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Determination of e/m for the Electron

Introduction

In this experiment you will measure e/m, the ratio of the charge of an electron to the mass of an electron. The currently accepted value for e/m is 1.758820 × 1011 C/kg. When an electron enters a region in which there is a uniform magnetic field, B, perpendicular to the velocity, v, of the electron (Caution: The capital letter V will be used below to represent voltage. Don't confuse v with V!), the electron experiences a force, F, with a magnitude given by the following equation.
( 1 )
F = evB 
The force is perpendicular to both v and B and its direction can be found by using the right-hand rule. The force will cause the electron to move in a circular orbit with radius r (uniform circular motion). Equating this force to the mass times the centripetal acceleration, we have the equation below.
( 2 )
evB = m
v2
r
 
Solving Eq. 2
evB = m
v2
r
 
 for e/m, we get the following equation.
( 3 )
e
m
=
v
Br
 
If the radius of the cyclotron orbit is measured, we can calculate e/m from Eq. 3
e
m
=
v
Br
 
. All we need is the velocity of the electron. In our experiment, the electron is accelerated by a set of plates with a potential difference, V, between them. The velocity of the electron can therefore be derived from conservation of energy.
( 4 )
1
2
mv2 = eVacc 
Solving Eq. 4
1
2
mv2 = eVacc 
 for the velocity, we get the following equation.
( 5 )
v =
2eVacc
m
 
Finally, by substitution of the velocity from Eq. 5
v =
2eVacc
m
 
 into Eq. 3
e
m
=
v
Br
 
 and solving for e/m, we obtain Eq. 6
e
m
 =
2Vacc
B2r2
 
 relating e/m to the potential difference, the magnetic field, and the radius of the electron's circular orbit.
( 6 )
e
m
 =
2Vacc
B2r2
 

Apparatus

Figure 1

Figure 1

The set up of the equipment used in this experiment is shown in Fig. 1. The major items on the e/m-unit are the three-element electron tube and the Helmholtz coils. The electron tube, Helmholtz coils, and the power supplies for the tube filament and for the accelerating voltage are mounted in a single base called the e/m unit. Controls and connectors on the front panel of this unit are (from left to right): The current to the Helmholtz coils is supplied by an external DC power supply. A multimeter is used to measure the current to the Helmholtz coils. The electron tube and the Helmholtz coils are described in detail below.

The Electron Tube

An "electron gun" is mounted within the electron tube with its centerline coincident with the vertical axis of the tube. The electron gun has three elements:
Figure 2

Figure 2: The Electron Gun (cutaway view)

The electron beam is projected vertically through a small hole in the center of the disk. The disk is mounted horizontally on the upper end of the electron gun. Four circles, with centers coincident with the hole and of radii 0.50, 1.0, 1.5, and 2.0 cm, are marked on the upper face of the disk. The bulb and disk are coated with a material that fluoresces when struck by electrons. The tube contains a trace of an inert gas that aids in focusing the electron beam as well as causing the beam to make a visible trace.

The Helmholtz Coil

The magnetic field is provided by a pair of identical circular coils arranged so that the distance between the coils is equal to the radius of the coils. Such an arrangement, called the Helmholtz Coils, provides a highly uniform magnetic field in the region at and near the center of the coil pair.
Figure 3

Figure 3: The Helmoholtz Coils

The magnetic field produced by the Helmholtz coils is proportional to the electric current, I, in the coils.
( 7 )
B = jI 
At the exact center of the Helmholtz coils, the value of j is given theoretically by:
( 8 )
j = μ0
4
5
3
N
R
 
where: The values of N and R for the Helmholtz coils used in this lab are: N = 196 turns, and R = 0.105 meters. Substitution of these values into Eq. 8
j = μ0
4
5
3
N
R
 
 gives: j = 1.678 × 10–3 T/A, or 1.678 mT/A. Eq. 7
B = jI 
, with the above calculated value of j, will be used to determine the magnetic field from measured values of the current.

Procedure

Connect the apparatus according to the arrangement shown in Fig. 1 and start taking data following the steps outlined below:
Figure 4

Figure 4

It may not be possible for the beam to reach the 2.0 cm diameter line.
Be sure that you and your TA each initial your data sheets, and that you hand in a copy of your data before you leave the lab.

Analysis

An important modification must be made to Eq. 6
e
m
 =
2Vacc
B2r2
 
 before it can be used to calculate e/m. The modification is needed because the cathode where the electron beam originates is situated a distance a = 3.2 ± 0.5 mm below the exit hole.
Figure 5

Figure 5

As the figure above shows, the diameter, d, that is measured is not the actual diameter, d', for the electron's orbit. The actual diameter can be derived by using the Pythagorean equation shown below.
( 9 )
d' =
d2 + a2
 
Substitution of r = d'/2 and Eq. 9
d' =
d2 + a2
 
 into Eq. 6
e
m
 =
2Vacc
B2r2
 
, we get the following equation.
( 10 )
B2 =
8V
e/m
1
d2 + a2
 
Note: the use of the letter 'V' in this equation is to indicate a variable for voltage. This does NOT mean that you calculate B using 8 volts, but by using 8 times the voltage measured!
1
Using a spreadsheet, calculate a table of d, 1/(d2 + a2), average I, uI, B, uB, B2, and uB2 from the data for the lower accelerating voltage (≈50V).
2
Plot B2 vs. 1/(d2 + a2) and do a linear least square fit. Determine the slope and the intercept of the fitted curve.
3
From the slope, determine the value for e/m in C/kg using Eq. 10
B2 =
8V
e/m
1
d2 + a2
 
. Be sure to use the appropriate units for the variables B and d. The uncertainty in e/m can be calculated using the upper-lower bound method.
4
Repeat steps 1, 2, and 3 for the data obtained from the higher accelerating voltage.
5
Take the weighted average of the two values of e/m obtained by using the formula:
e/m = w1(e/m)1 + w2(e/m)2.
The weights w1 and w2 are taken to be proportional to the inverse of the square of the uncertainty of each e/m respectively. In this way the value of e/m with the smaller uncertainty will be given more weight. Thus,
( 11 )
w1 = k
1
u12
and w2 = k
1
u22
 
where the proportionality constant, k, is chosen to normalized the weights so that w1 + w2 = 1. The value of k is given by the equation below.
( 12 )
k =
1
1
u12
+
1
u22
 
Calculate the values of w1 and w2 and check if w1 + w2 = 1. Calculate the weighted average of e/m.

Discussion

Compare your experimental value of e/m to the accepted value. Discuss quantitatively the possible sources of error.
Remember to pledge your work.