DEFLECTION OF ELECTRONS BY AN ELECTRIC
FIELD
Ken Cheney
ABSTRACT
The deflection of electrons by a constant electric field is investigated using a CRT. The
dependence of the deflection on the accelerating voltage, deflecting voltage, and geometry is
compared with theory.
PHYSICS INVOLVED
The physics consists of applications of mechanics, the force on a charge in a uniform electric
field, and the definition of electric potential.
Electrons are accelerated by an accelerating voltage Va and are moving in the x direction with
a velocity vx . The electrons move between two deflection electrodes of length l in the x direction.
The electrodes have a voltage Vd between them and are separated by a distance d in the y direction.
The resulting force on the electrons gives them a velocity v y in the y direction. The electrons
leave the deflecting electrodes and travel in a straight line to a screen located a distance L from
the deflecting electrodes. The velocity in the y direction results in a deflection Y in the y direction
relative to where the electrons strike the screen with no deflection.
The initial velocity in the x direction is obtained by equating the energy the electrons obtained
by going through the accelerating voltage to the kinetic energy they acquire:
1
qVa = mvx2
2
The electric field due to the electrodes is calculated from the energy required to move a charge
q from one electrode to another against the electric field. i.e.
⌠ Eqdy
⌡
Vd ≡
= Ed
q
E = Vd /d
So
The force acting in the y direction is calculated from the electric field in the y direction:
F = qE = q(Vd /d)
This force gives an acceleration in the y direction:
F qVd
a= =
m dm
The time the electrons spend between the plates is given by:
t = l/vx
and the velocity acquired in the y direction is given by:
v y = at
LB1CED.DOC 1.064, May 3, 2002
Ken Cheney
1
The distance the electron moves in the y direction while being accelerated by the electrodes is
given by:
1
y = at 2
2
substuting from above
2
1  qVd   l 
 
= 
2  dm   vx 
 1 l 2  Vd
=

 4 d  Va
Equation 1
After leaving the deflecting electrodes the electrons travel in a straight line until they strike the
screen. The electrons have traveled a distance L in the x direction and a distance D in the y
direction. vx and v y form a triangle similar to that formed by L and D:
D/L = v y /vx
Solving for D and substituting for the velocities from above gives:
vy
D =L
vx
 lL  Vd
= 
Equation 2
 2d  Va
Adding the y motions for Equations 1 and 2 gives a total deflection in the y direction of Y:
 1 l 2 lL  Vd
Y =
+ 
 4 d 2d  Va
Equation 3
APPLICATIONS
From the point of view of someone applying electron deflection the most interesting part of
Equation 3 is the simple linear dependence of deflection on the deflecting voltage. If you want
to use an oscilloscope as a voltmeter this linear dependence makes life very easy. Once the
oscilloscope is calibrated for one voltage it is correct for all voltages.
If you want to direct a beam of charged particles by deflecting plates no elaborate calculations
are required, twice the deflecting voltage produces twice the deflection. This is useful from
scanning electron microscopes to Star Wars type applications.
Technology is full of applications of beams of charged particles, usually electrons. The most
familiar of these is the oscilloscope. Others examples are electron microscopes, electron
welding, cyclotrons, linear accelerators, etc.
In all cases the design and operation is made easier by the linear relation between the deflection
voltage and the deflection.
EXPERIMENT
EQUIPMENT
The central piece of equipment is a Cathode Ray Tube, CRT, as is found in an oscilloscope.
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DEFLECTION OF ELECTRONS BY AN ELECTRIC
FIELD Ken Cheney
A separate high voltage power supply provides the power for the filament that heats the
electron source (6.3 v AC), and two high voltages which accelerate and focus the beam of
electrons. The accelerating voltage is the sum of these two high voltages.
N.B. The two high voltages both accelerate and focus the electron beam. Therefor when
the accelerating voltage is changed the beam may be defocused and turn into an ugly blob
instead of the neat dot desired. Do not panic, just adjust the high voltages a bit and the dot
will return. At really low accelerating voltages nothing will eliminate the blob. The repulsion
of the electrons is simply blowing up the beam before it reaches the screen.
Three D cells, flashlight batteries, are used to supply a negative bias to avoid drawing too
much current from the electron source and damaging the screen.
A last power supply provides the deflecting voltage. This power supply can be reversed to
deflect the electrons in either direction. This voltage is measured by a separate Digital Volt
Meter (DVM or DMM) and is therefore known very accurately.
The sketch included shows how the wires are actually connected. Notice that the CRT’s
deflecting electrodes are connected to flat TV lead in wire. Don’t forget to ground the
unused electrodes.
The connections to the rear of the CRT are rather fragile, be careful.
PROCEDURE
The dependence of the deflection of the electron beam on the accelerating and deflecting
voltages is to be checked.
Use at least four accelerating voltages; as small as practical, as large as practical, and voltages
in between.
For each accelerating voltage use as wide a range of deflecting voltages as possible, i.e. use
the whole width or height of the CRT screen.
It is rather hard to read the screen accurately, the screen is quite small. It is easier to adjust
the deflecting voltage so it puts the dot produced by the electron beam on one of the lines
on the CRT screen and then read the deflecting voltage accurately from the DMM.
ANALYSIS EXPECTED
DEFLECTING VOLTAGE
Plot the deflection versus the deflecting voltage for each of the accelerating voltages. It is
most convincing if you use FITS to get a good idea of just how linear the relation is. FITS
will also give you the best fit equation for the deflection which you will want for the next
part.
ACCELERATING VOLTAGE
To appreciate the effect of the accelerating voltage of the deflection it is useful to put all
of the above plots on the same graph. You will notice that the higher the accelerating
voltage the less the deflection, i.e. the beam is said to be "stiffer".
If you believe computers should do the graphing but are unsure exactly how, you could look
at the sample program DATAPLOT.PUB.PHYSCI on the HP3000. This program shows
how to plot your data and the theoretical curves that FITS gives you.
DEFLECTION OF ELECTRONS BY AN ELECTRIC FIELD Ken Cheney
3
Theory says that for constant deflecting voltage the deflection should vary as 1 over the
accelerating voltage. You can check this prediction with the data you already have by
chosing any deflecting voltage and using the best fit equations to calculate the deflection
for each accelerating voltage. This will give you the data to check the dependence of
deflection on accelerating voltage. You can use FITS to do this or just plot deflection versus
1 over accelerating voltage.
DETAILED CALCULATION
For many applications it is enough to know that deflection is a linear function of deflecting
voltage. Still it would be more satisfying to know that we can calculate exactly what the
deflection will be from first principles, i.e. Equation 3. Try this.
Notice that the real electrodes are not simple parallel plates but diverge on the downstream
end. This divergence allows a greater deflection of the electrons than would be possible
with parallel plates where the electrons might hit the electrodes. You will need to use some
average separation for the electrodes.
WRITE UP EXPECTED
A formal write up. You can omit the introduction. Include enough sketches to identify all
measurements used.
GRADING CONSIDERATIONS
Pay particular attention to:
Linearity
Why the deflection may not match the detailed prediction well.
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DEFLECTION OF ELECTRONS BY AN ELECTRIC
FIELD Ken Cheney