The Mass of the Electron
Theoretical context

 
A moving charge in a magnetic field will feel a force given by F  qv  B . If the
velocity of the charge is perpendicular to the B field this reduces to F  qvB , where the
force is perpendicular to both the velocity and B field. Since the force is perpendicular to
the charge’s velocity it does no work on the charge and the charge has a constant velocity
and moves on a circular path. The force balance equations on the charge are as follows:
Centripeta l Force  Magnetic Force
mv 2
 qvB
R
mv
 qB
R
If the charge, in our context the electron, is accelerated through a electric potential
difference of VAccelerating to establish its velocity, the velocity of the charge will be given
by an energy balance analysis as shown below:
Work done by the Accelerati ng Potential  Kinentic Energy of the Charge
qV Accelerating 
v
1 2
mv
2
2qV Accelerating
m
Combining the expressions from the force balance and energy balance, a relationship for
the mass of the charge can be derived that does not explicitly contain the velocity.
mv
 qB
R
m
v  qB
R
m 2qV Accelerating
 qB
R
m
2m 2 qV Accelerating
 q2B2
2
R m
2mVAccelerating
 qB 2
2
R
2m
R2 
V Accelerating
qB 2
This relationship is testable by plotting R2 as a function of VAccelerating. The plot should
be linear for constant B field and have a zero intercept. The slope of the plot gives the
mass of the electron if the electronic charge and the B field are known. Taking the
logarithm of each side of this relation yields another testable equation:
2m
R2 
V Accelerating
qB 2
 2m

log R 2  log  2 V Accelerating 
 qB

 2V Accelerating 

2 log( R )  log( m)  log 
qB 2


 2V Accelerating 
  log( m)
2 log( R )  log 
qB 2


y  mx  b
 
A plot of 2log( R ) as a function of 2VAccelerating/qB2 will be linear with a slope of 1.0 and
an intercept of log(m), where m is the mass of the electron.
The magnetic field will be generated using Helmholtz coils. Helmholtz coils provide a
spatially uniform B field within their central volume. The value of the B field can be
calculated from the following expression that is usually covered in a 2’nd year Electricity
& Magnetism course:
32NI
B Helmholtz  3
5 2 10 7 a
where, N is the number of turns of wire in each coil, I is the current through the coil in
amperes, and a is the separation of the two coils in meters (a is also the radius of each
coil.)
Error Analysis:
The largest error will be in the radius of the electron’s trajectory - dR. However, we will
account for the following errors: current through the Helmholtz coils - dI, separation of
the Helmholtz coils - da, accelerating potential – dV. If the electron mass were
computed for each individual measurement using the relation
R 2 B 2q
m
2V Accelerating
the percent errors in R, B and VAccelerating add to a percent error in m. (How is the percent
error in B calculated?) If the value of the electron mass is determined graphically, then
its uncertainty will be given by the extreme fits of lines through the data ponts.
Purpose:
To derive an estimate of the mass of the electron with an experimental uncertainty and to
compare this experimentally determined value with the accepted value.
Procedure:
Record values of VAccelerating and RTrajectory for several different BHelmholtz Fields
Analysis:
You must devise a method by which you can separate the poor quality data from the high
quality data. This is left for you to decide on how you may identify poor quality data that
should not be included in the analysis.
You must include at least one graph that verifies the basic relationships. This graph
could help you identify the poor quality data, as well. Fundamentally, you want a graph
of some function of RTrajectory as a function of some function of VAccelerating. I leave it to
you to decide the most appropriate course.
In you conclusion you must state the validity of out basic relationships from your graph,
state the best value of the mass of the electron with an uncertainty, and compare your
derived mass to the accepted mass of the electron with a percent difference.