Charge Particle in Magnetic Field
8. Charge Particle in a Magnetic Field*
Objective: The electric charge (-e) and the mass (m) of every electron is exactly the same. They
are characteristic properties of this particular kind of particle: the electron. So it is too with other
elementary particles, such as protons. The objective here is to measure the charge to mass
ration e/m of the electron.
The learning objectives are the following:
1. To be able to explain how an electron gun works and to calculate electron speed from the
accelerating voltage.
2. To know what Helmholtz coils are and how they work.
3. To understand how, under certain circumstances, a charged particle moving in a magnetic
field will follow a circular orbit at constant speed.
4. To put these ideas together (“integrate” them) and develop a method for measuring the
charge to mass ratio e/m of an electron or any other type of charged particle.
(All of this material is presented in your textbook. You use several interesting pieces of
apparatus and get the flavor of an actual modern physics experiment.)
Reading assignment: Review uniform circular motion, work-energy theorem and conservation of
energy. Study sections of magnetic field, currents create fields, magnetic force, the right-hand
rules, paths of charged particles.
Knight, Jones and Field (162): 24.2 The Magnetic Field, 24.3 Electric Currents Also Create
Magnetic Fields, 24.4 Calculating the Magnetic Field Due to a Current, 24.5 Magnetic Fields
Exert Forces on Moving Charges
Serway and Vuille (212): 19.6 Motion of a Charged Particle in a Magnetic Field, 19.9 Magnetic
Fields of Current Loops and Solenoids
Serway and Jewett (252): 29.2 Motion of a Charged Particle in a Uniform Magnetic Field , 29.3
Applications Involving Charged Particles Moving in a Magnetic Field
Right-hand rule for the direction of
C
A B :
C
B
A
I. Right-hand rule for a vector product: when you write
C  A B
A
1. Put the vectors tail to tail.
2. A is the “first vector” because it is left of the “  ”.
3. Extend fingers of right hand along the first vector.
4. Curl them into second vector ( B in this case).
5. Thumb points along the direction of the cross product
(which is
C
in this example).
______________________________________________________________________________
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*© William A Schwalm 2012
Charge Particle in Magnetic Field
One use of the right-hand rule above is the magnetic force
F  qv  B .
Pre-laboratory problem assignment: Complete these before coming to lab. The instructor will
check them as you enter. Finally they will form part of your report.
1. Suppose that in an electron gun, electrons are emitted from a filament f and are accelerated
by an electric field to an opening o, as shown. The charge on an electron is (– e) where e =
-19
1.6 x 10 C.
(a) If the voltage (same as electric potential) at f is – 140 Volts, what is the potential energy
of an electron at f ?
(b) The electrons come off the filament with very low velocity, hence with very little kinetic
energy. So what is the total energy of an electron at f ?
(c) After the electrons exit the gun at o the voltage is zero. Thus, what is the potential
energy of an electron at o ?
(d) Given that the mass of an electron is m = 1.9 x 10
going when it exits the gun?
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kg, how fast must an electron be
2. When an electron moves subject only to a magnetic force, its kinetic energy does not change.
State the work-energy theorem (see text) and then explain why the magnetic force does not
change the electron’s kinetic energy, and hence the speed remains constant.
f
3. A beam of electrons makes a circular orbit in the plane perpendicular to an applied magnetic
field. If the electron speed is 15 mega meters per second, how strong must the field be if the
radius of the circle is 12 cm? Show your work, starting with uniform circular motion.
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o
Charge Particle in Magnetic Field
4. Define an electron volt (unit of energy).
Work group assignment: A new fundamental particle (particle X) is expected to have a certain
charge and mass. This particle will be produced in a high-energy collision in a super collider.
Your work group is attached to a team working on the charge-to-mass determination part of the
project. To test a method of determining the charge to mass ratio of the X particle, your particular
group will apply it first to the electron.
