The first is the experiment of Joseph John Thomson, who first demonstrated that atoms are actually composed of aggregates of charged particles. Prior to his work, it was believed that atoms were the fundamental building blocks of matter. The first evidence contrary to this notion came when people began studying the properties of atoms in large electric fields.

If a gas sample is introduced into the region between two charged plates,
a current flow can be observed, suggesting that the atoms have been
broken down into charged constituents. The source of these charged particles is
a heated cathode that, in fact, causes the atoms of the sample to
ionize. These were known as *cathode rays*.
In 1897, Thomson set out to prove that the cathode rays produced from
the cathode were actually a stream
of negatively charged particles called electrons.
(See Figure 1.8 in the textbook for Thomson's experimental setup).
From Maxwell's
theory, he knew that charged particles could be deflected in a
magnetic field. A schematic of the experimental setup is shown below:

We now zero in on the field region and set up a coordinate system as shown in the figure below:

In this coordinate system, electrons enter the region between the plates with an (unknown) velocity in the -direction. In order to determine this velocity, electric and magnetic fields are both applied, and each gives rise to a force on the electron. These forces are in the -direction. The electric force , where is the magnitude of the electric field, and the magnetic force is , where is the magnitude of the magnetic field, and is opposed to the force on the electric field.

If these forces balance, then there will be no deflection
of the electron in the -direction, i.e. all of the
electrons' motion will be along the -direction, which
was the initial direction when they entered the field region.
If the forces balance, then the total force on the
electrons will be zero, that is or

from which the unknown velocity can be determined as

Next, the magnetic field is switched off, so that the total force is due entirely
to the electric field.
Since the force is non-zero, if the charge carriers can be
deflected by the force, this provides evidence for their
being fundamental particles. If they *are* fundamental
charged particles, then they should have a well defined
mass and charge. In this second part of the
experiment, the specific trajectory
followed by the particle will be used to determine
the ratio of the charge to the mass of the particle.

When there is only an electric field, then there is a
nonzero force in the -direction but no
force in the -direction. Thus, this problem is exactly
the same as that of a projectile in a gravitational field.
As can be done in the projectile problem, the and
motion of the electrons can be analyzed separately and
independently.

In the -direction, the motion is very simple because there
is no force in this direction. The electrons simply move
with a constant velocity , which we already determined
has the value . Note that this value is correct
even though there is no magnetic in this part of the
experiment! It is just the velocity we determined from
the previous part of the experiment, and this value
has not changed. Thus, as a function of time , the
-position of the electrons is

The force in the -direction is a constant, hence motion in the -direction
is analogous the gravitational force. The constant force
gives rise to an acceleration , and
the -position at time is then

The electric field is tuned such that the particle traverses the entire plate region in the time required for it to strike the positive plate. Let the total distance travelled be , as shown in the figure. The time required to traverse the plate region () is

This is also the time required to move a distance in the direction:

Solving the above for the ration gives

Thus, using his experimental apparatus, Thomson was able to determine the charge-to-mass ratio of the electron. Today, the accepted value of is 1.758819610 Ckg.

In 1906, Robert Millikan was able to determine the value of the charge on the electron in his ``oil drop'' experiment. A schematic of his experiment is shown below:

Then using Thomson's value of , he calculated the value of . In his experiment, Millikan used a fine spray of ionized oil droplets, which he allowed to be acted on by gravity but to which he also applied an electric field in the direction opposite gravity, i.e. up. By tuning the electric field, he balanced the force due to the pull of gravity and the electric field force so that the drops remained suspended in space. Thus, if a drop has a positive charge and a mass , the force balance condition becomeswhere is the magnitude of the electric field. Solving for the charge-to-mass ratio, we have . In order to determine the mass, the drops are allowed to fall in the gravitational field without the influence of the electric field. In this case, there are still two forces acting on the drops. One is the gravitational force and the other is the frictional force due to air resistance. Since this force is proportional to the velocity, it vanishes when the drop is stationary, which is why it does not need to be taken into account when the electric field is on. However, when the drops are allowed to fall, the total force is , since friction opposes gravitational force. Here, is the coefficient of air resistance. Thus, according to Newton's law of motion

Assuming the drop starts initially at rest, the equation can be solved for , yielding

Since is very small and is relatively large, the exponential factor quickly decays to 0, leaving simply a constant velocity known as the

The currently accepted
values of and are: