Download experiment iii experiments with an electron beam

EXPERIMENT III
EXPERIMENTS WITH AN ELECTRON BEAM
An electron beam is a collection of free electrons, all traveling in approximately
the same direction with the approximately the same velocity. While it is
frequently difficult to follow where a single electron goes, it is verily easy to
follow where a beam of millions (or billions) or electrons go together. You will be
following such motion in a cathode ray tube (CRT).
The CRT is the basis for all electron display tubes: televisions, computer
monitors, and oscilloscopes. Although the CRT is being supplanted today by flatscreen displays (you can read the obituary of the CRT at
http://news.bbc.co.uk/1/hi/uk/1459817.stm!), they are still the basis of most
TV’s and monitors world-wide.
The operation of a CRT is illustrated in Figure 1. A beam of electrons is
extracted from the cathode, accelerated to a kinetic energy of ~300 eV, focused so
that they will all converge on a single spot when they reach the front screen,
deflected by two pairs of electrodes that enable you to steer the beam, and finally
strike a front phosphor screen where they produce a visible spot. By changing the
voltage applied across the two pairs of deflection electrodes the beam can be
scanned both horizontally and vertically.
During this experiment, you will model the pattern of electric potentials and fields that
are produced in each region of the CRT, and understand how those electric fields perform
the functions of each element in succession.
You will then power up the tube and use it to perform experiments with the beam of
electrons. In a first experiment, you will try to determine if the Earth’s gravity deflects a
beam of electrons by a measurable amount. Then you will apply voltages to a pair of
deflection electrodes, and measure the amount of deflection of the electron beam. You will
analyze the dynamics of the electron motion, and compare your observations of beam
deflection with your modeling of the potentials and electric fields and the electron motion
in those fields.
Figure 1. Beam optics in a cathode ray tube.
Experiment IV: experiments with an electron beam
We will discuss here the principles of operation of a CRT that you need to know to
understand the following experiments and use it to study the deflection of an electron.
You can read more about the structure and operation of a CRT at
http://www.tpub.com/neets/book6/21e.htm.
Let’s summarize the elements of the CRT that you will use in your deflection
experiments. They are shown schematically in Figure 2. The electrons start as conduction
electrons in the metal of a small cathode. The cathode is a small disk of metal foil that is
heated by a resistive filament so that conduction electrons can be extracted from the metal
surface by applying an electric field outside the surface. We will set V=0 to be the cathode
potential in all discussion of this experiment.
Question: which way should the electric field point in order to extract electrons from
the cathode?
The extraction electric field is provided by a grid: a metal foil disk with a hole in it,
which is maintained at a potential Vg. A potential Vg ~0-10 V typically suffices to extract
sufficient electron beam to produce a visible spot on the phosphor screen at the far end of
the CRT.
The beam is focused by the radial component of electric fields in the aperture of a
focusing electrode: a second metal foil disk with a hole in it, which is maintained at a
potential Vf ~ 80 V.
The deflection electrodes are maintained at potentials close to that of the phosphor
screen (~300 V), so that the electron beam has been accelerated to its full kinetic energy
before entering the regions between pairs of deflection electrodes that are separated by a
gap distance g transverse to the direction of the electron beam. If the two x electrodes of a
pair are set to voltages Vr and Vl that differ by some amount, then there will be a transverse
electric field
Ex =
(Vr − Vl )
g
r
r
The electrons in the beam will then experience a force F = (−e) E that will deflect the
beam in the transverse x̂ direction.
Finally the beam reaches the phosphor screen on the front of the CRT, where each
electron scatters from an atom of metal in the phosphor (the white coating on the inside
wall of the glass face of the tube), exciting one or more electrons in the atom. The excited
electrons then drop back into their original orbital in the atom, emitting a photon of light
that carries an energy equal to the difference in energies of the excited and ground state
orbitals.
Now let’s go into more detail about each of the CRT components in turn.
One final critical feature of the CRT is its vacuum enclosure – the glass tube. The
entire system of electrodes, and the trajectory of the electron beam, must be maintained in
a hard vacuum. If an electron scattered from an atom of gas in the space inside the tube as
it was traveling from cathode to screen, it would lose some of its kinetic energy and also
scatter to a different direction, and would therefore not make it to the screen.
Experiment IV: experiments with an electron beam
Question: Assuming that each gas molecule has a size of 5 Å2, calculate the maximum
pressure (as a fraction of atmospheric pressure) that can be inside the tube.
left
deflection
0
0
80
300 300 300
300
270 300 330
320 350 380
filament
cathode grid
focusing
right
deflection
phosphor screen
V(x)
x
Figure 2. Electrodes and their voltages in a CRT. Three electron trajectories are shown,
for a center ray, a left deflection, and a right deflection.
Cathode and filament
Let’s begin with the cathode, a small disk of metal foil that emits electrons into the
vacuum within the tube. To appreciate the subtlety of the cathode’s operation, you must
first understand that it is having to do something that ought not to happen! The conduction
electrons in a metal are free to move within the metal, but they are bound from leaving
from the surface of the metal. To understand this statement, consider what happens if a
conduction electron tries to leave a surface of the metal – the boundary where metal ends
and vacuum begins. From your last lab, you calculated the distribution of potential and
electric field of an electric dipole, as shown in Figure 3 (The solid curves are the lines of
force of the electric field, the dashed curves are equipotentials). But now consider the
pattern of fields produced by a conduction electron (charge –e) that is just above the
surface of its metal. Remember that the metal surface is an equipotential, so the lines of
force of electric field must enter perpendicularly into the metal surface. As a result the
pattern of electric fields and potentials outside the metal is exactly the same as it would be
if we replaced the metal by an image charge +e located the same distance inside the metal
as the electron is outside – the upper half of the pattern of fields and potentials shown in
Experiment IV: experiments with an electron beam
Figure 3. Of course the image charge is not really there. The pattern of fields is actually
produced by a distribution of + surface charge (totaling +e) that appears on the surface of
the metal.
The important thing for a cathode is that it shouldn’t work: as you can see in the figure
the electron should be dragged back into the metal as soon as it leaves! This retarding
effect is given a quantitative measure by the work function Φ, measured in volts, which is
the work/charge that must be done to drag a conduction electron out of a metal surface so
that it is free. For the metals used in cathodes (e.g. oxide-coated molybdenum), the work
function is ~1 eV.
So how do we get the electron out? First we heat the metal to ~900 oC (1200 oK) –
that’s red hot! Temperature is a measure of the random kinetic energy of each particle in a
material. The proportionality between temperature and kinetic energy per particle is
Boltzmann’s constant: k = 1.38x10-23 J/K = 8.6x10-5 eV/K. So by heating the metal, each
conduction electron has more kinetic energy as it approaches the surface. The average
kinetic energy is
KE = kT = 0.1eV
Alas, that is still a lot less than the work function. So how do we get conduction
electrons out of the metal surface? There are two components to the answer. First, the
above calculation of kinetic energy is the mean value of kinetic energy of the conduction
electrons, averaged over all them in the metal. Individual electrons can have significantly
more (or less) kinetic energy, according to a bell-shaped probability distribution. So a
small fraction of the conduction electrons can have many times the mean kinetic energy.
Second, we apply an external electric field that exerts a force upon any conduction
electron that manages to leave the surface. The potential distribution that results is shown
in Figure 4. If we take the potential of an electron to be zero inside the metal (to the left),
the potential steps up to Φ at the surface, then falls linearly outside (V = -Ex). So a
conduction electron inside the metal with a kinetic energy that puts him part-way up the
barrier would still have to ‘hop over’ the remaining potential barrier to escape.
In quantum mechanics this is actually possible! It is called tunneling. Qualitatively, if
a barrier is low enough and narrow enough, then a particle can ‘tunnel’ through the barrier
to the other side even though during the brief time that it is ‘inside’ the barrier energy is
not conserved. You can understand this in terms of the uncertainty principle, which states
that the uncertainty in energy times the uncertainty in time in any measurement must
always be greater than a fundamental limit, Plank’s constant: ∆E ⋅ ∆t > h . So long as the
potential energy barrier ∆E is shallow enough, and the time to tunnel through ∆t is short
enough, we cannot do an experiment to tell that energy was not conserved!
So if the time the electron spends inside the barrier is small enough, the uncertainty in
its energy is enough that it can tunnel through. If you want to learn more about quantummechanical tunneling, you can visit http://phys.educ.ksu.edu/vqm/html/qtunneling.html.
For the design of our CRT, we heat the cathode using a resistive wire filament, to
which we apply 6.3 V alternating current (a.c.) voltage from an external power supply.
Experiment IV: experiments with an electron beam
Figure 3. The electric field distribution of a dipole.
Figure 4. Potential distribution in the vicinity of a metal surface, in the presence of a work
function Φ and an external extraction field E.
Experiment IV: experiments with an electron beam
The width ∆x of the barrier can be controlled by changing the electric field Eg just
outside the cathode surface. Since the time the electron takes to penetrate the barrier is ∆t
= ∆x/v, this in turn controls the probability that an electron can tunnel. In Figure 4, a
stronger field (-E1) produces a steeper potential slope outside the work function barrier, so
that the width of the barrier is less. A weaker field (-E2) produces a shallower potential
slope, and a wider barrier.
Grid
This controlling electric field is produced by the grid: a metal foil disc with a hole on
its axis. The disk is located concentric with the cathode and at a distance zg from its
surface. The electron beam passes through the hole, traveling along the z axis. The grid is
maintained at a potential Vg by an external power supply. This voltage thus controls the
electric field Eg = Vg/zg. In order to extract electrons from the cathode, we apply a voltage
between the tube and the grid. It takes very little potential difference in order to extract
sufficient electrons to produce a visible beam spot on the phosphor screen.
Focusing electrode
The focusing electrode is a second metal foil disk with a concentric hole, spaced a
distance zf from the grid, and maintained at a potential Vf that is supplied by an external
power supply. The electron beam passes through the hole.
When Vf > Vg, there is an electric field in the region between the two electrodes. In the
r
region far from the hole, this field is approximately a uniform field E = (V f − V g ) / z f .
Near the hole, however, the electric lines of force on the axis bend outward to
terminate near the lip of the hole. In that region there is a radial component Er to the
electric field. You will calculate this field distribution in your first problem. This radial
field distribution acts as a lens on the beam. Electrons that are traveling parallel to the z
direction, but displaced from the axis, will experience a transverse force directed in the
radial direction. Depending upon the sign of Er, this force will either push them towards
the axis (focus) or away from the axis (defocus).
