Department of Physics, Middlebury College |
1992-93 |

Modern Physics Laboratory |

**Discussion**

In this experiment you will observe the behavior of electrons in a magnetic field and determine a value for the electron charge-to-mass ratio e/m. The apparatus consists of a large vacuum tube supported at the center of a pair of Helmholtz coils, as seen in the photograph of Fig. 1. The vacuum tube contains an electron gun which produces a collimated beam of electrons that is deflected by a magnetic field. An electron gun has two main parts: a filament that produces electrons through thermionic emission, and an anode that is placed at high positive potential so as to accelerate thermal electrons from the filament to the main region of the vacuum tube, as shown in Fig. 2. The magnetic field produced by the Helmholtz coils deflects the electrons into circular trajectories and these paths are made visible through collisions by the electrons with a trace amount of mercury vapor present in the vacuum tube. A complete description of the e/m vacuum tube may be found in Ref. 1.For two coaxial coils of radius a and separation distance d, the magnitude of the magnetic field B at the center of the arrangement is given by

^{2}

B

=

µ

_{o}N I a^{2}(1)

^{_______________}[ a

^{2}+ (^{d}/_{2})^{2}]^{3/2}where N is the number of turns in each coil and I is the current through each coil. In the Helmholtz configuration one chooses d = a so that the magnetic field in the central region is very homogeneous. The magnitude of the magnetic field in the central region is then given by

B

=

8µ

_{o}N I(2)

^{______________}(125)

^{1/2}aAfter being accelerated by a potential difference V in the electron gun, electrons execute circular motion in the Helmholtz magnetic field region. From the radius of curvature R of this motion one can compute the electron charge-to-mass ratio from

e/m

=

^{125}/_{32}a

^{2}V

(3)

^{________ }

^{_______}µ

_{o}^{2}N^{2}I

^{2}R^{2}

**Procedure**

(1) Familiarize yourself with the experimental apparatus by making a schematic diagram of the electrical connections. Do not turn on voltages until you understand the role of each of the power supplies.(2) Observe the electron beam as the Helmholtz field and anode potential are independently varied. What is the effect of reversing the Helmholtz field direction? Verify directly that the electrons in the beam have a negative charge. In your laboratory notebook, outline a method for determining the sign of the electron charge.

(3) With fixed anode voltages V of 20, 40, 80, and 120 V, determine the Helmholtz coil current I necessary to deflect the electron beam to each of the five cross bar pins. The location of these five pins is given in Fig. 2. These 20 data points will be referred to as "Data at Fixed Anode Potential".

(4) With fixed Helmholtz coil currents I of 2.5, 3.5, 4.5, and 5.5 A, determine the anode potential V necessary to deflect the electron beam to each of the five cross bar pins. These 20 data points will be referred to as "Data at Fixed Helmholtz Field".

(5) Each Helmholtz coil has N = 72 turns of wire and a center radius a = 33 cm. Record these values in your laboratory notebook.

**Analysis**

Using Eq. (3) it would be a simple matter to calculate 40 values of e/m from the 40 data points you have taken. Unfortunately, the earth's magnetic field is present in this experiment and failure to take proper account of it will lead to an inaccurate value for e/m.The e/m apparatus has been positioned so that the Helmholtz coils are coaxial with the earth's magnetic field. In Middlebury, the earth's magnetic field direction is toward the ground, making an angle of approximately 40

^{o}with the vertical. In your experimental configuration, this means that the earth's magnetic field partially cancels the Helmholtz field B. Because some of the Helmholtz field is simply offsetting the earth's field, your experimental values of the Helmholtz current I are slightly larger than they would be in the absence of the earth's field.In the absence of the earth's magnetic field Eq. (3) predicts that at fixed anode voltage V the Helmholtz current I is directly proportional to 1/R. Therefore, if you plot I versus 1/R you expect a straight line passing through the origin as in the dashed line of Fig. 3. However, if you plot your experimental "Data at Fixed Anode Voltage" for a single, fixed value of V, you will obtain a straight line much more like the solid line of Fig. 3. The solid line has the same slope as the dashed line determined from Eq. (3), but it does not pass through the origin. This failure to pass through the origin can be attributed directly to the presence of the earth's magnetic field in this experiment.

Note that when the earth's field is cancelled exactly, the electron trajectories are straight and their radii of curvature are infinite so that 1/R = 0. Thus, I

_{E}, the I-axis intercept of the solid line in Fig. 3, represents the Helmholtz coil current necessary to cancel the earth's magnetic field. You must thereforesubtractI_{E}from all measured data before computing e/m with Eq. (3).(1) Refer to Ref. 2 and derive Eq. (1), (2), and (3). Supply the necessary fundamental constants so that you have a simple numerical formula for reducing your experimental data.

(2) Using axes of I and 1/R, plot all 20 measurements of the "Data at Fixed Anode Voltage" set on a large (3 ft x 3ft) sheet of graph paper. For each of the four fixed voltages, carefully draw a straight line through your data points. The common intercept of these four lines on the I axis is I

_{E}. Use this value of I_{E}to correct all Helmholtz current measurements in this experiment.(3) Substitute your value of I

_{E}into Eq. (2) to compute the magnetic field B_{E}that the Helmholtz coils cancel. Compare your value of B_{E}to the magnitude of the earth's magnetic field at Middlebury.(4) After making the correction for I

_{E}discussed above, use your entire set of 40 measurements to obtain 40 values of e/m. Histogram these 40 values of e/m in an appropriate figure. Determine the mean of these e/m values and their standard deviation.(5) Make reasonable estimates of the systematic and random uncertainties in your measurements of the experimental quantities a, d, V, I, and R. Use your laboratory data to make a final determination of e/m along with an appropriate uncertainty estimate. Use the fundamental constants of Appendix A to compute the currently accepted value for e/m along with the currently accepted uncertainty. Compare this accepted value for e/m to your experimental value.

**References**

1. *Instructions for Use of No. 0623B e/m Apparatus*, manual
(Sargeant-Welch Scientific Company, Skokie, IL).

2. D. Halliday and R. Resnick, *Fundamentals of Physics*, 2nd
ed. (John Wiley & Sons, New York, 1981), pp. 566-567.