THE EXPERIMENT

Purpose

To measure the charge-to-mass ratio of the electron.

Equipment

1. Special vacuum tube designed specifically for electron e/m measurements;
2. power supplies and meters to operate the e/m tube;
3. Helmholtz coils and a separate power supply;
4. compass and inclinometer to find the direction or the earth's magnetic field.

Preparation and lab work

The e/m tube to be used in the lab contains an electron gun to produce a beam of electrons, a very low pressure mercury vapor atmosphere to render the beam visible, and some markers to permit measurements on the trajectory of the electrons making up the beam. The simple electron gun is pictured schematically in Fig. 2. The cathode is shown as a resistance inside

Figure 2

the cylindrical anode can, The filament is heated by the current flowing through it from the filament supply. The high temperature causes electrons to be emitted ("thermionic emission") by a special coating on the filament surface. These electrons are accelerated to the anode by the voltage V applied between the filament and the anode can. Most of these electrons strike the inner surface of the anode with a kinetic energy eV. Some, however, are allowed to pass out through a narrow- slot in the anode, forming a flat ribbon beam of electrons which we use for the experiment. The tube itself has a small amount of mercury vapor added to its otherwise good vacuum. A few of the beam electrons strike mercury atoms and ionize them; when the mercury re-combines, a characteristic visible radiation is given off, and we can thus, "see" where the electron beam is.

Figure 3

Another feature of this special e/m tube is that it contains a rod upon which a number of short posts are mounted. These posts serve as targets for the electron beam to strike. They and the electron gun are arranged as in Fig. 3. The posts are in a straight line which includes the electron gun, and all are parallel to the gun's cylindrical axis. The electron beam, however, is emitted in a direction perpendicular to this Post-gun line.

Now, if a uniform magnetic field is established throughout the tube and directed normally out of the paper, the electron beam will be bent into a circle, as shown. The radius of the circular electron path is related to the magnetic field and the electron energy in the following simple way. The electron experiences a magnetic force perpendicular to its velocity; this is just balanced by the centrifugal force due to its resulting curved motion, and so,

,

or

Now,

;

thus

,

and finally,

.

The distances between the posts and the filament are supplied with the tube. To the far side of the posts from the filament, they are:

   Post Number             Distance
        1                  0.065 meter
        2                  0.078   "
        3                  0.090   "
        4                  0.103   "
        5                  0.115   "

Each of these is, of course, 2r when the beam is just grazing the back side of the post.

In order to perform this experiment we must know r, V, and B. We have just been given r, and V is a simply applied and measured potential from a variable power supply. How do we establish and measure B?

We employ a uniquely useful arrangement of conductors called Helmholz coils. These, like shoes, are not useful except in pairs. They are only simple, thin, hoop-like coils which assume very special qualities when they are positioned on a common axis and spaced a distance equal, to one-half their common diameter. If one connects them in series -- and in the same polarity -- and passes current through them, the result is a magnetic field near the center or this array which is exceedingly homogeneous over a fairly large volume.

For those who might be motivated to try it, the calculation of the field on the axis (through the use of the Biot-Savart law) is not hard. You can even prove the virtues of the half-diameter spacing by computing the central field for some general spacing x, and finding x for which the second derivative of B_z with respect to z vanishes. We won't require any of that here, however. Without deriving the result, we will simply give the formula

for B:

Here, N is the number of turns per coil, a is the mean coil radius in meters, and = 4 x 10^-7 weber/ampere-meter. B is now in webers/m² .

For the coils you will use here,

N = 72

a = 50.33 m.

The complete apparatus for this experiment, in addition to the vacuum tube and Helmholtz coils, consists of:

1. Two power supplies which provide current for the filament and the coils,
2. A variable voltage power supply with current limiting for safety. This provides the accelerating potential for the electron gun.
3. Two meters -- an ammeter and a voltmeter. You will notice that these are large and have mirrors on their scales to assist you in obtaining accurate readings. They measure the coil current and accelerator potential, respectively. Filament current and coil current are also shown by crude meters on the dual power supply.

These components are to be connected according to the diagram of Fig. 4. The terminals of the tube elements and coils are on the coil frame. Observe the polarities indicated in Fig.4 and follow them. You may get the power leads to the coils in the wrong polarity, but this is easy to recognize and remedy once you begin the experiment.

Figure 4

Basic lab measurements

To begin your experiment, connect the tube anode to the positive terminal of the anode power supply, turn the supply on and set to 22 volts. Make sure that the filament supply controls are at zero, i.e. fully counterclockwise, before turning on the supply. Place the viewing hood over the tube, and fit your face snugly against it in order to exclude as much room light as possible. This is necessary because the beam is not bright enough to see in daylight.

Slowly advance the current control on the power supply until you see the beam emerge from the electron gun. It will probably be necessary to use 4 amperes or more; do not exceed 4.5 amperes in any event, since this seriously shortens filament life. Since the anode power supply is "current limited," you will need to select a filament current which gives a visible beam but not too much anode current (anode voltage drops if current excessive).

Notice that the beam has a curvature opposite to that we wish to give it. This is due to the terrestrial magnetic field. Turn on the coil current supply (set initially for zero current) and advance it very slowly until you see the beam straighten out. (If the beam only bends further in the "wrong" direction, reverse your connections to the Helmholtz coils to reverse their field direction.) When it looks perfectly straight to you, record the coil current required to do this at the actual accelerating potential as indicated on your voltmeter.

You can now increase the coil current until the beam curves around and touches one of the posts. Your setting should be such that the outside edge of the beam touches the rear of the post. There are various energy losses from the beam electrons, including scattering from mercury atoms, and those electrons which have suffered least loss or no loss have .he largest radii of curvature. (You may improve the beam visibility and sharpness by a slight adjustment of filament current, but do not exceed 4.5 amperes.) When you have measured the coil current required to touch each post, change the anode power supply to 44 volts and repeat the whole procedure. You may find it impossible to reach some of the posts, or to get a sharply defined beam reaching others; try to get as many measurements as you can.

In computing e/m, remember that the current required to cancel the earth's field must be subtracted from each recorded coil current, since only this difference has produced the field which is bending the electrons. Make a separate computation of e/m for each combination of V and r. Compute the average, and also use the scatter in these separate data to estimate the probable error assignable to this final value of e/m. The accepted value of e/m is 1.76 x 10¹¹ coulomb/kg.