### Video Transcript

An experiment is done such that a
beam of electrons enters a rotating cylinder and strikes the black target area, as
shown in the diagram. The angle between the black target
area and the slit is known, along with the cylinder’s radius and the speed of
rotation. Relying only on this data, which of
the following properties of the electrons can be determined from this
experiment? (a) Speed, (b) charge, (c) mass,
(d) magnetic moment, (e) spin.

Whenever we see a question like
this, there are three steps we should take. We should understand the
experimental apparatus. This includes all of the various
parts of the experiment and how they interact with each other. The next thing we need to
understand is what data and measurements and information we have about the
experiment. These data include known
information about the apparatus, things that we actually measure, and even
fundamental constants that are relevant to the experiment. Finally, the last step in a
question like this is to use physics to connect what we know about the setup and the
data to reach a conclusion. In this case, the question itself
is to determine which conclusions we could reach from our knowledge of the
experiment and physics.

Let’s start by understanding the
experimental setup. As shown in the diagram, there’s a
rotating cylinder with a target area and a slit. A beam of electrons enters the
cylinder through the slit, travels across the interior of the cylinder, and reaches
the other side. While the electrons are moving
through the interior of the cylinder, the cylinder is rotating. And it’s rotating in such a way
that the black target area rotates directly into the path of the electrons in time
for them to reach the back of the cylinder. Now, we understand how the
experiment works. Let’s make one more observation
that will be useful to us later. That observation is that the beam
of electrons only interacts with the cylinder at two points: when it enters the
cylinder through the slit and when it strikes the black target area at the back of
the cylinder.

Let’s now move on to understanding
the information that we know about the experiment. The first thing we’re told in the
question is that we know the angle between the black target area and the slit. We’ve drawn that angle on the
diagram and given it the symbol 𝜃 naught. Knowing this angle, we can also
determine the supplement of this angle, which is the angle that the black target
area moves through to intercept the beam of electrons. This angle is 𝜋 minus 𝜃 naught
radians because this angle plus the angle between the slit and the target area makes
one-half of a circle or 𝜋 radians.

The second thing we know about the
cylinder is its radius. We’ve added this radius to the
diagram and labeled it with the symbol 𝑟 for radius. Finally, the last thing that we
know about the cylinder is its speed of rotation. We’ve labeled this known speed on
the diagram with the symbol 𝜔. For a rotating object, it’s
appropriate to define the speed of rotation as the rate at which an angle about its
center of rotation changes. Symbolically, we can write that 𝜔
is equal to Δ𝜃 divided by Δ𝑡, where Δ𝜃 is the change in angle and Δ𝑡 is the time
interval corresponding to that change. The dimensions of 𝜔 are then angle
per time, or in units radians per second.

We now understand all of the
information that we have about this experiment. The only fundamental constant we
might expect to come into play is the charge on an electron. However, as we shall see in a
moment, the electron charge does not actually play a role in this experiment. We are now ready to use physics to
determine what conclusions we could reach from this experiment. Let’s first note one kind of
physics that is not going to come into play, and that is electromagnetism. Recall that the electrons only
interact with the cylinder when they enter through the slit and when they strike the
black target area, neither of which are explicitly electromagnetic interactions.

Furthermore, the information given
doesn’t tell us about any charge, electromagnetic field, or potential associated
with any part of the apparatus. Therefore, even though the
electrons themselves are charged, there is no possibility for electromagnetic
interactions with the cylinder itself. If there are no electromagnetic
interactions, we also cannot measure any electromagnetic properties, which means,
for example, that we can’t measure the charge as in choice (b) or the magnetic
moment as in choice (d). And choice (e), spin, is just the
intrinsic magnetic moment of the electrons, which is another electromagnetic
property. So we also couldn’t measure
that. So just by noting this lack of
electromagnetic interaction between the electrons and the cylinder, we’ve managed to
reduce our choices to (a), speed, and (c), mass.

Let’s now make an educated guess as
to which of speed or mass could be determined and then try to determine that
property to confirm our guess. The quantities we know are 𝜃
naught, so we have information about angle, 𝑟, so we have information about
distance, and 𝜔 which is angle per time, so we also have information about
time. Recall that speed is distance
traveled per time. Mass, on the other hand, is a
measure of how much matter is in an object and is itself one of the base
quantities. Looking back to our known
quantities, to determine speed, we would need to know distance and time. And we have available to us
distance and time. On the other hand, angle, distance,
and time do not contain the base quantity mass. So it’s a good guess that speed of
the electrons is the property that can be determined from this experiment.

So let’s try to determine speed
from what we know. We’ll need a distance traveled by
the electrons and the corresponding time that it took to travel that distance. There is only one distance that we
know for sure the electrons traversed in this experiment. And that’s from the slit to the
back of the cylinder where they struck the black target area. But we know the size of this
distance. It’s just the diameter of the
cylinder, which is twice the radius. Now, we need to know the time that
it took for the electrons to traverse this distance. We don’t know directly what this
time is, but we do know the angle through which the cylinder rotated in that time
and the speed of rotation. We can therefore use our definition
for the speed of rotation to determine the time interval corresponding to this angle
of rotation.

Plugging in, we have 𝜔, the speed
of rotation, is equal to 𝜋 minus 𝜃 naught, the angle rotated through, divided by
Δ𝑡, the unknown time that we’re looking for. To solve for Δ𝑡, we multiplied
both sides by Δ𝑡 divided by 𝜔. On the left-hand side, 𝜔 divided
by 𝜔 is one, and we’re left with Δ𝑡. And on the right-hand side, Δ𝑡
divided by Δ𝑡 is one, and we’re left with 𝜋 minus 𝜃 naught over 𝜔. And this is exactly the formula
that we’re looking for, the time for the electrons to traverse the cylinder in terms
of the information we have about the experiment.

Thus, we have found the distance
traveled by the electrons and the time that it took to travel that distance both in
terms only of the information we have about the experiment and numerical
constants. Plugging this distance and time
into the definition of speed, we would have an expression for speed only in terms of
known quantities. This in turn means that the
knowledge that we have is sufficient to determine the speed of the electrons. And thus, choice (a), the speed, is
a property of the electrons that can be determined from this experiment.