# Lesson Video: Identifying Particles in Detectors Physics

In this video, we will learn how to determine the charge and mass of particles from the paths that they follow through particle detectors.

14:11

### Video Transcript

In this lesson, we’re going to learn how some properties of charged particles determine their expected motion in the presence of magnetic fields. We’ll then work backwards, starting with an observation of a particle’s motion, and determine from that properties of the particle like charge, mass, and charge-to-mass ratio.

Let’s start by considering the force acting on a particle as it moves in a magnetic field. Here we have a charged particle and its direction of motion is perfectly horizontal towards the right. We’ll call the charge of the particle 𝑄 and its speed 𝑣. We’ve represented a magnetic field in this diagram by an array of circles with x’s in them. The x’s represent that the magnetic field is pointing away from us. In other words, we’re looking in the same direction that the magnetic field is pointing. We’ll use the letter 𝐵 to represent the strength of this magnetic field.

Since the direction of the particle’s motion and the direction of the magnetic field are perpendicular, the force on this object when it’s inside the magnetic field is the charge of the object times its speed times the strength of the magnetic field. The direction of the force is perpendicular to both the direction of motion and the direction of the magnetic field, so either directly up or directly down. The particular direction depends on the sign of the charge of the particle. If the particle has a positive charge, then in this configuration, the force will be up. And if the particle has a negative charge, in this configuration, the force will be down. Regardless of whether the charge is positive or negative, the force is always perpendicular to the direction of motion.

There is another situation where an object moves subject to a force that is perpendicular to its direction of motion. And that’s an object moving in a circle. For an object moving in a circle, the force is perpendicular to the motion, and in this case, the force has a special name. It’s called the centripetal force. The reason there needs to be a force acting on an object moving in a circle at all is because the object is changing direction, and direction changes require a force. For a charged particle in a magnetic field, the force is perpendicular to the direction of motion, just like for an object moving in a circle. Therefore, a charged particle in a uniform magnetic field will also move in a circle with the force on the particle always pointing towards the center of the circle.

If the particle were negatively charged, the trajectory would curve downward in keeping with the direction of the force. Our goal is to relate this circular motion to the force that the charge experiences because of the magnetic field. To do this, recall that the centripetal force on an object moving in a circle is the mass of the object times the square of the object’s speed divided by the radius of the circular path. Since the force on our charged particle due to the magnetic field is in fact the centripetal force that our particle is experiencing, we can equate these two expressions. This gives us that the mass of our charged particle times the square of its speed divided by the radius of its path as it moves through the magnetic field is equal to its charge times its speed times the strength of the magnetic field.

In this context, we refer to the radius of the path as the cyclotron radius because the circular paths made by charged particles moving in magnetic fields are known as cyclotron motion. Let’s carefully examine the quantities that we have in this equation. We have the strength of the magnetic field, which describes the environment of the particle but not the particle itself. We have the speed of the particle and the radius of its path, which describe its motion. And we have the mass and the charge, which are properties of the particle itself. Remember, our goal was to relate particle properties to a description of its motion. So let’s collect all of the quantities that describe the motion on one side of the equation and all of the quantities that are properties of the particle on the other side.

We’ll accomplish this by dividing both sides by the mass and speed of the particle and also the strength of the magnetic field. On the right-hand side, 𝑣𝐵 divided by 𝑣𝐵 is just one. And on the left-hand side, 𝑚 divided by 𝑚 is one and one factor of 𝑣 in the denominator cancels one factor of 𝑣 in the numerator. This gives us exactly the sort of expression we’re looking for. We have information about the particle’s motion on one side of the equation and information about the particle itself on the other side.

Before doing anything further with this equation, let’s observe that the right-hand side 𝑄 divided by 𝑚 is the charge-to-mass ratio of the particle. The fact that the only appearances of charge and mass in this equation are as part of the charge-to-mass ratio tells us several things. Firstly, it tells us that two particles may have different charges and masses but undergo the same motion in a magnetic field because the ratio of their charge to their mass is the same. Furthermore, this equation tells us that we need some kind of independent measurement to establish the particular charge or particular mass of a particle since knowledge of just its trajectory — that is, its velocity, the radius, and the strength of the magnetic field — is only enough to specify the charge-to-mass ratio.