In-class response: You will work some or all of these prediction or measurement questions out
as a group and the reporter of your group will record them on the white board. The instructor will
ask one or more of your group members to comment and explain.
1. Look at your apparatus. The large coils of wire are called Helmholtz coils. Inside the
spherical glass tube there is a partial vacuum with a small amount of mercury vapor. At the
lower center of the tube is the electron gun.
(a) Figure out which direction electrons will be moving just as they exit the gun. Then, which
way (toward or away from you) will the magnetic field from the Helmholtz coils need to
point to make the electrons go in a circular orbit? Be ready to explain.
(b) Based on what you have in part a) which way must the electric current circulate in the
Helmholtz coils? Be ready to explain.
2. During the experiment, the lights will be turned down. You will observe the electron beam as
a purple green glow. This is caused when a small fraction of the electrons strike some of the
very low density of mercury atoms in the residual gas of the partial vacuum. The beam
electron knocks one of the atomic electrons into a higher energy orbit momentarily. When it
comes back down to its ground state, the colored light is emitted. Why does the beam still
seem to go in a circle, if the light only comes when an electron collides with an atom and
hence is scattered off in some random direction?
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Charge Particle in Magnetic Field
3. Suppose in a given magnetic field strength Bo an electron (charge –e, mass m) with kinetic
energy of 400 electron volts circulates counterclockwise in a circle of radius 3.5 meters. Now
suppose the X particle has charge +2e and unknown mass M. Then in the same magnetic
field strength a 200 electron-volt X particle travels in a circle of radius 10.2 meters, what is
M? Explain your reasoning.
Exploring the equipment:
Helmholtz coils
High-voltage
Power supply
Low-voltage power
supply
Knob to adjust the
current in the
Helmholtz coils
e/m apparatus
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Charge Particle in Magnetic Field
As usual, the company provides a description of how the equipment works, but you have to figure
out how to use it make the required measurements.
In the figure above you see two meters, the spherical glass vacuum tube where the electrons will
move and a pair of Helmholtz coils that can produce a magnetic field that is more or less
constant. The box on the right high-voltage power supply.
Filament heater 6.3 VAC
from power supply
Current through the coils must not exceed two amps. 
e/m (up) position
Focus beam
Focus beam
Front panel on the e/m apparatus
Electron gun accelerating voltage
V0 from
high-voltage power supply
Above is the front panel on the experiment apparatus.
Helmholtz coils provide the B field. On the panel the leftmost pair of terminals are for current
supplied to the Helmholtz coils. They would be connected to the DC power distribution system
via the wall receptacle.
Next to the coil terminals on the front panel is the knob that controls the Helmholtz coil current.
Full counter clockwise is for minimum current, which is a good place to set this knob initially.
One of the two meters shown in the figure on the previous page is an ammeter that would monitor
the current in the coils. It must be placed in the circuit in such a way that the current passes
through it. Carefully choose an appropriate scale.
Current through the coils must not exceed two amps. 
Electron gun. The next thing you see on the panel is a knob that will control focus of the electron
beam as it emerges from the electron gun. This actually controls a voltage drop between the first
and second anode inside the gun. During the experiment, whenever you readjust the
accelerating voltage, you may need to readjust the focus.
Next there is a toggle switch that should be in the e/m position.
The next pair of terminals connects to the high voltage power supply for the electron gun. This
voltage can range up to +400volts. Although you can get a minor shock if you aren’t careful, it is
like a static electricity and not very dangerous .
The next two terminals, labeled “deflection plates” are not used. They would be for a different
type of experiment.
Next you have two high-voltage terminals where you connect the high voltage power supply that
provides the accelerating voltage for the electron gun.
Finally you have the connections for 6.V to 6.3V AC supplied to the gun filament. This should
never vary during the experiment.
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Charge Particle in Magnetic Field

Focus beam
Current to coils

Filament heater 6.3 VAC
from power supply
Below is a schematic diagram indicating the way the apparatus is wired and what the controls do.
