Department of Physical Electronics
Faculty of Science, Masaryk University, Brno
Physical laboratory 3
Task A Charge movement in electric and
magnetic ﬁelds
Tasks
1. Verify formula (2) for the focal length of a short magnetic lens. Plot a graph showing
Ua = f(I2
f ) and calculate the focal distance f from the slope.
2. Verify the validity of formula (10) for the magnetic deﬂection of the electron beam. Plot
graphs showing if the deﬂection magnitude y as a function of Iv and Ua corresponds to the
formula (10).
Theory
An electron beam is used in many diﬀerent electronic devices for various purposes. In such
devices, it is generally needed to focus or deﬂect the electron beam. An old-style cathode ray tube
(CRT) monitor is one of the simplest examples as it can also be used for viewing this beam on a
luminescent screen.
A charged particle beam can be focused by a short magnetic lens. A short magnetic lens is a
coil having a rationally symmetric magnetic ﬁeld set so that it is interacting with the beam over
a negligibly short part of its trajectory, focusing an originally divergent beam into a point on the
screen. The focal length of the short magnetic lens f can be calculated as
f = 98
r
n2
Ua
I2
f
, (1)
where r is the radius of the focusing coil, Ua is the acceleration voltage used for acceleration of
the charged particle beam, n is the number of the coil threads and If is the current going through
the focusing coil. In order to determine the focal length f and verify the validity of formula 1, we
can rearrange it into
Ua =
fn2
98r
· I2
f . (2)
If the position of the short magnetic lens does not change throughout the measurement, the graph
of the dependence of Ua on the square of the focusing current If (i.e. Ua = f(I2
f )) while keeping
the point focused will be a line. Now, if we know the number of the coil threads and the dimensions
of the coil, we can derive the focal length of the short magnetic lens.
The deﬂection of the electrons moving in a magnetic ﬁeld happens through the Lorentz force
(similarly to focusing) written as
F = −e(v × B). (3)
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Figure 1: Force interaction of the magnetic ﬁeld with an electron beam. The electrons enter the
deﬂecting ﬁeld B in time t0 = 0 and stay in it for the time t1 along the path L1. No further
deﬂection takes place along path L2 during time t2. The Lorentz force there is zero and the
electron path is a line.
Let us assume the magnetic induction is perpendicular to the electron movement and that it
interacts with the electrons only along the L1 part of their path (see 1). Now we can describe the
electron movement along the y axis by the following equation derived from equation (3)
d2y
dt2
=
e
m
· vxB. (4)
After integrating equation (4) we obtain:
dy
dt
=
e
m
· vxBt + C. (5)
Assuming that the derivation dy/dt = 0 in time t0 = 0, we obtain the integration constant
C = 0. The speed along the y axis that the electron gains after going through the magnetic ﬁeld,
therefore, is
vy =
dy
dt
|t=t1 =
e
m
vxBt1, (6)
where t1 is the total time of ﬂight of the electron through the deﬂecting ﬁeld. Substituting
t1 = L1/vx, where vx is the electron speed along the x axis, which we can obtain from the
accelerating voltage as:
eUa =
1
2
mv2
x => vx =
2eUa
m
1
2
, (7)
we obtain
vy |t=t1 =
e
m
· B · L1. (8)
The deﬂection of the electron beam on the screen will be approximately
y = vyt2, (9)
where t2 is the time of ﬂight from the deﬂecting systems to the screen, t2 = L2/vx.
Substituting vy from equation (8) into equation (9) , we obtain the formula for the electron
beam on the screen
y =
e
2m
L1L2
B
√
Ua
. (10)
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Figure 2: Schematic drawing of the used CRT with magnetic focus and magnetic deﬂection. K cathode,
W - Wehnelt cylinder, A1 a A2 - anodes, FC - focusing coil, V C - two pairs of deﬂection
coils, S - screen.
The magnetic induction of the deﬂecting ﬁeld B is directly proportional to the current ﬂowing
through the deﬂecting coil Iv. For the veriﬁcation of formula (10) we will measure the dependence
y = f1(Iv)
at constant accelerating voltage Ua and the dependence
y = f2(Ua)
at constant deﬂection current Iv.
Our study will be conducted using a CRT screen, which is, in essence, a multi-electrode
electron tube with an electron jet and a luminescent screen – see Fig. 2. The electron source is a
hot cathode. These electrons are repulsed by a negative potential of a grid (the so-called Wehnelt
cylinder) into a space in the cylinder, creating a narrow axially divergent beam. This beam then
passes through the centre of two anodes A1 and A2, which are at a high positive potential of up
to 20 kV with respect to the cathode. The anodes accelerate the electron beam to a velocity high
enough to cause excitation of the luminescent layer on the screen. The anodes also take part in
the (this time electrostatic) focusing of the beam.
