Bainbridge Experiment to Determine the Charge-to-Mass
Ratio of an Electron
Physics 205 - Modern Physics
Introduction
The Bainbridge apparatus will be used to measure the charge-to-mass ratio e/m of
the electron. The apparatus consists of an evacuated glass chamber, containing an
electron source and electrode plates for accelerating the electrons. A small amount of
mercury vapor in the chamber is ionized by the passage of the electrons, allowing the
electron beam to be observed visually as a faint blue-white track. The apparatus is
situated at the center of a Helmholtz pair, an electromagnet consisting of two coaxial
current-carrying coils (see Figs. 1 and 2). The electrons are released by thermionic
emission from a hot filament contained in a small cylindrical enclosure at the base of the
glass chamber. A voltage applied to a grid inside the cylinder allows the electrons to be
accelerated to a desired final velocity. The electron beam then emanates from a thin slit
in the cylindrical enclosure. In the presence of the magnetic field, the electron trajectories
are circular arcs. The radius of the circular arcs can be accurately measured by adjusting
the accelerating voltage or magnetic field so the electron beam just passes through one of
a set of crossbars mounted on a wire within the chamber. The distances from each
crossbar to the slit has been accurately measured. With knowledge of the accelerating
potential, the magnetic field, and the radius of the electron trajectory, it is possible to
determine the charge-to-mass ratio e/m to a few percent accuracy with this apparatus.
The distances from each crossbar to the slit are shown in Table 1. The magnetic
field B is controlled by adjusting the current in the Helmholtz coils. The expression for B
in terms of the current I and geometric parameters of the coil arrangement is
B=
8  N I
125 a
(1)
where B is the magnetic field in Teslas (1 T = 104 Gauss), o = 4 x 10-7 H/m is the
magnetic susceptibility (MKS units), N = 72 is the number of turns on each coil, I is the
coil current in A, and a = 0.33 m is the mean radius of the coils.
The electron trajectory can be calculated simply from the Lorentz force law and
classical dynamics. Assume the electrons emerge from the slit with a kinetic energy K
given by
1
K = eVo = 2 m v2
(2)
where e is the electron charge, Vo is the accelerating voltage, m is the electron mass, and
v is speed. The subsequent trajectory of the electrons in a uniform magnetic field is a
circular orbit. The radius can be found by equating centripetal force and the Lorentz
force.
m v2
R =evB
(3)
By combining equations (2) and (3), one can solve for the charge-to-mass ratio e/m in
terms of experimentally measurable quantities.
Table 1: Slit-Crossbar distances
Crossbar number
Distance (cm)
1
6.48
2
7.75
3
9.02
4
10.30
5
11.54
Cros sb ars
Electron b eam
B (o ut of pag e)
Slit
Accelerating p lates
To filament
curren t su pply
To accelerating voltage s upply
Figure 1.
N=72 turns
Helmholtz pair
R
B
I
I
Figure 2.
Procedure
1. Turn on the Labvolt power supply. Set the filament current to 2.7 A. Do not exceed
3.0 A or the filament will overheat and break! Wait a few minutes for the filament to
heat up; you should see a white glow through the slit in the small cylinder inside the glass
chamber.
Caution: The apparatus contains exposed metal terminals at fairly high voltages.
Exercise appropriate caution.
2. Set accelerating voltage Vo to 60V. Turn off the room lights and use a flashlight. You
should see a blue-white glow of the electron beam trajectory inside the chamber,
emanating from a slit on the right side of the cylinder.
3. Turn the coil current I up until the beam bends into a circular arc, passing through the
highest crossbar on the vertical wire mounted in the chamber. Adjust I so that the outer
(upper) edge of the beam just touches the crossbar. (In all measurements, adjust so that
the outer edge touches the crossbar-the outer edge of the beam contains the most
energetic electrons, which have come from the surface of the filament and have negligle
transverse velocities). The distances from crossbars to slits are given in Table 1. Using
equation (1), find the B field corresponding to the value of I you used. Find e/m (see
Question 1). Compare with the accepted value e/m = 1.759x1011 Coulombs/Kg.
4. Set coil current I = 3.0 A. Adjust the accelerating voltage Vo until the beam hits each
crossbar in succession. Make a table of Vo vs. R, where R is the radius of the circular
electron orbit.
5. Set the accelerating voltage Vo to a fixed value, and vary coil current I so that the
beam hits each crossbar in succession. Make a table of I vs. R. You will use this table to
make a plot (you decide what to plot on each axis). From the plot you will deduce e/m.
Think carefully what value you should set Vo at to obtain the best results for e/m.
Hint: Remember the earth's magnetic field, which is ~0.5 Gauss = 5  10-5 T.
Questions
1. Using Eqns (2) and (3), derive an expression for e/m in terms of experimentally
measurable quantities.
2. Plot Vo vs. R2, using the data from Procedure 4. Determine e/m from the slope of your
plot. Calculate the error in your e/m value, compared to the accepted value of
e/m=1.759x1011 C/Kg.
3. Plot the data from Procedure 5. Decide what to plot on each axis so that you can most
easily determine e/m from your plot. Compare your value of e/m with the accepted one.
Discuss why you chose the value you chose for the accelerating potential Vo.
4. You have measured e/m three times (Procedure 3, 4, 5). Which measurement was the
most accurate? Why do you think this is so?