Experimental Verification of Filter Characteristics Using

Report
Experimental Measurement of
the Charge to Mass Ratio of the
Electron
Stephen Luzader
Physics Department
Frostburg State University
Frostburg, MD
Outline of topics
• Purpose of the experiment
• Some theory about the motion of an
electron in a magnetic field
• Helmholtz coils
• The experimental apparatus
• How to analyze the data
2
Purpose
• The electron is a fundamental constituent of
all matter in the universe. It is important to
know the properties of the electron,
including its mass and electric charge.
• The purpose of this experiment is to
measure the ratio of the electron’s charge to
its mass, a quantity referred to as “e/m”.
3
Some theory on the motion of an
electron in a uniform magnetic field

B

v

If an electron with a velocity v
pointing right enters
 a uniform
magnetic field B pointing out of
the page, the resulting force

 
F  ev  B
will cause the electron to move in
a counterclockwise circle.
4
The radius R of the circle can be calculated using the
definition of centripetal force:
mv2
F
R
where m is the mass of the electron.
Since the velocity is perpendicular to the magnetic field,
the magnitude of the centripetal force is
F  evB
Combining these equations gives an expression for e/m:
e
v

m BR
5
If the electron achieved its speed v as a result of being
accelerated through a potential difference V, then its kinetic
energy is
mv 2
 eV
2
Solving this equation for v and substituting into the
equation for e/m gives
e
2V
 2 2
m B R
If we can devise an experiment to make electrons orbit in a
magnetic field, we can obtain a value for e/m.
6
A special arrangement of two coils called
Helmholtz coils is used to create a uniform
magnetic field over a fairly large region of
space
a
a
It can be shown that if two circular
coils are separated by a distance
equal to their radii, the magnetic
field will be uniform over a large
region between the coils. In the
diagram, a is the radius of the coils.
7
The magnitude of the magnetic field on the axis of a coil
at a distance x from the center is
B
0 a 2 NI
x
2
a

2 3/ 2
where I is the current in the coil and N is the number of
turns in the coil.
At the midpoint of the Helmholtz coils, which is a distance
x = a/2 from the center of each coil, the magnitude of the
total field from both coils is
80 NI
B
a 125
8
Summarize what we have deduced so far:
1. An electron whose velocity is perpendicular to a
uniform magnetic field will move in a circular orbit.
2. The value of e/m can be calculated if we can measure
the accelerating potential, the radius of the orbit, and
the magnetic field.
3. By using a special arrangement of coils called
Helmholtz coils, we can calculate the magnetic field
if we know the radius of the coils, the number of
turns in each coil, and the current through the coils.
9
The Experimental Apparatus
A glass bulb containing a small
amount of mercury vapor is placed
between a pair of Helmholtz coils.
An electron gun in the bulb shoots a
beam of electrons into the mercury
vapor.
Some of the electrons strike mercury
atoms, causing them to emit light.
This light allows us to see the orbit
of the electrons so we can measure
its radius.
10
The Experimental Procedure
1. Measure the accelerating potential.
2. Measure the radius of the electron orbit for at least
3 different currents in the Helmholtz coils.
3. Calculate the value of B for each value of I.
4. Calculate an average value of e/m (with standard
deviation) from the measured values of V, R, and B.
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The radius of the orbit is determined by observing the
circular path of the electrons against a scale. The radius
can be calculated from the measured values x1 and x2 of
the right and left sides of the circle:
+
x2
x1
For example, if x1 = 3.0 cm and x2 = -4.1 cm, we conclude
that R = 3.55 cm.
12
We also need the following information about the
Helmholtz coils, which is provided by the
manufacturer:
a = 15 cm
N = 130 turns
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For your analysis, you must carry out the following steps.
1. Prepare a table with all important data, including
accelerating potential, Helmholtz coil currents, positions
of right and left sides of electron orbits, number of turns,
and coil radius.
2. Calculate the radius of each orbit.
3. Calculate the magnetic field for each orbit.
4. Calculate a value of e/m for each orbit.
5. Calculate an average value for e/m, including
uncertainty.
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