Charge to mass ratio for the electron
Introduction
As you have seen throughout physics 1, conservational theorems are used appropriately
to solve many types of problems. One of the well-known conservational approaches involves the
mechanical transformation of potential energy to kinetic energy. This can be seen for a sliding
mass on a frictionless incline, and even circuit analysis. In some cases, the transformation
equations of energy from one form to another supplement other equations in calculating some
quantity.
Lab objective
In this lab we will use the Helmholtz coil equation combined with energy conservation to
determine the charge per mass ratio of the electron. We will use standard analysis to compare
with the accepted value of the ratio.
Theory
The Helmholtz coil configuration is displayed in the figure below. The system consists of
two equivalent coils of radius R and separated by a distance of L = R. Thus, the center of the coil
configuration is simply R/2. Both coils have identical number of (N) wire turns used to generate
the magnetic field. What is extremely useful about this coil configuration is that, for a given
current through each of the windings in both coils, an equation can be derived for the value of
the magnetic field at the very center of the coil configuration. Consequently, the magnetic field
at this point is uniform. With a predictable magnitude and direction, this can be used in
applications in the study of magnetic thin films and other applications.
The Biot-Savart Law may utilized to determine the value of the magnetic field at a point
P along the axis drawn from the center of a coil:
⃗ =
𝑑𝐵
𝜇0 𝐼𝑑𝑙 × 𝑟̂
.
4𝜋 𝑟 2
⃗ is generated with an infinitesimal current with
In the above equation, an infinitesimal field 𝑑𝐵
constant magnitude and direction 𝑑𝑙 . The radius vector, the unit vector 𝑟̂ , connects the point of
the current direction to the point P as seen in the figure below. Note that the net field at that point
will point in the direction shown. This is due to the symmetry in the figure. Note that 𝑑𝑙 → 𝑅𝑑𝜑,
and integrating around the entire loop from 0 → 2𝜋 yields
𝐵=
4
5√5
(𝑥 2
𝜇0 𝐼
.
+ 𝑅 2 )2
For a pair of coils with N-turns each, the magnetic field at a point P precisely in the center
between the coils is simply
𝐵=
(8.99 × 10−7 )𝑁𝐼
8 𝜇0
𝑁𝐼
=
.
(𝑥 2 + 𝑅 2 )2
5√5 𝑟 2
The unit of the magnetic field is a Tesla (T).
The principle of this experiment is that electrons who enter the above magnetic field at a
velocity which is completely perpendicular to the field will be trapped and move within the
circle with uniform circular motion. How is the velocity of the electrons to be determined? Since
we cannot utilize a speed detector for this lab, we will instead use conservation of energy, i.e. the
transformation of electrical potential energy to kinetic energy. This (classical) transformation
looks like
𝑒𝑉 →
1
𝑚𝑣 2 .
2
The potential is controlled by the power supply. How is this used in the final equation that takes
into account the information given in our field equation? The force on the electron is the Lorentz
Force.
⃗.
𝐹 = −𝑒𝑣 × 𝐵
The two vectors are perpendicular and thus the magnitude is 𝐹 = 𝑒𝑣𝐵. The figure below
illustrates this relationship between field, velocity, and magnetic force.
Recall that the magnitude of the force on the electron in uniform circular motion is given by
𝐹=
𝑚𝑣 2
= 𝑒𝑣𝐵.
𝑟
Note: Here, r is the radius of the circle. The figure below shows what happens when a charge q
enters perpendicular to a magnetic field.
From conservation of energy and the centripetal force, we seem to have two equations for 𝑣 2 .
Equating these two equations we get
𝑒
2𝑉
=
.
𝑚 (𝐵 ∙ 𝑟)2
Re-arranging the equation to put it in the form of 𝑦 = 𝑚𝑥 + 𝑏 yields
2𝑉
𝑒
= 𝑟 2.
2
𝐵
𝑚
2𝑉
In the above equation, 𝑦 = 𝐵2 and 𝑥 = 𝑟 2 . The slope should be the charge per mass ratio when
plotting the data and obtaining a linear graph.
Experimental set-up
1. The experiment is essentially set-up when beginning the lab. Do not re-arrange any cords
or meters. The equipment consists of a set of power supplies, the Helmholtz coil
configuration, the gas tube (filled with helium at 0.1 Pa of pressure), a filament to obtain
the electrons, and an electron “gun”. The filament voltage is to be maintained at 6.3 V.
Note that the number of turns in each coil are 130, and the radius of the coil is 15.8 cm.
2. On the tunable constant voltage power supply, set the voltage range to 0 – 200 V.
3. Turn both power supplies on. Set the constant voltage power supply to 120 V, and wait a
few minutes for the filament to heat up. An electron beam will emerge when it is warm.
4. You can adjust the constant voltage supply to optimize focus and brightness.
5. On the constant current power supply, increase the current to the coils until the electron
beam curves upward. Continue (gently) until the electron beam forms a closed circle.
Record the value of this current in the table below.
6. Calculate the magnetic field and place the value in your table.
7. Note: if you cannot form a circle, rotate the platform slightly to either the right or left to
align the coil field with the Earth’s magnetic field.
8. For each value of accelerating voltage, measure the radius of the circle. To measure the
radius, look through the tube at the mirrored scale. Make sure you estimate an uncertainty
of radius 𝛿𝑟. Note: To avoid parallax errors, move your head to align the electron beam in
the tube with the reflection of the beam as you see it in the mirrored scale. Measure the
radius of the electron beam as you see it on both sides of the mirrored scale, and then
average the values for the value you place in the table.
2𝑉
9. Open up Excel and make a plot of 𝐵2 𝑣𝑠. 𝑟 2 . Use LINEST to determine the slope and the
uncertainty.
1. Calculate the t-score using your value, the uncertainty, and the accepted value. Discuss the
type of error for this lab. Explain in detail.
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2. How could you minimize error in this lab?
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3. Draw a vector diagram of this experiment.
In this lab we measured the distance between nodes of standing waves at different
frequencies, and with the data recorded we calculated the wavelength and 1/2. We made a
graph of 1/2 vs the frequency and determined the velocity of the wave (slope of the graph)
and compared it to the expected velocity. The objective of the lab was to study the concept
of resonance by observing the standing waves pattern on a string. Standing waves don't form
under just any circumstances. They require that energy be fed into a system at an appropriate
frequency. That is, when the driving frequency applied to a system equals
its natural frequency. This condition is known as resonance. Standing waves are always
associated with resonance. Resonance can be identified by a dramatic increase in amplitude
of the resultant vibrations. The progression of wavelengths can be expressed by the following
mathematical equation: An = 2L/n n = 1, 2, 3 ... In this equation, An is the wavelength of the
standing wave, L is the length of the string bounded by the left and right ends, and n is the
standing wave pattern, or harmonic, number. For the fundamental, n would be one; for the
second harmonic, n would be two, etc. The resonant frequency can be found by using the
relationship between the wavelength and the frequency for waves as shown in the following
equation: v = Af. In this equation, v is the (phase) velocity of the waves on the string, 1 is the
wavelength of the standing wave, and f is the resonant frequency for the standing wave. The
results of this experiment were important since the experimental value for v and the value
calculated from the equation were different by less than a 10% margin of error. This means
the relationship established on these equations were correct.
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