Video: Visualizing and Understanding a Given Setup

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

07:16

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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.