There will actually be two such regions of electric field – one just beyond the grid
electrode, and one just before the focusing electrode. You will calculate the beam optics as
an electron passes through these fields.
Deflection electrodes
Two pairs of deflection electrodes are located along the beam axis, starting a distance
zd after the focusing electrode. Each pair of electrodes is centered on the axis, and has a
transverse gap d between the two electrodes. As an electron leaves the focusing electrode,
r
Vd − V f
that accelerates it to a kinetic
it will experience an accelerating force F = (−e)
zd
energy eVd. Of course the electron sees both of the two electrodes equally as it is being
Experiment IV: experiments with an electron beam
accelerated. The potential Vd that determines the beam kinetic energy is therefore the
V + Vr
.
average of the potentials on the two plates: Vd = l
2
The electron then enters the region between the pair of electrodes. There it experiences
r
V − Vl
a transverse force F = (−e) r
which deflects the beam direction. By controlling the
d
voltage between the pair of electrodes, you can control where the electron beam strikes the
phosphor screen.
You will notice that the deflection electrodes are not simply flat plates, but flare out at
the downstream end. You should think about why this is done: it affects both the pattern
of electric fields and the range of electron trajectories that can reach the screen.
Phosphor screen
When an energetic electron strikes an atom, it can scatter from orbital electrons,
exciting them to a higher allowed energy state (excitation) or even ejecting them from their
orbit altogether (ionization). We make use of this phenomenon to detect where the beam
strikes the phosphor screen.
Some kinds of atoms have an arrangement of atomic orbitals that have a high
likelihood for collisional excitation of an outer electron to an excited state that lies above
the ground state by an energy that is that of a photon of visible light (~1 eV). When a
beam electron excites such an atom, the de-excitation (when the electron drops back into
its ground state) produces a photon of visible light.
The phosphor screen is a film of white powder that is deposited on the front face of the
glass CRT tube. The powder must be of just the right thickness: thick enough that most
electrons will in fact collide with a phosphor atom, but thin enough so that the emitted light
is not absorbed in the powder that lies between an emitting atom and the front of the glass.
There is also a very thin metal film (less that a wavelength of light thick so it is
actually transparent!) so that we can maintain a potential Vp on the phosphor screen. We
will set Vp = Vr, the voltage on one of the two deflection electrodes. Thus the electron
beam travels through a region of constant potential (hence constant kinetic energy) as it
passes from the exit of the deflection electrodes to the screen.
PROBLEM 1
MODELING THE POTENTIALS AND FIELDS IN A CRT
You will use the Relaxation Method to numerically calculate the electric potentials and
fields in each of the elements of the CRT: the cathode, the focusing electrode, and the
deflection electrodes.
The deflection electrodes have Cartesian geometry, and can be modeled using the
EXCEL spreadsheet that you developed in the last laboratory.
The cathode and focusing electrode have cylindrical symmetry, however. In order to
calculate potentials in cylindrical geometry you must add a correction for the curvature of
the coordinate system. The details of this correction are presented in Appendix A of Lab 3.
A spreadsheet that embodies the correction is available as lapcyl.xls accompanying this
manual.
Measure the geometry
You should begin by preparing a measured drawing of the geometry of the electrodes
that form the potential distribution that the electrons see as they traverse the CRT from
cathode to screen.
These include the cathode (a flat disc), the grid (a disk with a hole in its center), the
focusing electrode (identical to the grid), the deflecting electrode pairs, and the front screen
(a curved conducting surface).
Your success in modeling the potentials will depend critically upon the accuracy in
measuring the dimensions of each electrode, the size of the hole in the grid and focusing
electrode, the spacing between elements, and the geometry of the deflection electrodes.
Use your measurements to make a neat drawing of the electrode geometry on grid
graph paper (use the kind that has 1 mm grid lines).
Calculate potential distributions in the regions of the CRT
From the drawing you will be able to choose a cell spacing and enter the geometry of
each region in turn into an appropriate EXCEL spreadsheet.
You should model the potentials in each of the three regions in a separate spreadsheet.
In each case you must choose the appropriate boundary conditions, and for the cylindrical
problems designate an axis.
Discuss the motion of an electron that starts on the axis at the cathode in terms of the
potential distributions that you obtain.
Focusing of the beam
Adjust the geometry you use in modeling the region between the grid and focusing
electrode so that you have enough cells (spacing h) in the radial region of the holes to
r
calculate the electric fields E = E r rˆ + E z zˆ in that region. You will need to calculate
Problem 1: modeling the potentials and fields in a crt
E r (m, n) = −
Φ m +1,n − Φ m −1,n
2h
,
E z (m, n) = −
Φ m ,n +1 − Φ m,n −1
2h
Now you will use these calculated electric fields to model the focusing action upon an
electron.
Consider an electron that passes through the hole in the grid, traveling straight from left
to right but displaced from the axis at a radius a equal to half the radius of the hole in the
grid and focus electrodes. Choose your scale of the problem so that this radius is a goodly
number n0 of cells from the axis.