Inside the CRT, the pressure is reduced to a value, where the mean free path of the electrons
is higher than the distance between the cathode and the screen. If this wasn’t the case, the
movement of the electrons would be aﬀected by their collisions with the present gas molecules. A
focusing coil (a short magnetic lens) is placed outside of the tube on its circumference. It focuses
the electrons to a point on the luminescent screen.
Two pairs of deﬂection coils are placed on the outside of the tube after the focusing coil. One
pair is for deﬂection in the vertical direction, and the second is for the horizontal deﬂection of the
electrons. Also, the electric ﬁeld deﬂection can be used in some devices such as oscilloscopes.
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1 2 3
4 5
Figure 3: Measurement apparatus:: (1) Potentiometers of the control of the acceleration voltage
(Ua), focus (If ) and deﬂection current (Iv). (2) AC power source for the acceleration voltage and
deﬂection current. (3) DC power source for the focus current. (4) Voltmeter for the measurement
of the acceleration (anode) voltage. (5) Screen.
Figure 4: Electrical drawing for electrical connection of the measurement apparatus.
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Experimental procedure
Connect the CRT to the power sources, voltmeters and ammeters as per Fig. 4. After connection
and checks of all the circuits, set all the potentiometers so that the supplied voltages are at a
minimum. Then carefully increase the voltage, one circuit at a time and check the working of the
individual circuits on the screens and meters.
When verifying eq. (2), change the anode voltage and focus the picture on the screen by
changing the focus current.
When studying the deﬂection, AC current ﬂows through the deﬂection coil giving rise to an
AC magnetic ﬁeld. Therefore an oblong track of the beam will be visible on the screen. Its length
is proportional to the amplitude of the deﬂection current Iv.
Owing to the luminescent screen’s inertia and the human eye, we do not see a moving point,
and we see a line. When measuring the y = f(Ua) we will start with the lowest values of the
Ua where the observed track is the widest when being still visible. We will set its length to be
nearly over the whole width of the screen by setting the appropriate focus current. Now, when we
increase the Ua, the track width will decrease (see (10)), and it will not go over the screen width.
The track will de-focus with any change of the Ua and it needs to be re-focused by changing the
focus current at every step. Measure the y = f(Iv) dependence for two diﬀerent constant values
of Ua and then the y = f(Ua) dependence for two diﬀerent constant values of Iv.
Tip for the report: When graphically verifying that the measured data correspond to the
theoretical formula, it can be advantageous to convert the tested formula to a linear dependence
of the tested parameter.
The focus coil has a radius of r = 2 cm an the number of threads of n = 1000.
!!CAUTION!! The completed high voltage rectiﬁer is capable of supplying a high current
that can cause injury when handled carelessly!
Real-world use
Deﬂection and focusing of the electron beams are necessities for the practical use of electron beams.
Let us take an example of a scanning electron microscope (SEM), which is in principle similar to
the studied CRT. It comprises an electron source (electron gun) and diﬀerent lenses that focus the
electron beam onto the sample. In the CRT, this electron beam lights up the luminescent layer
on the screen, while in the SEM, it interacts with the sample instead. The lens system reduces
the diameter of the electron beam on the surface of the sample to maximise the resolution of the
SEM. The practical beam diameter nowadays is approximately in the order of a few nanometres.
The optical system also governs the beam current (number of electrons impacting the sample per
unit of time) and its optical properties. A whole system of coils is used to achieve such control of
the beam, whereas only one coil is used in the laboratory CRT.
Similarly to the CRT also SEM moreover contains a deﬂecting coil system. Their task is to
change the beam’s position on the sample and acquire the image in a point-by-point manner.
In both the SEM and the CRT it is necessary to deﬂect the electron beam in two axes. The
maximum deﬂection current (in the case of magnetic deﬂection) governs the maximum deﬂection
and, therefore, the maximum magniﬁcation of the SEM. As diﬀerent beam energies (diﬀerent
acceleration voltages) are used in electron microscopy, it is necessary to know the dependence of
the deﬂection magnitude on the accelerating voltage. For example, when the operator images a
given viewﬁeld and changes the beam energy, the microscope system needs to change the deﬂection
current and the acceleration voltage to keep the same viewﬁeld.
References
[1] Fuka, Havelka: Elektřina a magnetismus, SPN, Praha 1965
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[2] Ondráček Z. : Elektronika pro fyziky, Skriptum Přír. fak. MU, Brno 1998
[3] Sedlák B., Štoll I. : Elektřina a magnetismus, Vydavatelství Karolinum, Academia Praha 1993