 
Electron gun
accelerating voltage
Vo from power
supply
e/m (up)
position


Voltmeter
1000 V DC
scale
Ammeter
10 amp
DC scale
DC power for magnet coils
from wall receptacle
Below is the front panel of the high-voltage power supply actually used.
This switch toggles both meters
between the two different voltage
sources. Each time you press it
switches.
6.V
Voltage
Current (mA)
50 volts
adjust
500 volts
adjust
DC
-50
V
+500 V
0
8-6
On
This dial
must stay at
6 volts (heater
filament)
Off
AC
Charge Particle in Magnetic Field
Connections The filament heater connection—rightmost in the upper figure—is made to the AC
power supply sector in the lower figure. It is essential that the dial on that sector remain at 6.V.
(Noted above.) The actual accelerator voltage input terminals—next to the rightmost in the upper
figure—connect to the terminals marked +500 V on the high-voltage sector in the lower figure.
The 0 terminal in the center is (-) and the one at the right is (+).
It’s pretty clear that you want the 500 volts-adjust knob to start at its minimum (Full counter
clockwise) before you turn on the power.
As you see, there is a switch that toggles the meters from the -50V to the +500 V supply. You
are only using the +500, so the meters should be switched there.
Voltage to the electron gun should not actually exceed +300 V. 
Make all the connections necessary, and when you believe the equipment is ready for operation,
call on your instructor for a pre-flight checkout.
Power up:
Remember that even with the room lights low you will probably need to cover the apparatus with
the black shroud and peak under the shroud in order to see the beam.
With all dials at full counter clockwise (except the filament dial, taped) turn on the main power at
the high-voltage power supply (HV). If you wait a few seconds, you should see the filament
glowing in the experiment tube.
Toggle the meter on the HV power supply to the +500V setting, and then turn the 500 volt adjust
up (not too far, not to 500V) until you see a beam coming out of the electron gun.
Toggle the meter over to the -50 Volt scale and turn up the focus voltage until the beam is
somewhat focused.
Turn up the current to the coils in order to bend the electron beam.
You may need to work back and forth amongst the previous three dials in order to get a
reasonably well focused circular orbit.
Your teacher may ask you what each of these dials does, so be ready to explain.
How the coils make a magnetic field: In your text you find (Ampere’s law and the Biot-Savart law)
how an electric current can make a magnetic field. In the case of the particular helmholtz coils
you have the magnetic field strength is
Bz  C I where the constant C (of course not capacitance) is worked out from Biot-Savart to be
3
N o
N o
 .71552...
a
a , where N is the number of turns of wire in each coil (N =
henry
o  4  107
meter is the
130, says the specifications), a the coil radius, and as always
4
C  
5
2
magnetic constant (permeability of free space).
One more in-class question: Look straight ahead, through the glass sphere where the electron
beam should be forming a circle. Behind the circular electron orbit there is a mirror with a scale
to measure length. Question: How can the mirror be used when you have to measure the radius
of the circular orbit? That is to say what is the purpose of the mirror?
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Charge Particle in Magnetic Field
Explore the operation of your apparatus to see how you can adjust the electron speed, magnetic
field and so on while still keeping a well-defiled circular orbit.
v at this
instant.
r
You have to come up with a formula relating the orbit
radius r to the magnetic field strength and the speed.
Then you can get a formula for r in terms of the current
I in the coils and the accelerating voltage V in the
electron gun. Write your results on the white board.
Then give your result here.
Measurement Plan: Through group discussion, work out how you can make measurements to
determine the charge to mass ratio? How will your crew be utilized? It is best if you can make a
graph that would have e/m as the slope of a straight line, but it’s up to you. Explain your method.
What will you measure? What data do you need? What will you plot? How will you analyze the
data, and finally, how will you estimate the error in your method? Outline a plan on the white
board and record it in your report.
Summary: During the last few minutes of class, your instructor will ask you questions relating the
work you’re doing to the objectives. Make note of this discussion in your report.
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