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Physics Department, University of Windsor
64-311 Laboratory
Experiment 1
Determination of e/m for an Electron (B)
Introduction
The ratio of charge to mass of the electron was first determined by J.J.
Thomson in 1897. For many years afterwards, only the ratio was known; the
charge and the mass being so small they eluded measurement. This
remained the case until Millikan measured e by using charged oil drops
(1911). In Thomson's classical experiment a stream of electrons was
deflected initially by a magnetic field; an electric field was then applied to
bring the deflected beam back to its original position. This current
experiment uses a different apparatus which was later developed by
Bainbridge (1938). Here, electrons of a well defined energy are directed into
a circular path by a uniform magnetic field. A measurement of the energy
together with the radius of curvature and the magnitude of the magnetic
J J Thomson
field yields e/m. This experiment will also serve as a useful introduction to electron beams.
Description
The e/m vacuum tube is a spherical bulb placed in the plane midway between a Helmholtz coil
pair (two parallel coils of radius a, separated by a distance equal to a) supported in a rigid frame. Each
coil has 130 turns and a mean radius a = 15 cm. Inside the e/m tube (see Figure 1) is a simple electron
gun consisting of an indirectly heated cathode (What is that?) surrounded by a coaxial grid, together
with a cylindrical anode. When the filament is hot, the emitted electrons are accelerated towards the
anode and those emerging through its aperture form an electron beam. The beam is made visible by the
presence of helium vapour in the tube; the electron collisions giving rise to green fluorescence. With a
suitable adjustment of the applied magnetic field B and accelerating voltage V, the electron beam
emerging from the aperture can be made to travel in a circular / helical path.
The magnetic field B at the centre of the Helmholtz coils is given by:
⎛ 4⎞
B=⎜ ⎟
⎝5⎠
3/ 2
μ 0 NI
a
=
32πNI
5 5a
x10 −7 T
(1)
where N is the number of turns in one coil. This equation, together with: FB = evB = mv 2 / r and
eV = mv 2 / 2 , will enable you to determine e/m from the experimental variables.
Method
Make electrical connections to the tube and coil as shown in Figure 2. Initially leave the filament
wires disconnected from the 6.3V AC supply, and turn on the 600V DC supply – first setting the
voltage to 200V. Although the power supply provided is capable of delivering 600V; ensure you do not
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exceed 300V – else the equipment will be damaged. Turn the Helmholtz coil ‘current adjust’ dial on
the apparatus (NOT power supply) to fully clockwise. Ensure the current and voltage dials on the
Helmholtz coil power supply are set to zero before turning it on. Now turn on the Helmholtz coil power
supply and turn the voltage to ~7V DC – take care that the current does not exceed 2 A. You can now
control the coil current using the ‘current adjust’ dial on the apparatus.
Figure 1 A schematic view of the apparatus (above),
together with details of the ‘triode’ electron gun
configuration.
Figure 2 Wiring diagram for the electron gun and Helmholtz coils.
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Now connect the filament current leads to the 6.3 V AC supply; wait a minute and observe what
happens in the tube. At this point you should see an electron beam either moving in a spiral or striking
the side of the tube. Slowly (and carefully) rotate the tube in its cradle (i.e. around a vertical axis).
What happens? Explain what you observe. Rotate the tube in its cradle so that the beam describes a
circle. Adjust the Helmholtz coil current and then the accelerating voltage in the 150V -300V DC
range. Are the trends in the observed electron beam trajectory as you would expect?
Adjust the ‘focus’ dial the electron beam is both bright and sharp, then leave it fixed throughout
the experiment.
First, adjust the Helmholtz coil current so that you can get a circular trajectory for a wide
accelerating voltage range (i.e. 150 – 300V DC). Keeping the current fixed, measure the diameter of
the circular trajectory as a function of accelerating voltage (in 10V intervals). This requires some care
as (a) you need to measure the diameter, not a chord, and (b) you must move your head to align the
electron beam with its reflection in the mirrored ruler to minimize parallax errors. It is advisable to
measure the ‘outside’ diameter of the circular ring, rather than the ‘inside’ diameter – why is that?
Second, choose an accelerating voltage so that you can change the circular diameter for a wide
range of Helmholtz coils currents (≤ 2 A). Take a set of measurements for the radius, r, as a function of
coil current, I. If, by altering the accelerating voltage, you can access a range of diameters not covered
by your present value, then repeat the experiment accordingly.
Both sets of data can be processed to give values of e/m.
Treating the second set of data first: plot the magnetic field, B, versus 1/r, where r is the radius of the
electron trajectory. The points should fall on a straight line given by:
⎡ 2V ⎤
B=⎢
⎥
⎣ (e / m) ⎦
1/ 2
⎛1⎞
⎜ ⎟ + B Local
⎝r⎠
(2)
which you should derive. The last term is the component of BLocal parallel to B. Calculate the slope and
the error in the slope for each plot by using the least squares method of best fit. Using propagation of
errors, now find your best value for (e / m ) ± Δ(e / m) and the offset BLocal.
Assuming BLocal to be negligible, which it may not be (!), the equation can be easily manipulated
to give:
B
B
B
V=
32μ 02 N 2 I 2 (e / m)
125a
2
r2
(3)
Using the first series of data, plot V verses r2 and gain determine the gradient using a least squares
fitting procedure and hence find (e / m ) ± Δ(e / m) .
How do your values compare to the accepted value of 1.75882 x 1011 Ckg-1. Which method gives
the most accurate result? Why? Would it have made a difference if you had incorporated your value of
BLocal (±ΔBLocal) into (3)? [Find out!]
B
B
Questions
Discuss any potential systematic errors.
One obvious source of error lies in assuming that the magnetic field calculated at the center of
the Helmholtz coil pair is uniform over the path of the electron beam. While this is probably not quite
true, the magnetic field can be shown to be quite uniform in a fairly large region around the center.
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References
Ainslie D.S. Am. J. Phys. 26, 496 (1958)
Bainbridge K.T. American Physics Teacher 6, 35 (1938)
Cacak R.K. and Craig J.R. Rev. Sci. Instrum. 40, 1468 (1969)
Christy R.W. and Davis W.P. Jr. Am. J. Phys. 28, 815 (1960)
Ficken G.W. Jr. Am. J. Phys. 35, 968 (1967)
lona M., Westdal H.C. and Williamson P.R. Am. J. Phys. 35, 157 (1967)
Appendix A: Internal Circuit Diagram
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