Now create a spreadsheet in which you are going to numerically calculate the motion
of the electron just as you did last semester in your computer problems. The spreadsheet
should contain three regions, each containing the number of (m,n) cells that you had in
your model of the grid and focusing region and each bounded by colored cells to mark the
boundaries clearly. In two of the regions, calculate the radial and axial field components
as indicated above.
In the other region, you will model the dynamics of the electron beam: how each
electron moves in response to the force that acts upon it.
In this dynamics region of the spreadsheet, create columns for all the quantities that
you will need.
t, z, vz, az, r, vr, ar, dt
When you modeled kinematics in the past, you typically used equal time steps. But your
electric field data is available in equal space steps. To match the data to the problem, we
will need to keep track of the velocity v = v r2 + v z2 , and calculate the time step between
steps in the code to be that needed to advance the electron one column in the spreadsheet (a
dt = h/vz.
distance h);
Now you are ready to begin tracking the electron.
Set up a first row of the calculation with initial conditions:
Step1: t = 0, z = 0,
vz1 = v0,
az1 = -eEz(1,n0)/M, r1 = n0h,
vr1 = 0,
ar1 = -eEr(1,n0)/M,
dt1 = h/vz1
There is a bit of difficulty with the starting velocity v0 that the electron has as it enters
the focusing region. Since we will give the grid almost 0 voltage, the electrons pass into
this region with very little velocity. We need to start them with non-zero velocity, so that
the first calculation of dt does not diverge. We will set choose Vg = 10 V. You should
calculate the velocity of an electron with a kinetic energy of 10 eV, and enter it for v0.
where e = 1.6x10-19 C and M = 9x10-31 kg.
Problem 1: modeling the potentials and fields in a crt
Now enter the iteration for the next step:
Step 2: t = t1+dt1 ,
v z 2 = v z1 + a z1 ⋅ dt1 ,
a z = a z1 − eE z 2 / M
v r 2 = v r1 + a r1 ⋅ dt1 ,
a r 2 = a r1 − eE r 2 / M
Now here we must take care about how to get the correct values of the electric field
components. Our data was calculated on a mesh of steps of size h. We choose the time
step dt so that the electron advances by exactly one step in z, but what about r? the values
of Ez2 and Er2 in the above expression refer to a point at a radius that is not in general a
multiple of h!
We deal with this difficulty by interpolation. We first determine the radial index that
is next smaller than the current radius of the electron:
N = TRUNC(r1), δ =
r1 − Nh
, E z 2 = (1 − δ ) E z (2, N ) + δ ⋅ E z (2, N + 1)
h
and similarly for Er2. TRUNC is an EXCEL function that returns the truncated value of a
number (drop the decimal fraction). You will need to add columns in your spreadsheet to
accommodate the above calculations for each z step.
Now you are ready to copy this last line of code into as many z steps as your region
contains, and graph the electron trajectory (r,z). Each row will correspond to a new step
forward in z.
It’s time to play with this new tool and learn about beam focusing. Try changing the
starting parameters for r and vr. Change the strength of focusing by changing the value of
Vf.
If you consider an electron entering the focusing region with vr = 0 and a radial offset
a, the distance from the focusing electrode to the point z = F where the electron crosses the
axis is called the focal length, just as with light optics. Use your new tool to find the focal
length for electrons having different values of a, and see if they all share a common focal
length.
Deflection Electrodes
Now repeat the above procedure to calculate the trajectories of electrons passing
through the deflection electrodes. You will need to set the initial velocity to correspond to
the kinetic energy that an electron will have as it enters the pair of electrodes at potentials
Vr, Vl (r and l subscripts refer to the right and left electrodes of a pair). Since the beam
starts at potential 0, the kinetic energy of an electron entering the deflection electrode pair
on axis (a central ray) will be
T = e(Vr + Vl ) / 2 .
Obtain the asymptotic angle that this central ray will have as it leaves the region of the
deflection electrode. You will have to include a goodly piece of space beyond the
electrode in order to take account of its fringe fields.
Problem 1: modeling the potentials and fields in a crt
This angle will be the deflection angle for electrons as they next head for the screen.
Since the deflection electrodes and screen are set to about the same potential, the velocity
will be roughly constant on this last leg of its journey. From the dimensions of your CRT,
calculate the sensitivity α [cm/Volt], which is the change in the deflection distance of the
spot of light at the screen per change in deflection voltage. Later you will use this
measurement of α to obtain a measurement of the charge/mass ratio of the electron!
PROBLEM 2
DEFECTION OF AN ELECTRON BY A GRAVITATIONAL FIELD
You are working in a research laboratory that is attempting to make a better electron
microscope. The key to advancing the project is the precise control of a beam of electrons.
For your study of electron control you decide to use a Cathode Ray Tube (CRT), the same
device that is the basis of most TV sets. In the CRT, electrons are emitted at one end of an
evacuated glass tube and are detected by their interaction with a phosphorous screen on the
other end. You know that every object in flight near the Earth's surface is subject to the
gravitational force. From your physics experience you also know that the acceleration of
all objects in free fall is the same, independent of their mass. Even though an electron has
a small mass, it has the same gravitational acceleration as a baseball or a bullet. You
worry that the gravitational force will deflect the electron from its path giving it the
parabolic trajectory that you studied in the first semester of physics.
?
How does the gravitational force due to the
Earth affect the motion of the electrons in the
CRT?
