Download Experimental Analysis

Eric Cotner
4/22/11
Phys 2DL: Prof. Branson
Lab: Wed 1:00
e/m Ratio Experiment
In 1897, JJ Thomson was the first to discover the charge to mass ratio of the electron by firing a
cathode ray tube through a uniform magnetic and electric field and measuring its deflection. In this
experiment, we use a uniform magnetic field from two parallel Helmholtz coils to curve a beam of
electrons accelerated through a known potential to certain
velocity.
Setup and Experimental Design
The apparatus consists of two Helmholtz coils with a
variable current on either side of a spherical glass tube filled
with helium gas (pictured to the right). Inside the tube is an
electron gun (also at right) that utilizes a heater to expel
electrons from the cathode through the grid towards the
anode, which accelerates electrons through a variable potential,
exiting the gun downwards. Once the electrons have been
accelerated to a sufficient speed, the magnetic field will begin
to exert its influence, and because it is oriented perpendicular
to the electron beam, will act on it with a force parallel to the plane of the Helmholtz coils, curving the
beam back to its origin on the other side of the electron gun.
While in transit around the beam tube, the electrons strike
helium atoms, exciting them and emitting a greenish-blue glow
that allows an observer to measure the radius of curvature. This
energy exchange is also a source of error due to the fact that
the electrons lose kinetic energy and velocity to the helium,
resulting in larger measured radii and a smaller derived value
for the e/m ratio.
Equations Used
To reach calculations of the e/m ratio, we must perform the following. Equating the magnetic
force to the centripetal force experienced by the beam, we can derive an expression for the e/m ratio.
For the force:
Rearranging the kinetic energy:
and for the kinetic energy:
and plugging into the force equation:
Solving the Biot-Savart equation for the field of the Helmholtz coils, we get:
for the axial field strength.
Assuming the distance from either coil is half the radius, the final field strength is
.
Substituting this into the e/m ratio equation, we get the useful expression
,
where V is the accelerating voltage, R is the coil radius, N is the number of coils, I is the coil current, and
r is radius of curvature of the electron beam. For this experiment, R was found to be .149 m by
averaging the coil diameter along several axes, and N is specified to be 130 by the lab manual.
Data
The following tables catalogue the data of the experiment:
Accelerating Coil Current Diameter of
Accelerating
Voltage (V)
(A)
Curvature (cm)
Voltage (V)
200
1.08
10.5
400
200
1.14
10.0
400
200
1.20
9.5
400
200
1.28
9.0
400
200
1.42
8.5
400
200
1.52
8.0
400
200
1.62
7.5
400
200
1.74
7.0
400
300
2.76
5.5
500
300
2.54
6.0
500
300
2.36
6.5
500
300
2.20
7.0
500
300
2.04
7.5
500
300
1.92
8.0
500
300
1.80
8.5
500
300
1.72
9.0
500
500
Coil Current
(A)
1.64
1.72
1.80
1.90
2.00
2.10
2.24
2.38
2.86
2.68
2.52
2.38
2.24
2.12
2.02
1.94
1.84
Diameter of
Curvature (cm)
11.0
10.5
10.0
9.5
9.0
8.5
8.0
7.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
Analysis
Based on the data gathered from this experiment and using the equation derived above,
, it is possible to derive the e/m ratio for each measurement. These figures illustrate
the derived e/m ratio for each individual measurement. Strangely, the first 4 readings are skewed largely
from the rest of the data, corresponding to an average e/m of 1.995x10 11 C/kg, a 13.4% deviation from
the currently accepted value of 1.758820x1011 C/kg. The mean value of the remaining data is
approximately 1.64083x1011 C/kg, a 6.7% deviation.
e
e
m
m
2.1 1011
2.1 1011
2.0 1011
2.0 1011
1.9 1011
1.9 1011
1.8 1011
1.8 1011
1.7 1011
1.7 1011
1.6 1011
1.6 1011
0
5
10
15
20
25
30
Test
1.5 1011
150
200
250
300
350
400
450
500
Voltage
If we rearrange the e/m equation so that it reads
, it is possible to use the
curve-fitting software of Mathematica to fit a value to e/m for each voltage series using this model by
plotting the radius of curvature versus the coil current.
Voltage
(V)
200
(green)
300
(yellow)
400
(magenta)
500
(blue)
e/m (C/kg)
1.766x1011
σe/m
(C/kg)
5.063x108
11
8
Deviation from
Accepted e/m
0.38%
Radius
0.10
0.08
1.656x10
5.153x10
5.83%
1.607x1011
6.940x108
8.63%
1.593x1011
7.328x108
9.43%
0.06
0.04
1.5
2.0
2.5
3.0
Current
These results are quite close (within 10% error) to the currently accepted e/m value of 1.75882x1011
C/kg, especially the 200 V data series, which is surprising considering the data that was thrown out was
from this series.
Conclusion and Error Analysis
Perhaps the largest source of error in this lab is due to the velocity of the electrons. The majority
of electrons do not have the full kinetic energy supposedly given to them by the accelerating potential
due to nonuniformity in the accelerating field. This is reflected by higher deviations from the accepted
Error Percentage
value at higher kinetic energies, where the
15
nonuniformities in the accelerating field become
more apparent. To the right is a diagram of the
percent error of the data as a function of voltage,
10
which is clearly increasing as voltage increases. In
the top left corner you can make out the 4 data
5
points in the 200 V series with the exceptionally high
percent error. The source of this error is a little
vague, as the values are higher than the expected
0
Voltage
150
200
250
300
350
400
450
500
value, meaning that either the accelerating voltage
or current coil was higher than the one reported on the voltmeter/ammeter, or the radius of curvature
was measured incorrectly.
In conclusion, the data from this experiment, which concludes the e/m ratio is about 1.64x1011
C/kg, supports both the theory of electrons as particles with a quantized charge and mass, and also
supports the accepted value of the e/m ratio for electrons, which is currently 1.758820x1011 C/kg, from
which there is a 6.7% deviation, well within acceptable limits for such a simple experiment.