Electron Charge to Mass Ratio
Jacob Benner, Lauren Hoffman
Department of Physics and Astronomy, Augustana College, Rock Island IL 61201
Abstract: When a particle which charge is subjected to a magnetic field it feels a force. We use this
knowledge in this experiment. By using the Helmholtz apparatus we can measure the force on an electron
beam by a magnetic field of known strength. Since the beam creates a circle due to its centripetal
acceleration we can figure out the ratio of electron charge over mass. We measure the radius and use
the mathematical tools to find the relationship between it and the e/m ratio. The accepted value for an
electron is 1.75E11 c/kg. When graphing the results we found a e/m ratio of 4.27E11 but when
measurements were taken for each individual voltage results were much closer to the theoretical. The
reason for this is the large error present.
Introduction: In this experiment we sought to find the ratio of charge to mass of an electron. We know that the
electron has a charge, because of this we know the particle will be affected by a magnetic field. This is exactly what
the Helmholtz device does; it creates a magnetic field that bends the beam into a circle. See figure 12. The field is
created by a solenoid of wires. We know the strength of the magnetic field because we control the potential through
1
the solenoid V. We also know the velocity of the beam v by using the equation eV= mv2 . Since we know B and v
2
we know the force of the magnetic field. We also know since the beam is moving in circle it has a centripetal
acceleration 1. When we add all these elements together we get an equation for the e/m ratio. The equation is
dependent on the radius the beam makes in the B field and so that is what we measured for different potentials1. It is
important to keep in mind that the accepted value for the charge of an electron is 1.6E-19 coulombs and the accepted
value for its mass is 9.11E-31 kg. Therefore the accepted value for the e/m ratio is 1.75E11 coulombs/kg.
Figure 1: A diagram of the coils and electron gun of the Helmholtz apparatus2.
Procedure: When setting up the Helmholtz apparatus the heater was set at 6.3 V, the Helmholtz coil voltage was set
at 9 V, the Helmholtz coil current was set as 1 A. The Electrode Voltage then we kept between 150 and 300 V 1.
Once the beam was on for a little while we started to see a blue-green circular beam as seen in figure 2. Once the
apparatus is ready we can start measuring our radius of the electron beam. To do this we start with the voltage of the
Electrode at 300 V. We then measure the radius of the beam by looking at where the edges of the electron beam are
on the ruler. However, this measured radius is incorrect due to parallax. The parallax occurs because there is a
distance between electron beam and the ruler. This causes the radius to appear bigger than it actually is due to the
fact that things farther away look smaller. To compensate for this we use the fact that the ruler is also a mirror. We
know from optical physics that the imaginary image produced on the mirror is at the same distance from the mirror.
This smaller radius is then averaged with the real image to get the actual radius. We originally measured the values
by eye and then switched to the camera. The camera helped to keep the image at the center of the ruler. We repeated
this process with 8 other electrode voltages. We then calculated the Strength of the B field and the e/m ratio at every
voltage. We then plotted 2V on the Y-axis and B2r2 on the X-axis to get e/m as the slope. We use these values
𝑒
2𝑉
𝑚
𝐵2 𝑟 2
because of the equation ( ) =
when the denominator is moved to the other side it follows the standard
y=mx+b. where 2V is y, B2r2 is x and b is 0.
Imaginary Image
Ruler and
mirror
Real Image
Electron Gun
Figure 2: the electron beam being acted on by the B field. This photograph was taken when V was at 210 V.
Results: When the calculations were done on the B field we found it to be 7.813E-4 T with an uncertainty of 1.11E5 T. After the radii were measured and each was calculated for the e/m ratio we arrived at figure 3. After the graph
was made we see a linear fitting with a slope or e/m ratio of 4.271E11 coulomb/kg with an uncertainty of 3.528E10
coulomb/kg. This can be seen in figure 4.
V
N
302
280
275
250
230
225
210
200
176
130
130
130
130
130
130
130
130
130
Mu
I
1.26E-06
1.26E-06
1.26E-06
1.26E-06
1.26E-06
1.26E-06
1.26E-06
1.26E-06
1.26E-06
a
r
1
1
1
1
1
1
1
1
1
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.01
0.0015
0.0535
0.053
0.0515
0.049
0.048
0.047
0.0465
0.046
0.045
B
e/m
error
actaul
0.000781372 1.72816E+11 15.79233548 1.75631E+11
0.000781372 1.63264E+11 15.06021463
0.000781372 1.69826E+11 16.12173097
0.000781372 1.70542E+11 17.01579757
0.000781372 1.63505E+11 16.65346539
0.000781372 1.66829E+11 17.35359243
0.000781372 1.59074E+11 16.72479729
0.000781372 1.5481E+11 16.45344731
0.000781372 1.42355E+11 15.46591029
error
1
0 0.00E+00
0.3 1.10503E-05
Figure 3: all variables used to calculate the electron to mass ratio of the electron. Note: the diffrerent values for different voltages.
Also V is in volts, N is unitless, 𝜇 is in T*m/A, I is in Amps, a is in meters, r is in meters, B is in T, and e/m is in coulomb/kg.
2V
Linear Fit of Sheet1 2V
Graph of slope as e/m
600
2V (V)
500
Equation
y = a + b*x
Weight
Instrumental
Residual Sum of
Squares
400
623.15621
Pearson's r
0.97695
Adj. R-Square
0.94792
Value
2V
Intercept
2V
Slope
Standard Error
-146.82976
51.93229
4.27135E11
3.52773E10
300
1.20E-009
1.40E-009
(B*R)^2 (T*m)
1.60E-009
1.80E-009
Figure 4: Graph with slope as e/m. slope value is 4.27E11 c/kg with an uncertainty of 3.527E10 c/kg.
Discussion: After analyzing the data I noticed quite clearly that the value for e/m of the graph is off from the actual
value and differs by around 2.5E11 c/kg. This is a significant difference. Also the difference is not covered by the
uncertainty of the slope of 3.527E10 c/kg. When we look at the separate individual voltages however, we see much
closer numbers. The error again does not cover the difference in values but the difference this time is between 1 and
3E10 c/kg. In a sense the experiment worked out in that we got values for the ratio that was very close to the
accepted value. Overall our values were lower than the accepted value which leaves me to believe there is a
systematic error in one or more variables that are leading to lower numbers. It is hard to say which measurement
involved this error. It was apparent though that there is a fairly large error in the measurement of the radii of the
electron beam. It is very hard to look at the beam and see the marking it is corresponding too. This could cause a
fairly significant error in our results. The rest of the error came from the other variables.
References:
1 “Electron Charge to Mass Ratio” lab manual. Dr. Vogel
2 School of Physics at Georgia Tech. Advanced lab Physics. Web. 11 Dec. 2013.
<http://advancedlab.physics.gatech.edu/labs/waveparticle/wave-particle-3.html>.