Reyes
Charge and mass of the electron
Background:
Electric Force
(AKA Coulomb’s Force,
F
e
) is defined as:
(1)
Electric Field
E
(defined as electric force per unit of charge) is:
(2)
1
C
is 1 C
oulomb
Replacing the expression for the
electric force (1)
in to the
electric field (2)
we get:
(3)
Using the electric field (3), we can calculate the electric force (1) exerted upon any charge as:
(4)
Wh
en electric charges move, they generate a magnetic field which exerts an additional force, therefore, the total force
over a charged particle is:
(5)
The additional magnetic force depends on the charge itself (
q
), its speed (
v
)
and the magnitude and direction of the
magnetic field (
B
). It has been found that
F
m
is perpendicular to
v
and to
B
,
and it can be calculated as:
(5)
(
θ
is the angle formed by the vectors
and
)
If
and
are perpendicular (
θ
= 90°
), then
the magnetic force
:
(6)
(
charge
x
speed
x
magnetic field
)
Replacing the equation (4
)
for
electric force,
and (6)
for magnetic force
on equation (
5
)
for the total force
, we see that
the electromagnetic force acting upon a charge
q
will be given by:
(7) (AKA Lorentz’s Force.)
Joseph John Thomson’s experi
ment:
From the cathode, electrons with mass
m
and charge
e
are emitted. Some pass through the hole in the anode with a
speed
v
. If there is no other interaction, they would show a linear path. If an electric field is applied, then the electrons
would be de
viated towards the opposite charge of the electric field.
A magnetic field would also exert a force on the electrons. If the magnetic field is homogeneous, the electrons are
forced into a circular trajectory with radius R. In the circular motion, the part
icle feels the centripetal acceleration:
(8)
If we apply Newton’s 2
nd
law
(
F
=
ma
)
, the magnetic force (6) has to be equal to the mass, multiplied by the
centripetal
acceleration (8):
(9)
Rearranging this
expression (9) we get the relation
e/m
for the particles found in the cathode rays:
(9a
.
)
Unfortunately, the initial velocity
v
is not known, but if we apply the electric and magnetic field at the same time and
balance them, then
the particles return to a
straight pattern, and that is
the case:
or
(10)
From which:
(11);
t
herefore
, eq 9a may be rewritten as
:
(12).
Since the radius
R
, the electric and magnetic fields (
E
and
B)
can be measured, measuring
e
or
m
the other one could
be obtained. This led to
Robert Andrews Millikan’s Experiment
: Mullikan observed oil drops electrically charged
and suspende
d on air. An electric field would tend to make the
m
go up, and the gravitational field would pull them
down; when both forces balance each other:
(13)
Since
F
g
= m
g
(weight equals the mass times the acceleration
of gravity
), then
:
(14)
;
where
m
is the
mass of a drop and
q
its charge. Millikan found that the charge of the droplets
q
was always a multiple of:
e
= 1.591 x 10

19
C
or
q = ne
(
15
)
;
(where
n
= 1, 2, 3…
)
The value currently accepted is:
e
= 1.6022 x 10

19
C, and then
, from (12)
m =
9.1095 x
10

28
g.
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