you can measure
the radius of the beam path without parallax error.
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Figure 1: The Pasco e/m apparatus and a close-up of the tube
Figure 2: The electron “gun”
Figure 3: The control panel of the e/m apparatus and required connections
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3.1 Measuring e/m
1.
If you will be working in a lighted room, place the hood over the e/m apparatus.
2.
Flip the toggle switch up to the elm MEASURE position.
3.
Turn the current adjust knob for the Helmholtz coils to the OFF position.
4.
Connect your power supplies and meters to the front panel of the e/m apparatus, as
shown in Figure 4.
5.
Adjust the power supplies to the following levels:
Heater:
6.3 VAC or VDC
Electrodes:
150 to 300 VDC
Helmholtz Coils:
6-9 VDC
6. Slowly turn the current adjust knob for the Helmholtz coils clockwise. Watch the
ammeter and take care that the current does not exceed 2 A.
7. Wait several minutes for the cathode to heat up. When it does, you will see the
electron beam emerge from the electron gun and it will be curved by the field from
the Helmholtz coils. Check that the electron beam is parallel to the Helmholtz
coils. If it is not, turn the tube until it is. Don't take it out of its socket. As you
rotate the tube, the socket will turn.
8.
Carefully read the current to the Helmholtz coils from your ammeter and the
accelerating voltage from your voltmeter. Record the values.
9.
Carefully measure the radius of the electron beam. Look through the tube at the
electron beam. To avoid parallax errors, move your head to align the electron beam
with the reflection of the beam that you can see on the mirrored scale. Measure the
radius of the beam as you see it on both sides of the scale. Then average the results.
10. Repeat this step about 5 times for different values of I and V.
3.2 Analysing the results
The magnetic force F acting on a charged particle of charge q moving with velocity v
in a magnetic field B is given by Eq. [1]. Since the electron beam in this experiment is
perpendicular to the magnetic field, the equation can be written in scalar form as:
F  Bev ,
[2]
where e is the charge of the electron. Since the electrons are moving in a circle, they
must be experiencing a centripetal force of magnitude mv2/r and so, combining these,
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e
v

.
m Br
[3]
Therefore, in order to determine e/m, it is only necessary to know the velocity of the
electrons, the magnetic field produced by the Helmholtz coils, and the radius of the
electron beam. The electrons are accelerated through the accelerating potential V,
gaining kinetic energy equal to their charge times the accelerating potential. Therefore
eV = 1/2 mv2. The velocity of the electrons is therefore:
v  (2eV / m)1 / 2 .
[4]
The magnetic field produced near the axis of a pair of Helmholtz coils is given by
the equation:
B
 0 NI
(5 / 4)3 / 2 a
[5]
A derivation for this formula can be found in most introductory texts on electricity
and magnetism. Alternatively, see the Experiment “Magnetic Field Gradients”, also in
the EM Laboratory. Equations [4] and [5] can be plugged into equation [3] to get a final
formula for e/m:
e
v
2V (5 / 4)3 a 2


m Br
( 0 NI ) 2 r 2
[6]
1.
From the readings you took in the previous section, work out values of e/m using
Eq. [6].
2.
A better way of analysing the data for systematic errors is to check the dependence
of r on I and V. First take a set of readings of r for different values of V, with I held
constant. Then take a similar set of readings with I varying and V held constant.
Plot a graph to verify the following equations:
r 2  k1V with k1 
2(5 / 4) 3 a 2 m

(  0 NI ) 2 e
and
k2
2V (5 / 4)3 a 2 m
r  2 with k2 

I
(0 N )2
e
2
In each case, calculate a value of e/m from the gradient of your graph.
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[7]
6.3 Two simple demonstrations
1 . Instead of using the Helmholtz coils to bend the electron beam, you can use a
permanent magnet to show the effect of a magnetic field on the electron beam.
Just provide the following power to the e/m apparatus: HEA'IER: 6.3 VAC
ELECTRON GUN ELECTRODES: 150–300 VDC When the electron beam
appears, use your permanent magnet to bend the beam. Draw an appropriate
diagram to show that this motion is in accordance with Eq. [1], i.e., with Fleming’s
Left Hand Rule.
2.
The socket for the e/m tube is designed so that the tube can be rotated 90 degrees.
The tube can therefore be oriented so it is at any angle, from 0–90, with respect to
the magnetic field from the Helmholtz coils. By setting up the equipment as for
measuring e/m, you can rotate the tube and study how the beam deflection is
affected. Write down an equation for the trajectory of the electrons when the tube is
rotated through and angle .
4. The Cyclotron — Principle of Operation
In the cyclotron, which was an important machine in the early days of nuclear and
particle physics, protons, alpha particles or other, heavier, charged particles move inside
an evacuated enclosure containing two hollow D-shaped cavities (see Figure 1), which
are conducting. A potential difference can be applied between the cavities and a
magnetic field is applied perpendicular to the plane of the page. Throughout the
subsequent sections, let us assume a system of co-ordinates as in Fig. 4, on which all
the relevant variables are also defined.
The principle of operation of the device is as follows:
(i) The particle starts off at rest (or moving slowly) at the centre. A voltage is applied
between the two D’s and the particle feels a force Eq, which causes it to accelerate
(either up or down, depending on the sign of its charge) towards one of the D’s.
Assuming that the particle starts off with negligible x-velocity, show that when it
reaches the entry to the D, it has a final y-velocity given by
vy final  vy2 initial  2qEd / m ,
[8]
where d is the distance in the electric field through which the particle “falls”.
(Assuming that the gap between the D’s is very small, it is valid to ignore the effect
of the magnetic field during this period.)
(ii) While the particle is inside the D’s, it does not feel any electrostatic force, because
inside any enclosed conducting surface (i.e., that of the D), E = 0. The force felt is
simply that due to the magnetic field, which causes the particle to move in a
circlular orbit. Show that the radius of this part of the trajectory is
r
mv
.
Bq
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[9]
(iii) After performing exactly half of a full revolution of its circular orbit, the particle
finds itself at the straight edge of the D again. By this time, the potential difference
between the two D’s has been reversed in sign. The particle is thus accelerated in
the opposite direction and gains energy once again as in Eq. [8]. It then passes
through the second D on a new semi-circular orbit, whose radius is determined by
the new velocity.
(iv) The whole process is repeated many times. Each time the particle passes through
the gap between the D’s, it is accelerated by the electric field, gaining in velocity
and energy. When it is inside the D, the force due to the magnetic field acts at right
angles to the direction of motion and so there is no change in the energy of the
particle.
As described above, the potential difference between the two D’s must oscillate, so
that every half revolution of the particle, the voltage has changed sign. Show that,
as long as relativistic effects are not significant (for v less than about 0.1c), the
required (angular) frequency of oscillation is

