Electron Beam Deflection by a Magnetic Field
and Measurement of q/m for the Electron
Introduction:
In this experiment you will study the effects of a magnetic field on a beam of electrons.
A reasonably well-known field is produced by using Helmholtz coils. Since the accelerating
voltage for the electrons is known, their velocity can be calculated and their charge-to-mass ratio
determined from a measurement of the radius of their circular path in the region of the magnetic
field.
A. The Magnetic Field
The simplest way to produce a uniform magnetic field is to use two identical coils of
wire, as shown below. When the coils are separated by a distance R equal to the radius of each
coil, an especially uniform field is produced near the center of the region between the coils.
When coils are arranged in this way, they are called Helmholtz coils.
When a current I flows in a
wire, a magnetic field with strength
proportional to I is produced. (If I is
doubled, B will double also.) Aside
from this simple feature, the
magnetic field produced by a bent
wire or a coil is generally
complicated and hard to calculate. A
fairly difficult calculation is required
to determine the field at an arbitrary
point between the Helmholtz coils,
but for points in the center near the
axis the result is a simple expression.
If there are n turns of wire in each
coil and a current of I amperes flows
in each coil, then the field in the
central region between the coils has a
magnitude
B (Tesla) = 2
0 I n R2
2(R2 + R2 /4)3/2
=
0 I n
R(1.25 )3/2
-7
= 8.99 x 10 n I
R
(1)
where, for each coil, R is the radius (in meters). For this special geometry, both dB/dx and
d2B/dx2 are zero in the middle of the coils (where x is the distance along the axis from the
middle point), which insures that B is quite uniform along x near the center.
B. Circular Motion of Electrons in a Uniform Magnetic Field
According to the Lorentz force law, the magnetic force on a moving charged particle is
perpendicular to both the particle's velocity v and the magnetic field B . The magnitude of the
4-1
force is given by F = qv  B = qvB, where v is the component of the particle's velocity
perpendicular to B . The magnitude of v for electrons in a beam can be calculated from a
measurement of the accelerating voltage Va in the electron gun. In a uniform magnetic field, the
force is perpendicular to the particle velocity and the magnetic field, and is constant in
magnitude. If v is perpendicular to the field B so v = v, the orbit of the particle is a circle of
radius r, and the force is the centripetal force. Applying Newton's second law,
2
F = ma = mv = q v B.
r
(2)
Since q = e and eVa = (1/2) mv2, we have
e = v2 = e2 B2 r2/2V
a
m 2Va m2
which reduces to
e 2Va
m = B2r2
(3)
m
Equipment:
Voltmeter, ammeter, electron beam tube and Helmholtz coils, 6.3V A.C. filament power
supply to heat beam tube filament and produce free electrons, adjustable 0-300V D.C. power
supply for electron beam acceleration in beam tube, adjustable 0-2A D.C. power supply for
current to Helmholtz coils to produce magnetic field. Computer running Graphical Analysis. The
required circuit will be connected in advance.
Procedure:
Students will work in groups of three or four.
measurements.
Each student should make some of the
IMPORTANT: Be sure to record n and R for the Helmholtz coils you use.
A. Setup
Turn on the power supplies. Allow the filament of the electron tube to warm up for a
minute or so. Set the current I in the Helmholtz coils to some value around 1.0A. When the
filament is hot, slowly turn up the accelerating voltage Va and observe the path of the electrons.
(DO NOT EXCEED 300V.) If the path of the beam is not perpendicular to the magnetic field,
grasp the lowest part of the base of the tube and rotate it, being VERY CAREFUL not to pull or
break any wires. Vary the current in the coils (not to exceed 2.0A) and observe the bending of
the beam. Reverse the current to the coils and observe the effect. (Then go back to the original
polarity.) Observe the path of the electrons when the tube is rotated slightly away from the
perpendicular position and record your observations.
B. Charge-to-Mass Ratio of the Electron: Orbit Radius as a Function of B for fixed Va
4-2
Position the electron beam perpendicular to the magnetic field. Now set Va at a value
of ≈175V and record Va. Keep Va at this same value for all of part B. The magnetic field B is
varied by changing the current I, and the mean radius r of the electron beam's circular path is
determined
by measuring the positions of the inner and outer edges of the beam, x1, x2, x3, x4. (See figure on
next page.) When you measure these values, try to line up the image of the beam in the mirror
with the actual beam, so as to be sure you are viewing the scale perpendicularly to avoid reading
errors due to parallax. The inner radius r1 and outer radius r2 of the beam are then given by:
r1 
x2  x 3
2
The mean radius of the beam is
and
r=
r1 + r2
2
B. 1. The current should be set at ≈ 1A
(or the lowest value to give a full
4-3
r2 
x1  x 4
2
or
r
.
x1  x 2  x 3  x 4 .