EQUIPMENT
You will be using the Cathode Ray
Tube (CRT) described in the introduction
and in Appendix A.
θ
PREDICTION
Calculate how far an electron falls during its flight within the CRT when the CRT is
horizontal (θ = 0o). Will this electron deflection increase, decrease or stay the same as the
angle is increased from horizontal (θ = 0o) to vertical (θ = 90o)? Sketch a graph of the
distance of the electron from the center of the CRT when it hits the screen (its deflection)
versus the angle of incline of the CRT from the horizontal. Explain your reasoning.
Problem 2: Defection of an electron by a gravitational field
METHOD QUESTIONS
To test your prediction, it is useful to have an organized problem-solving strategy such
as the one used below:
1. Draw a picture of the CRT in the horizontal position. Do not include the deflection
plates shown in Figure 2 since they will not be used in this problem. Be sure you have all
the other components in your picture and you understand the function of these parts.
2. Draw the electron's trajectory from the time it leaves the electron gun until it hits the
screen. Label all of the important kinematic quantities in the problem. Label all of the
important forces on the electron. The target quantity for this problem is the electron beam
deflection at the screen. The quantities you can measure in this problem are the position of
the electron beam spot on the fluorescent screen, the initial electron accelerating voltage
(Vf in the diagram in Figure 2 and Appendix A), the distance from the end of the electron
gun to the CRT screen, and the angle the CRT makes with the horizontal (which is
currently zero degrees).
3. What physics principles will you use to solve this problem? To use kinematics, a
motion diagram will probably be useful. Choose a convenient coordinate system and
describe your reason for choosing it. How is the motion along one axis of your coordinate
system influenced by the motion along the other axis? During what time interval will you
analyze the motion of the electron? To determine the motion of the electron, you need to
know its acceleration.
4. Dynamics (Newton’s second law) will allow you to determine the acceleration of the
electron from the forces on it. What are the forces on the electron in the time interval you
have chosen? What is their direction? From the forces, determine the acceleration of the
electron during that time interval along each of the coordinate axes you have chosen.
5. If the acceleration is constant along any axis of your coordinate system, you can use
constant acceleration kinematics to describe the motion along that axis. If not, determine
the behavior of the electron’s acceleration as a function of time and use calculus. Write
down two equations (one for each coordinate axis) that describe the electron’s position at
the screen in terms of its motion (positions, velocities, accelerations, and times).
6. The magnitude of the electric field (in Newtons per Coulomb) between two equally
charged parallel plates is equal to the voltage between the two plates (in Volts) divided by
the distance between the plates (in meters). What is the direction of the electric field
between the two accelerating plates? How much work is done on the electron by this
accelerating field? Using conservation of energy, determine the electron’s velocity as it
leaves the electron gun in the CRT. What is the direction of the electron as it leaves the
accelerating field? What assumptions have you made?
7. Examine your equations giving the electron’s position at the screen. You can solve
them if the number of unknowns in your equations is equal to the number of equations. Is
Problem 2: Defection of an electron by a gravitational field
it? If it is, solve your equations algebraically for the distance an electron will fall while
traveling through a CRT. If it is not, write down additional equations that relate some of
the unknown quantities in your equations to quantities that you know. Complete your
solution by using the actual numbers that describe your situation.
8. Does your solution make sense? You can check by determining the time of flight of
the electron. In that amount of time, how far would a ball drop in free fall? If the solution
does not make sense, check your work for logic problems or algebra mistakes.
9. Now return to step 1 and solve the problem for a CRT pointed upward and at an
angle from the horizontal. What difference will this make to your solution? To simplify
your procedure, you might want to think of the relationship between the distance of
separation between the accelerating plates and the total distance from gun to screen. Are
the two of comparable magnitude? If not, what does that say about the relative magnitudes
of the gravitational deflections in each part of the CRT?
Problem 2: Defection of an electron by a gravitational field
EXPLORATION
WARNING: You will be working with equipment that
generates large electric voltages. Improper use can cause
painful burns. To avoid danger, the power should be turned
OFF and you should WAIT at least one minute before any
wires are disconnected from or connected to the power
supply. Never touch the conducting metal of any wire.
Follow the directions in Appendix A for connecting the power supply to the CRT.
Check to see that the connections from the power supply to the high voltage and the
filament heater are correct, before you turn the power supply on. The power supply is set
up internally to supply a constant voltage of 300 V between the cathode and the screen
(you cannot change this voltage using the controls on the power supply). Using the
controls illustrated in Appendix A, you can control the voltages applied to the grid, the
focusing electrode, and one deflecting electrode of a pair. The other deflecting electrod of
the pair is connected to the same 30 V potential as the screen. You can use the grid voltage
to adjust the beam current (thereby controlling how bright is the spot of light on the
screen). You can use the focus control to bring all electron trajectories to a common focus
on the screen (a small bright spot on the screen).
Do you expect the gravitational deflection to vary as a function of the angle of the CRT
with the horizontal? Try different orientations in the horizontal plane to see if you can
observe any difference. Does the qualitative behavior of the electron deflection agree with
your prediction?
For what orientation of the CRT is it impossible for the gravitational force to deflect
the electron? This is the location of the beam spot when there is no gravitational effect on
the motion of the electrons.
