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Figure 1.

[Click on image to enlarge it.]

To meet this objective we will use a vacuum tube capable of producing a visible beam of electrons as shown in Figure 1. (The beam is visible because it excites the low-pressure gas contained in the tube.) When immersed in a magnetic field perpendicular to the beam, the negatively charged electrons will be deflected according to the magnetic force, .

In this experiment, we will be able to determine the e/m ratio by measuring the electrons' potential energy and amount of deflection, and the strength of the magnetic field. Once we have determined e/m, we will use Millikan's value for the electron charge to calculate the electron's mass.

[Click on images to enlarge them.]
Figure 2.	Figure 3.	Figure 4.	Figure 5.
Vacuum tube.	Vacuum tube power supply.	Undeflected electron beam.	Digital multi-meter (DMM).

The Vacuum Tube
The vacuum tube is connected to a power supply and the electron beam is formed in the following way:

An electric current is applied to the tube's filament and electrons are released from the filament. (By increasing the current, more electrons are "burned off" and the beam becomes brighter.) The electrons are accelerated upward toward the anode plate when a potential difference, , is applied between it and the cathode plate. The same power supply that applies current to the filament also supplies the potential across the anode and cathode. In this experiment the voltage drop is measured by a digital multi-meter (DMM), which is connected across the anode and cathode inputs.

This potential difference imparts a change in the electrons' potential energy, , where is the charge of an electron. Due to the conservation of energy, this causes a change in the kinetic energy, . Since the electrons are at initially rest at the cathode where the potential is zero, the conservation of energy may be written as

Eq. 1

Here, is the anode potential and the equation's right-hand side is the familiar kinetic energy of a particle of mass, , and speed, . In this experiment the power supply may be used to vary the anode potential, thereby altering the speed of the electrons.

[Click on images to enlarge them.]
Figure 6.	Figure 7.	Figure 8.	Figure 9.
Helmholtz coils.	Current source.	An electron beam bent due to an external magnetic field.	A top-down view of the beam impacting the surface plate.

The Helmholtz Coils
Once the beam is visible in the vacuum tube, the experiment may proceed. The beam is deflected by applying a magnetic field, , perpendicular to the electron beam. Such a field is created by a set of Helmholtz coils. Helmholtz coils are comprised of two coaxial loops of wire, each with an identical number of turns. By definition, the coils are separated by a distance equal to their radii.

The coils are connected to a variable current source and an electric current is applied to the coils. It is important to note that the current travels in the same direction in each coil. The current loops create a magnetic field between the coils. This field is oriented perpendicular to the plane of the coils, along their common axis. In the center of the Helmholtz coils (where the vacuum tube is located) the magnetic field is given by the formula,

Eq. 2

where is the applied current, is the radius of one of the Helmholtz coils, and is constant known as the permeability of free space ().

Each electron in the beam, then, experiences a magnetic force, , where is the charge of the electron (). In this experiment the electron velocity is perpendicular to the magnetic field, so the magnitude of becomes

Eq. 3

If the strength of the magnetic field is large enough (i.e., enough current is passed through the coils) the electron beam will be bent into a circular path. The radius of the path may be determined by noting where the beam makes contact with the surface plate. Etched onto the plate are four concentric rings centered on the beam's exit hole. Each ring is separated by a distance of 0.50 cm. Note that the distance between the exit hole and the beam's impact point is twice that of the beam's radius of curvature.

Electrons moving in a circular path experience a centripetal force equal to the product of its mass, , and its centripetal acceleration:

Eq. 4

where is the radius of the electrons' circular path. Combining Equations 1, 3 and 4 we find

Eq. 5

Using a digital multi-meter it is possible to measure the voltage drop, , experienced by the electrons. The beam's deflection radius, , is measured visually by noting the location of the impact of the beam with the vacuum tube's surface plate. Finally, the magnetic field, , is determined from Equation 2 using the geometry of the Helmholtz coils and the current applied to the wires. (The current source is equipped with a panel meter which displays the electrical current, as shown at the right.)

Figure 10.

[Click on image to enlarge it.]

Exercise 1: Constant kinetic energy, variable magnetic field.

1. The manufacturer of the Helmholtz coils engraves the number of turns of each coil, , onto the base of the apparatus. Record this value in the Data Sheet.

   Number of turns, N [0.052 Mb]

2. Measure the diameters of one of the coils and then calculate its radius, . Record the value of the radii. In the video below, one of the coils has been removed for clarity purposes only.

   The coil diameter is measured. [0:33, 6.16 Mb]

3. Connect the power supply to the vacuum tube with wire leads being sure to match the colors of the banana jack outlets with those of the vacuum tube apparatus. In the video below, the black wire is connected to ground, red to the anode, blue to the filament. (The yellow wire is connected to the grid, which helps focus the beam, but was not used in this experiment.)

