Laboratory Exercise 7.
Determining
e
m
of the Electron
Cathode tube
The cathode tube (Figure 1) is an essential part of any TV, monitor, oscilloscope, etc. It consists
of a glass vessel which has a heated electrode (the cathode) at the one end, a second electrode (anode)
having positive potential relative to the cathode, and a screen at the other end. The cathode emits
electrons, which accelerate in the electric field between the cathode and the anode. The distance
between the anode and the screen is traveled with constant (usually very high) speed after which the
electrons bombard the screen. The screen itself is covered with a special material, scintillator, which
emits light when the high-energy electron hits the surface. For the observer, this process is represented
as the appearance of a light spot at the place where the electron impacts with the screen.
Figure 1: A cathode tube with a beam of electrons, curved due to a magnetic field
The potential difference between the cathode and the anode in the cathode tube is denoted by U .
U has value greater than 0, that is, the anode is positive with respect to the cathode.
To find the speed of an electron, which has been accelerated in an electric field between two points
having potential difference U , we can use conservation of kinetic plus potential energy:
1
Ek = mv 2 = Ep = eU
2
Here Ek is the final kinetic energy, while Ep is the initial potential energy. The charge of the
electron is denoted by e . We derive the following expression for the speed:
r
v=
2U
e
m
(1)
If the beam of electrons moves in a homogeneous magnetic field perpendicular to it, the Lorentz
force causes it to curve. In our case the homogeneous magnetic field causing the curving is the earth’s
magnetic field.
1
v
F
Figure 2: The Lorentz force acting on a charged particle in magnetic field. The field is perpendicular
to the white sheet.
The Lorentz force
The Lorentz force F~ , which acts on electrons that move with velocity ~v perpendicular to the
~ is perpendicular to both ~v and B
~ (see Figure 2). Its magnitude is:
magnetic field B,
F = evB
The Lorentz force F~ causes the electrons to move along a piece of a circle with centripetal accel2
eration vR . Therefore the radius R of the circle is related to v , B and me :
F = evB = m
v2
R
so
e
v
=
m
RB
(2)
From equations (1) and (2) we derive that
e
2U
= 2 2
m
R B
The radius R and the magnetic field B will be determined experimentally (see below).
Determining the radius R experimentally:
Any point on a circle has coordinates, (relative to a coordinate system centered at the center of the
circle) satisfying x2 + y 2 = R2 . In our case x = l , where l is the distance between the anode and the
screen and y = R − ∆y (Figure 3). Then we get
R2 = l2 + R2 − 2R∆y + (∆y)2
Therefore the radius is expressed in terms of things that we can measure as follows:
R=
l2 + (∆y)2
2∆y
2
Figure 3: Determining geometrically the radius of the eletronic beam
In practice the determination of ∆y is done by rotating the cathode tube around its axis, while keeping
it perpendicular to the earth magnetic field. The bright spot on the screen should describe a circle
whose radius is ∆y .
Measuring the earth magnetic field
In order to measure the earth magnetic field in our lab we use a rectangular frame with a coil of
wire wound around it. The frame is made as a torsion balance. The frame must be orriented in such
a way that two of the sides (those parallel to the axis of rotation) are perpendicular to tha magnetic
field and at the same time the plane of the frame is parallel to tha magnetic field. As it turns out, the
magnetic field of the earth at our lattitude goes at an angle roughly 45o to the horizontal, its direction
being from South, high, to North, low. When electric current is run through the coil the forces on the
two opposite sides create torques that cause the frame to turn slightly. By the amount of rotation we
can estimate the torque and from this the forces. The force on a straight piece of wire of length L ,
~ , when electric current I is running along it, is:
perpendicular to the magnetic field B
F = BIL
This can be easily derived from the basic expression for the Lorentz force.
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ANSWER SHEET
Laboratory Exercise 7.
DETERMINING
e
m
OF THE ELECTRON
Name:
Date, hour:
Instructor’s signature:
RESULTS
Determining R experimentally:
Value of l:
Measured value of the radius of the circle on the screen, ∆y:
Calculated value of the radius of the electron beam R:
Measuring the earth magnetic field:
Number of coils N :
Length of the two sides of the frame, perpendicular to the magnetic field:
Electric current in the coil:
Experimental value for the torque of the magnetic force (torque is the product of the force and the
distance to the axis):
Experimental value for the magnetic force F~ :
~
Calculated value for B:
Calculated value for
e
:
m
DISCUSSION:
What would happen to the radius of the electronic beam if we increse the accelerating voltage U
10 times? In a contemporary color TV the accelerating voltage is about 25000 V. If the tube is 0.5 m
long, what will be the deviation of the beam hitting the screen, due to the earth magnetic field? (Using
~ that we measured.)
the value for B
What is the error of your experimental value of me compared to the known one (as a percentage of
the known one. What do you think is the main source of error in this experiment?
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