If you observe a deflection of the electron beam, determine if this deflection is or is not
caused by the gravitational force. If it is not, what does this mean and how can you
minimize the effect of that force on your measurements? Is the deflection different if you
move the CRT to a different position in the room?
Devise a measuring scheme to record the angle of the CRT and the position of the
beam spot.
Write down your measurement plan.
Problem 2: Defection of an electron by a gravitational field
MEASUREMENT
Measure the position of the beam spot at an orientation of the CRT for which you
expect the gravitational deflection to be zero and the position at an angle for which the
gravitational deflection should be maximum. Make measurements at several intermediate
angles as well.
Note: Be sure to record your measurements with the appropriate number of significant
figures and with your estimated uncertainty. Otherwise, the data is nearly meaningless.
ANALYSIS
Make a graph of the position of the electron beam spot as a function of the angle that
the CRT makes with the horizontal.
If you observe a deflection, how can you tell if it is caused by the gravitational force?
If the deflection is not caused by gravity, what might be its cause? How will you decide?
Use your data to determine the magnitude of the deflection of the electron.
CONCLUSION
Did your data agree with your predictions? Did you observe any deflection of the
electron beam? Was it in the direction you expected? What could account for any aberrant
behavior? How can you arrange your CRT to minimize the aberrant behavior?
Does the Earth's gravitational force affect the motion of the electrons in the CRT in a
way that you can measure? State your results in the most general terms supported by your
data. Based on your results, do you think you need to take gravitational deflection into
account when using the CRT? Why?
PROBLEM #3
DEFLECTION OF AN ELECTRON BEAM BY AN ELECTRIC FIELD
You are continuing your attempt to design a better electron microscope. To precisely
control the beam of electrons you will use electric fields in the two directions
perpendicular to the original direction of the electrons. Before you can test the sensitivity
of the electron microscope design, you will need to determine how an applied electric field
affects the position of the beam spot. For your study of electron control you decide to use
a Cathode Ray Tube (CRT) in which electrons are emitted at one end of an evacuated glass
tube and are detected by their interaction with a phosphorous screen on the other end.
Inside the tube, the electrons pass between one set of parallel plates in the horizontal
direction and another set in the vertical direction.
?
How does the deflection of the electron beam
vary when passing through an electric field
perpendicular to the velocity of the electron
beam?
EQUIPMENT
You will be using the Cathode Ray Tube described in the introduction and Appendix
A. The fluorescent screen has a centimeter grid in front of it so you can measure the
position of the beam spot. The applied electric field is created by connecting the internal
parallel plates to a battery or power supply. Note that you will be using the deflection
plates as described in Figure 2 and Appendix A.
PREDICTION
Calculate how the electric field between the horizontal deflection plates affects the
position of the electron beam spot. Repeat for the vertical deflection plates.
Use this equation to make a graph of deflection as a function of the electric field
strength for each set of deflection plates.
Lab II- 17
PROBLEM #3: Deflection of an electron beam by an electric field
METHOD QUESTIONS
1. Draw a picture of the important components of the CRT. Only include one set of the
deflection plates shown in Figure 2. Draw a coordinate axis on this picture. Draw the
electron trajectory. On the trajectory, draw and label arrows representing the electron’s
velocity and acceleration in regions of different acceleration. Draw the electron trajectory
if there were no electric field between the plates. The difference between where these two
trajectories hit the CRT screen is the deflection.
2. Decide on the time intervals in which the electron has differing accelerations. What
forces cause these accelerations? Draw an arrow representing each force on your picture.
What forces are you assuming are negligible. In each region of different acceleration make
a motion diagram showing the electron's trajectory and the electron’s velocity and
acceleration when it enters the region, is in the region, and leaves the region. Qualitatively
describe the shape of the trajectory of the electron in each of the regions.
3. Determine the velocity of the electrons as they leave the electron gun in the CRT.
(see method question 6 of Problem #4)
4. Determine the acceleration of an electron as it travels through the region between the
deflecting plates by using dynamics (Newton’s second law). You can determine the force
on the electron in this region if you know the electric field. Is the electric field constant in
the region between the deflecting plates? What does that tell you about the acceleration of
the electron in that region?
The magnitude of the electric field between two equally charged parallel plates (in Newtons per
Coulomb) is equal to the voltage between the two plates (in Volts) divided by the distance between the plates
(in meters).
5. Use your drawing from step 1 and kinematics to determine the position and direction
of the electron as it enters the region between the deflection plates and when it leaves that
region. Write down the equation giving the electron’s position as it emerges from the
deflecting plates. Write another equation giving the electron’s direction.
6. Use your drawing from step 1, the position and direction of the electron as it leaves
the deflection plates, and geometry to write down an equation giving the position of the
electron when it hits the screen. This equation should give the total deflection during the
electron’s motion through all the regions of the CRT.
7. Examine your equations giving the electron’s position at the screen. You can solve
them if the number of unknowns in your equations is equal to the number of equations. Is
it? If it is, solve your equations algebraically for the deflection of an electron. If it is not,
write down additional equations that relate some of the unknown quantities in your
equations to quantities that you know. Complete your solution by using the actual
numbers that describe your situation.
8. Does your solution make sense? If not, check your work for logic problems or
algebraic mistakes.