4
circular beam). Each of the students
from the group at each apparatus must
make independent measurements Va, I,
x1, x2, x3, x4 and the data from all these
measurements should be tabulated using
Graphical Anaylsis.
B. 2. Change I and repeat part B. 1. for
two additional I values (not to exceed
2.0A). (Each student in the group should
take readings at least once.) When this
is complete, each group should have a
table with at least 3 sets of values for
each of 3 different values of I, all for the
same value of Va.
C. Charge-to-Mass Ratio of the Electron: Orbit Radius as a Function of Va for fixed B
Set I back to one of the values used in part B. Keep I at this same value for all of part C. Make
and record sets of independent measurements of Va, I, and x1, x2, x3, x4 as in part B.1) for a total
of three Va values different from the value used in part B. (Do not exceed Va = 300V.) When
this is complete, each group should have at least 3 sets of measurements for each of 4 different
values of Va (including the value from part B) in a data table, all for the same value of I.
Tabulation is to be performed using graphical analysis.
Analysis:
Use your data from parts B and C to calculate r (the average value of r) and ( r ) (the
standard deviation of r ) at each set of values of I and Va. Using r and ( r ) in meters, find r 2,
 r 2, 1/ r and (1/ r ). Remember, (1/ r ) = | r / r 2| with  r = ( r ).
B.
Find B and B for each value of I. Using
Graphical Analysis, plot 1/ r (in m-1) versus B (in
Tesla). Include vertical error bars (± (1/ r )) and
horizontal error bars B (see NOTE in Section C
below). By rearranging Eq. (3), we can see that the
graph should give a straight line with slope
S =  1/r / B = e / 2mVa
(4)
Eq. 4 may be solved to give e/m = 2 VaS2. Treating
Va as a constant, standard techniques of error analysis
may be used to show that (e/m) = 2(e/m)(S/S). Use
your results from Graphical Analysis for S and S to
find values for e/m and (e/m).
4-4
C.
Use your results for data taken with the same I to make a plot of Va (in V) versus r 2 (in
2
m ) using Graphical Analysis. Include vertical error bars Va and horizontal error bars  r 2.
The plot should include four points. By rearranging Eq. (3), we can see that the graph should
give a straight line with slope S' = Va / r2 = (B2 / 2)(e/m) . Are your results consistent with Eq.
(3)? Eq. 4 may be solved to give e/m = (2 / B2)S'. Treating B as a constant, standard techniques
of error analysis may be used to show that (e/m) = (e/m)(S'/S'). Use your results from
Graphical Analysis for S' and S' to find values for e/m and (e/m). Compare your results for e/m
from parts B. and C. with each other and with the accepted value of 1.759 x 1011 C/kg.
NOTE: Refer to the Extech multimeter chart to determine the accuracy of your measurements:
This information can be used to determine B in Part B. and Va in Part C.
Questions for the report:
1)
In your own words, describe the path of the electrons which you observed during the
setup of the experiment when the tube was intermediate between the perpendicular and parallel
positions. Explain why the path is of this form.
2)
Were your various results for e/m in reasonable agreement with each other and with the
accepted value of e/m? If they differ from the accepted value, can you think of any possible
reasons?
3)
What do you see as the relative advantages and disadvantages of the different methods
which you used to obtain values of e/m? What method do you think should be preferred? Why?
4)
Can you think of an assumption made in obtaining Eq. (3) which is not exactly true?
Would you expect this to affect your measured value of e/m significantly?
5)
The earth has a magnetic field of its own; the magnitude of this field is typically on the
order of a few times 10-5 T. What effect might the presence of this field have on your
determinations of e/m?
NOTE: Before you leave the lab, all measurements must be complete and each lab partner must
have complete data tables plus two graphs initialed by your TA.
4-5
Data Tables Electron Beam Deflection
Helmholtz Coil:
n=
R=
B. Charge - to - Mass Ratio of the Electron: Radius as Function of B for fixed Va .
Va
(volts)
Va
(volts
)
I
(Amps)
I
(Amps)
I
(Amps)
x1
(cm)
I
(amps)
B
(Tesla)
r1
(cm)
x2
(cm)
r2
(cm)
B
(Tesla)
4-6
x3
(cm)
( r)
(cm)
r
(cm)
1/ r
(m-1)
x4
(cm)
r
(m)
(1/ r)
(m-1)
( r)
(m)
C. Charge - to - Mass Ratio of the Electron: Orbit Radius as a Function of Va for fixed B .
Va
(volts)
Va
(volt)
Va
(volts)
I
(Amps)
Va
(volts)
x1
(cm)
I
(amps)
Va
(volts)
r1
(cm)
x2
(cm)
r2
(cm)
r2
(m2)
4-7
x3
(cm)
( r)
(cm)
r
(cm)
(r2)
(m2)
x4
(cm)
r
(m)
( r)
(m)