Bq
m
[10]
and that this is independent of the particle’s velocity. (This fact is important,
because if the oscillation frequency depended upon velocity, one would have to
change it continually as the particle accelerated, i.e., synchronise the field with the
stage in the cycle of the particle. Machines based on the latter technique do exist
and are called sychrotons for this reason. A major difference between the two types
of machine is that cyclotrons, in which the accelerating voltage frequency is
constant, can produce continuous beams of particles, whereas synchrotrons give
rise to pulsed beams, in which single bunches of particles are started at the same
time and taken through a complete acceleration cycle.)
(v) The particle finally hits a target placed at a given distance from the centre of the
spiral. Note that the final energy of the particles will be determined by the position
at which the target is placed.
5. Cyclotron Simulation
The operation of the cyclotron can be simulated by a simple computer program,
which you are asked to write. Both the D’s and the particle trajectory itself are
generated in the form of a series of (x,y) points, which can be used as an input to a
graph-plotting program such as xvgr. When plotted on the same diagram, a complete
picture is obtained. Figure 1 was generated by this general method.
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Figure 4 : Schematic diagram of cyclotron
Generating the D’s
The format of the cyclotron as a whole is based around a circle of radius a, which is
cut in half a slice. The two semi-circular halves are now separated by a distance d<<a.
The curved part of the upper D is thus a circular arc, centre (0,+d/2), generated by the
parametric equations x = a cos, y = a sin + d/2. The straight part of the D is a line
running from (ad/2ato (a, d/2Using a program in C, FORTRAN or any other
desired language, enerate one data-set consisting of, say, 50 (x,y) points for the circular
arc and a second data-set — this needs only two points ! — for the line. Do the same for
the lower D and display the results in xvgr.
The Particle Track
The particle trajectory can be simulated in four stages: (i) acceleration upwards, (ii)
circular motion in the upper D, (iii) acceleration downwards and (iv) circular motion in
the lower D. We shall make the simplifying assumptions that d << a, which means that
in the region between the D’s, we need to take account only of the electric field, and
that v << c, so that relativistic corrections are unnecessary. The exact path taken will
depend on a number of factors: the particle’s mass and charge, the accelerating voltage
and the magnetic field. These should all be built into the simulation, as parameters
which can be varied.
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(i) Assume that vx is zero when the particle exits the lower D at point x0 but that vy is
non-zero. When the particle gets to the upper D, it will have a new y-velocity given
by Eq. [2], whilst vx remains zero. The trajectory which you need to plot for this is
simply (x0, d/2) to (x0, +d/2).
(ii) The circular orbit starts at (x, d/2), which corresponds to angle on a circle with
centre (x0+r, d/2) and radius r, where r is given by Eq. [9]. It continues through to
angle zero. Simulate this with, say, 50 points around the curve, of the form
(x0+r[1+cos], d/2+rsin) . Explain the origin of these expressions for the
coordinates. Update the x-coordinate.
(iii) Since, ideally, no energy is gained or lost during the circular motion, the particle
ends this phase with the same velocity downwards that it had going up at the end of
(i). This velocity is increased during the transition across the gap into the lower D.
(iv) This step works in the same way as (ii).
Having simulated the four steps, the program should loop through them a number of
times, at each stage adding more points to the cumulative “path” file. As initial
conditions, place the particle with zero velocity at the point (0,0). Test your program
with alpha particles, an accelerating voltage of 1 kV and a magnetic field of 0.008 T.
Calculate the velocity and energy which the alpha particles attain after five revolutions.
(Suitable values to choose for a and d are 3 m and 20 cm respectively.)
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