Lab II - 18
PROBLEM #3: Deflection of an electron beam by an electric field
EXPLORATION
WARNING: You will be working with equipment that
generates large electric voltages. Improper use can cause
painful burns. To avoid danger, the power should be turned
OFF and you should WAIT at least one minute before any
wires are disconnected from or connected to the power
supply. Never touch the conducting metal of any wire.
Follow the directions in Appendix A for connecting the power supply to the CRT.
Check to see that the connections from the power supply to the high voltage and the
filament heater are correct, before you turn the power supply on. The power supply is set
up internally to supply a constant voltage of 300 V between the cathode and the screen
(you cannot change this voltage using the controls on the power supply). Using the
controls illustrated in Appendix A, you can control the voltages applied to the grid, the
focusing electrode, and one deflecting electrode of a pair. The other deflecting electrod of
the pair is connected to the same 30 V potential as the screen. You can use the grid voltage
to adjust the beam current (thereby controlling how bright is the spot of light on the
screen). You can use the focus control to bring all electron trajectories to a common focus
on the screen (a small bright spot on the screen).
Before you turn on the electric field between the deflection plates, find the CRT
orientation that gives no deflection of the electron beam. In this position the effect of all of
the outside forces on the electron is negligible.
Now apply a voltage across one set of deflection plates, noting how the electron beam
moves across the screen as the voltage is increased. How will you adjust the voltage level
and how will you measure it? Write down the range of voltages for which you can make a
good measurement. Repeat this procedure for the perpendicular set of deflection plates.
The electric field between two equally charged parallel plates (in Newtons per
Coulomb) is equal to the voltage between the two plates (in Volts) divided by the distance
between the plates (in meters).
Devise a measuring scheme to record the position of the beam spot. Be sure you have
established the zero deflection point of the beam spot.
How will you determine the strength of the electric field between the deflection plates?
What quantities will you hold constant for this measurement? How many measurements
do you need?
Write down your measurement plan.
Lab II - 19
PROBLEM #3: Deflection of an electron beam by an electric field
MEASUREMENT
Measure the position of the beam spot as you change the electric field applied to the
deflection plates. Make sure you take at least 2 measurements at each point for averaging.
Note: Be sure to record your measurements with the appropriate number of significant
figures and with your estimated uncertainty. Otherwise, the data is virtually meaningless.
ANALYSIS
Draw a graph of your Prediction equation of the deflection of the electron beam as a
function of the applied electric field.
Draw a graph using your measurements of the deflection of the electron beam as a
function of the applied electric field.
CONCLUSION
How does the graph based on your data compare to the graph based on your
prediction? If they are different explain why.
How does the deflection of the electron beam vary with the applied electric field?
State your results in the most general terms supported by your data.
Lab II - 20
PROBLEM #4
EFFECT OF VELOCITY ON DEFLECTION OF AN ELECTRON
You are continuing your attempt to design a better electron microscope. In the
Cathode Ray Tube (CRT), electrons are emitted at one end of an evacuated glass tube and
are detected by their interaction with a phosphorous screen on the other end. While
inside the tube, the electrons pass between two charged “deflecting” plates that can
change the path of the electron beam. Before you can test the sensitivity of the electron
microscope design, you will need to determine how the velocity of the electron leaving
the electron gun region of the CRT affects the position of the beam spot.
?
How does the deflection of the electron beam
vary with initial electron velocity?
EQUIPMENT
You will be using the Cathode Ray Tube described in the introduction and in
Appendix A. The applied electric field is created by connecting the internal deflecting
plates to a power supply.
PREDICTION
Calculate how the deflection of the electron beam spot changes as the initial velocity
of the electrons change.
Use this equation to make a graph of the deflection of the beam spot as a function of
the initial velocity of the electrons.
Lab II- 21
PROBLEM #4: EFFECT OF VELOCITY ON DEFLECTION OF AN ELECTRON
METHOD QUESTIONS
These questions are similar to those for Problem #5. If you have already completed
Problem #5 you should review your answers to those method questions. If not, you
should answer the following method questions.
1. Draw a picture of the important components of the CRT. Only include one set of
the deflecting plates shown in Figure 2. Draw a coordinate axis on this picture. Draw the
electron trajectory. On the trajectory, draw and label arrows representing the electron’s
velocity and acceleration in regions of different acceleration. Draw the electron
trajectory if there were no electric field between the plates. The difference between
where these two trajectories hit the CRT screen is the deflection.
2. Decide on the time intervals in which the electron has differing accelerations.
What forces cause these accelerations? Draw an arrow representing each force on your
picture. What forces are you assuming are negligible. In each region of different
acceleration make a motion diagram showing the electron's trajectory and the electron’s
velocity and acceleration when it enters the region, is in the region, and leaves the region.
Qualitatively describe the shape of the trajectory of the electron in each of the regions.
3. Determine the velocity of the electrons as they leave the electron gun in the CRT?
(see method question 6 of Problem #4)
4. Determine the acceleration of an electron as it travels through the region between
the deflecting plates by using dynamics (Newton’s second law). You can determine the
force on the electron in this region if you know the electric field. Is the electric field
constant in the region between the deflecting plates? What does that tell you about the
acceleration of the electron in that region?