   Also connect a digital multi-meter (DMM) across the vacuum tube's anode and ground leads. The DMM will be used in step 5 to accurately measure the anode voltage.

   The wire leads are connected to the vacuum tube. [0:32, 6.58 Mb]
   The digital multi-meter leads are connected across the anode voltage. [0:16, 2.98 Mb]

4. Turn on the power to the vacuum tube power supply and adjust the filament current so that the beam is sufficiently bright. In our example, the filament current is set to 0.6 amps and is held constant throughout this exercise.

   The filament current is set and the beam appears. [0:22, 4.19 Mb]

5. Adjust the anode voltage, , on the power supply to impart a kinetic energy to the electrons. In this exercise the anode voltage is set to an arbitrary value of 38.4 volts. Since we want the electrons to have a fixed kinetic energy, the anode voltage is not adjusted again for the duration of the exercise. You should record the anode voltage in the Data Sheet below.

   The anode voltage is set. [0:11, 2.22 Mb]
   The DMM displays the anode voltage. [0.047 Mb]

6. Connect the Helmholtz coil to the variable current source with wire leads. The polarity of the leads is not an issue. What will happen if the leads are inverted?

   
   Leads from the current source are connected to Helmholtz coils. [0:15, 2.82 Mb]

7. In this step we will use the Helmholtz coils to create the magnetic field that is used to deflect the electron beam. To do so, power up the variable current source and apply enough current to the coils so that the resulting magnetic field is strong enough to bend the beam into a circular path. The magnetic field should be large enough to cause the beam to impact the surface plate. Record the value of the current, . Also calculate the magnetic field strength, , and record this value in the Data Sheet.

   During this step, you must measure the beam's radius of curvature, , by carefully noting where the beam impacts the surface plate. Use the concentric circles imprinted on the plate as reference points to help you make the measurements. Recall that the circles are separated by a distance of 0.50 cm, and that the distance between the exit hole and the beam's impact point is twice that of the beam's radius of curvature.

   The beam is bent and the first deflection is measured. [0:41, 7.81 Mb]
   All deflection measurements are played here. [1:32, 17.3 Mb]

8. Repeat step 7, varying the strength of the magnetic field by varying the current applied to the Helmholtz coils. Take great care in measuring the electron deflection and its radius of curvature. A small error in your deflection measurement, say ± 0.05 cm, can cause a 10% error in your final calculation of the electron's mass. This measurement is especially sensitive when is small.

   All deflection measurements are played here. [1:32, 17.3 Mb]

   The second deflection measurement may be made. [0:08, 1.51 Mb]
   The third deflection measurement may be made. [0:08, 1.57 Mb]
   The fourth deflection measurement may be made. [0:09, 1.81 Mb]
   The fifth deflection measurement may be made. [0:09, 1.87 Mb]
   The final deflection measurement may be made. [0:10, 2.01 Mb]

   All deflection measurements are played here. [1:32, 17.3 Mb]

9. Use the values entered into the Data Sheet below to determine the ratio . You can accomplish this in two ways:
   
   1. Determine the ratio using measurements from each trial and then find the average ratio. (Students with little previous laboratory experience my need to use this method.)
   2. Or you may graph the appropriate data along the x- and y-axes and then analyze the resulting curve. The graph may be drawn by hand or created by a spread sheet application like MS Excel, for example. (For additional help, see our tutorials on Plotting experimental data, Creating a graph, and Using MS Excel.)

10. In 1913, Robert Millikan determined from his Nobel Prize-winning oil-drop experiments that the charge of an electron has a value of . Use your experimental results and Millikan's value to determine the electron mass, .

11. Calculate the percent error between your value for the electron's mass and the accepted value of .

12. For safe keeping, you may e-mail the data directly to yourself or to your TA by entering the data into the form below and then clicking The Send Button.

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Data Sheet

Exercise 1 Data Sheet
Your name:
Your e-mail address:
Number of coils,
Helmholtz coil diameter		Choose units ... cm m mm in
Helmholtz coil radius,		Choose units ... cm m mm m/2
Filament current		Choose units ... A V Ohms C
Anode voltage,		Choose units ... A V Ohms C
Current, I (A)	Magnetic Field, B (T)	Radius of Curvature, r (m)
Quantity plotted on x-axis	Choose quantity 1 / R^2 B^2 V e/m
Quantity plotted on y-axis	Choose quantity 1 / R^2 B^2 V e/m
Slope of your graph		Choose units ... T^2 / m^2 (m T)^-2) C / (2 V kg) C / (V kg)
Experimental value of		Choose units ... q / kg C / m C / kg C / g
Accepted value of
Experimental value of		Choose units ... kg g m cm
Accepted value of
Percent error		%
Please enter any comments you have about this experiment here:
For your records, you may send the above data to your own mail box, provided you entered the correct email address above. For your work to be graded, you must fill-in the printed form and hand it in at the end of class.