Download Hoffmann_em - Helios

Electron Charge to Mass Ratio
Mark Hoffmann, Peter Draznik, and Christian Montibrand
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
Abstract: We determined the electron charge to mass ratio to be 1.98 + .24 x 1011 C/kg. This
was accomplished using an e/m tube and Helmholtz coils. An electron gun is used to shoot
electrons into a magnetic field, thus causing the electrons to travel in a circle due to the
Lorentz force. The radius of the circle is measured as the voltage to the electron gun is
increased. Using these parameters, the charge to mass ratio of an electron can be determined.
More information on the experimental setup can be seen in the Experimental Setup section.
Our determined value fell within error to the accepted value of 1.7588 x 10 11 C/kg.
I. Introduction
The electron charge to mass ratio was first calculated by J. J. Thompson in 1897. His experiment provided
information proving that the electron was, in fact, a particle with measureable mass. In this experiment an
electron gun produces electrons in a direction perpendicular to an external magnetic field. This causes the
electrons to travel in a circle due to the Lorentz force. Helmholtz coils surrounding the e/m tube that the
electrons are confined to produce the magnetic field. The electrons collide with helium gas within the e/m tube,
which produces the green color they appear. The radius of the circle is measured as the voltage of the electron
gun is increased. Using these parameters, the charge to mass ratio of an electron can be determined. More
information on the experimental setup can be seen in the Experimental Setup section. We determined the
electron charge to mass ratio to be 1.98 + .24 x 1011 C/kg, falling within error to the accepted value of 1.7588 x
1011 C/kg.
II. Experimental Setup
The experimental setup consisted of Helmholtz coils, an e/m tube with electron gun, a ruler, a camera, and an
external power source. The Helmholtz coils surrounded the tube and were a distance of 15 cm apart. Putting a
current through these coils produces a very close to uniform magnetic field straight through the center of the
coils. This external magnetic field is important, as it is what causes the electrons to accelerate in a circle due to
the Lorentz force. The electron gun is within the e/m tube and shoots electrons in the direction perpendicular to
the external magnetic field. When the electrons collide with the internal helium gas molecules, it produces a
green hue. The external power source can control the voltage and current input to the electron gun. Increasing
the voltage and keeping the current constant increases the radius of the electron beam. The setup can be seen in
Figure 1.
A camera was placed on the same plane as the center of the circle the electrons produced as well as the ruler
behind the glass bulb. Pictures were taken at each voltage increment to calculate the radius of the circle. The
parallax of our perspective had to be taken into account to correct for the real radius of the circle. This method
is further discussed in the Discussion section. An example of the electron beam measurements can be seen in
Figure 2.
(a) Schematic of entire setup except for camera
Figure 1: Experimental Setup [2]
Figure 2: Sample Measurement of Electron Beam Radius
(b) e/m tube, Helmholtz coils, and scale
III. Results
The force felt by an electron is equal to q*v x B where q is the charge of an electron, v is the velocity and B is
the magnetic field. Because the external magnetic field is perpendicular to the electron velocity, F = evB,
where e is the charge of an electron. Because the electrons are moving in a circle, they have acceleration equal
𝑒
𝑣
to v2/r, where r is the radius of the circle [2]. Setting evB = mv2/r, it can simplify to = . To find v in terms
𝑚
𝐵𝑟
of a measureable quantity, we know the kinetic energy of an electron eV = ½ mv2. Combining this equation
3
𝑒
2 2
𝑁𝜇𝑜 𝐼 4 2
( ) ,
𝑎
5
with the previous one results in the relationship 2𝑉 = ( ) 𝐵 𝑟 . Since 𝐵 =
where N is the number
𝑚
of turns on the Helmholtz coil, 𝜇𝑜 is the magnetic permittivity of free space, I is the constant current, and a is
𝑒
the radius of the Helmholtz coil, B is constant. This allows us to graph the 2𝑉 = ( ) 𝐵2 𝑟 2 function with the
𝑚
increasing voltage and radius to determine the slope representing the electron mass ratio. The number of turns
in our Helmholtz coil is 130 and the radius is 15 cm. The magnetic permittivity of free space is known as
1.26x10-6 T*m/A. The constant current run through the electron gun is 1.27 + 0.03 A.
Figure 3: Electron Charge to Mass Determination graphing 2V (V) versus B2r2 (T2m2)
IV. Discussion
The electron charge to mass ratio to be 1.98 + .24 x 1011 C/kg, falling within error to the accepted value of
1.7588 x 1011 C/kg. The value we measured was obtained by a weighted least squares fit algorithm with
uncertainty in both the x and y coordinates [4]. Because we had a relatively large uncertainty in the x direction
for each data point, it was important we used this linear fit versus one that only takes into account the
uncertainty in the y direction [3]. A least squares fit in both dimensions forcing a y-intercept of 0 was also
tried, however that did not fall within error to the expected value. The reason the fit without forcing 0 may have
worked is because there could be some systematic error in our voltage. One possible explanation is that the
voltage displayed could be assuming infinite parallel plates to accelerate the electrons even though the plates are
finite. Another possible explanation is that we could have been systematically measuring the radius of the
electron beam too short or too long. To correct the setup for parallax, we measured the distance from the
camera to the electron beam as well as the distance to the ruler. We then used similar triangles to determine a
correction factor of .780 + .001 for the measured radii. To obtain a better value for the election charge to mass
ratio, we could have taken more data points by only increasing the voltage by 5 volts instead of 15 volts. Also,
we could have taken multiple measurements of the radius for each measurement. Since the green electron beam
had a width, it was difficult to measure the exact radius.
References
[1] J. Taylor, An Introduction to Error Analysis. Sausalito: University Science Books, 1997, pp. 181-207.
[2] E/M APPARATUS Instruction Manual. Roseville: Pasco Scientific, 1987, pp. 1-10
[3] C. Reed, Linear least squares fits with errors in both coordinates. Halifax: American Journal of Physics,
1989, pp. 642-646
[4] Bertrand, G. 2001, “Linear Least Squares Regression.” Web. http://web.mst.edu/~gbert/wLS/wls.html,
January 2014.