The magnitude of the electric field between two equally charged parallel plates (in Newtons
per Coulomb) is equal to the voltage between the two plates (in Volts) divided by the distance
between the plates (in meters).
5. Use your drawing from step 1 and kinematics to determine the position and
direction of the electron as it enters the region between the deflection plates and when it
leaves that region. Write down the equation giving the electron’s position as it emerges
from the deflecting plates. Write another equation giving the electron’s direction.
6. Use your drawing from step 1, the position and direction of the electron as it leaves
the deflection plates, and geometry to write down an equation giving the position of the
electron when it hits the screen. This equation should give the total deflection from all of
the regions of the CRT.
7. Examine your equations giving the electron’s position at the screen. You can solve
them if the number of unknowns in your equations is equal to the number of equations.
Is it? If so, solve your equations algebraically for the deflection of an electron.
Lab II - 22
PROBLEM #4: EFFECT OF VELOCITY ON DEFLECTION OF AN ELECTRON
Otherwise, write down additional equations that relate some of the unknown quantities in
your equations to quantities that you know. Complete your solution by using the actual
numbers that describe your situation.
8. Does your solution make sense? If not, check your work for logic problems or
algebra mistakes.
EXPLORATION
WARNING: You will be working with equipment that
generates large electric voltages. Improper use can cause
painful burns. To avoid danger, the power should be turned
OFF and you should WAIT at least one minute before any
wires are disconnected from or connected to the power
supply. Never touch the conducting metal of any wire.
Follow the directions in Appendix A for connecting the power supply to the CRT.
Check to see that the connections from the power supply to the high voltage and the
filament heater are correct, before you turn the power supply on. The power supply is set
up internally to supply a constant voltage of 300 V between the cathode and the screen
(you cannot change this voltage using the controls on the power supply). Using the
controls illustrated in Appendix A, you can control the voltages applied to the grid, the
focusing electrode, and one deflecting electrode of a pair. The other deflecting electrod
of the pair is connected to the same 30 V potential as the screen. You can use the grid
voltage to adjust the beam current (thereby controlling how bright is the spot of light on
the screen). You can use the focus control to bring all electron trajectories to a common
focus on the screen (a small bright spot on the screen).
TAKING EXTREME CARE, change the grid voltage, and determine the range of
values for which the electron beam has enough current to produce a visible spot on the
screen. Changing this voltage changes the width of the potential well at the cathode, and
dramatically affects the flux of electrons that manage to tunnel out.
The electric field between two equally charged parallel plates (in Newtons per
Coulomb) is equal to the voltage between the two plates (in Volts) divided by the
distance between the plates (in meters).
Before you turn on the electric field between the deflection plates, find the CRT
orientation that gives no deflection of the electron beam. In this position the effect of all
of the outside forces on the electron is negligible.
Lab II - 23
PROBLEM #4: EFFECT OF VELOCITY ON DEFLECTION OF AN ELECTRON
Now apply a voltage across one set of deflection plates, noting how the electron beam
moves across the screen as the voltage is increased. Find a voltage across the deflection
plates that allows the deflection for the entire range of initial electron velocities to be
measured as accurately as possible.
Devise a measuring scheme to record the position of the beam spot. Be sure you have
established the zero deflection point of the beam spot.
How will you determine the strength of the electric field between the deflection
plates? How will you determine the initial velocity of the electrons? What quantities
will you hold constant for this measurement? How many measurements do you need?
Write down your measurement plan.
MEASUREMENT
Measure the deflection of the beam spot as you change the initial velocity of the
electrons in the beam but keeping the electric field between the deflection plates constant.
Note: Be sure to record your measurements with the appropriate number of
significant figures and with your estimated uncertainty. Otherwise, the data is virtually
meaningless.
ANALYSIS
Draw a graph of your Prediction equation for the deflection of the electron beam as a
function of the initial electron velocity.
Draw a graph of your average measurements of the deflection of the electron beam as
a function of the initial electron velocity. How do your uncertainties affect your graph?
Compare the sensitivity α that you measure to the value that you obtained from
Problem 1.
Lab II - 24
PROBLEM #4: EFFECT OF VELOCITY ON DEFLECTION OF AN ELECTRON
CONCLUSION
Did your data agree with your prediction of how the electron beam would deflect due
to the initial electron velocity? If not, why not?
How does the deflection of the electron beam vary with initial electron velocity?
State your results in the most general terms supported by your data.
Lab II - 25
APPENDIX A
CONNECTION OF THE CRT TO THE POWER SUPPLY
The cathode ray tube in your experiments is a type 5CP1A. The full specifications
for the tube are available at the course web site:
http://faculty.physics.tamu.edu/mcintyre/courses/phys208H/labs/5cp1a.pdf .
The tube requires connection to several power supplies that provide the voltages for
the filament, the grid, the focusing electrode, and the deflection electrodes.
The figure below shows the connections from the tube to the power supply. All
voltages are provided from this single supply. Before attaching any leads to the power
supply, you should attach the tube base to a demo tube that has its envelope removed, and
use your multimeter to trace the connections from the color-coded leads to the electrodes
of the tube. You can then make a clear association between the voltages that you are
applying to the tube and the electrodes to which the voltages are connected.
6.3 Vac
300 V
0-80 V
0-100 V
screen
focus
deflection
filament
Lab II- 26