Finally, recall that negatively charged particles will move in the opposite direction to positively charged particles. Therefore, since particles and antiparticles have the same magnitude for their charge-to-mass ratio but opposite signs, they will follow trajectories of the same shape but opposite direction. Let’s illustrate these ideas by showing how the motion of a particle, as observed in a particle detector, can be used to determine information about its charge-to-mass ratio.

This picture shows an example of what we might observe in a particle detector. The black rectangle is the total area in which we can observe particles, and the white spiral is the path followed by one such charged particle. Detectors typically have uniform magnetic fields, so we expect that the paths followed by charged particles will be circles. However, instead of the circle that we expect, our picture clearly shows that the particle is moving in a spiral. This is because when charged particles accelerate, including by changing direction, they emit electromagnetic radiation. This radiation carries energy. So, by conservation of energy, as these particles start to circle around in a magnetic field, they emit radiation, lose energy, and therefore slow down.

Looking back at our equation that describes the motion of these particles, the charge-to-mass ratio is constant, as is the detectors magnetic field. Because those other quantities are constant, this tells us that the speed of the particle divided by the radius of the path is also a constant quantity. So as a particle’s speed decreases, the radius of its path also decreases, and so our particle follows a spiral instead of a circle. So, charged particles in detectors with uniform magnetic fields tend to follow spiral not circular paths.

Spirals do not have a single well-defined radius. Luckily, if we measure from the center of the spiral to the outer edge, we get approximately the radius that the particle would have if it was moving in a circle without emitting electromagnetic radiation. This is not fully accurate for quantitative calculations, but it is certainly good enough to qualitatively compare the tracks of two particles. For the particle that we’ve drawn without any knowledge of its speed or its charge-to-mass ratio, the only thing that we can really determine is the radius of its path. Luckily, information like speed and the charge is often available from other measurements.

Furthermore, if we detect two particles, we can often compare them qualitatively using only a little bit of extra information. Here, the detector is showing the paths of two charged particles. We’ll call them one and two. Now, let’s say that we’ve also determined that these two particles have the same initial speed. So by some external measurement, the particles have the same initial speed. And since they’re in the same detector, we know they’re experiencing the same magnetic field. Since these two quantities are the same, if we can compare the radii of their paths, we can compare their charge-to-mass ratios.

When we draw the radius of each path on the diagram, we can clearly see that the two particles actually follow paths with the same radius. But this means that all three quantities on the left-hand side of our equation, speed, magnetic field, and radius, are the same for both particles. This means that at least in magnitude, the two particles have the same charge-to-mass ratio.

There’s even more information that we can determine from this picture. The path of particle two and the path of particle one curve in opposite directions. Since they clearly started in the same place traveling in the same direction, the force on particle two must have pointed in the opposite direction to the force on particle one. But for two particles moving in the same direction in the same magnetic field, they will only experience forces pointing in opposite directions if they have opposite signs on their electric charge. So from only the detector output, we know the relationship between the sign of the charges of these two particles.

Furthermore, from a little bit of information about their initial speed, we’ve also managed to use the detector output to determine the relationship between the size of their charge-to-mass ratios. Putting these two pieces of information together, we know that particle one and particle two have charge-to-mass ratios with the same size but opposite sign, which is characteristic of the relationship between a particle and its antiparticle. In fact, seeing this particle one and particle two seem to have spontaneously appeared at the same position in space, it is almost guaranteed that they’re the particle–antiparticle product of pair production.

This means that in addition to charge and charge-to-mass ratio, we’ve also managed to relate the identities of particle one and particle two. If we later discovered that, for example, particle two was a positron, we would then conclude that particle one, its antiparticle, is an electron. So from just a picture of the paths followed by two particles in a uniform magnetic field and also a relationship between their initial speeds, we worked out several further relationships between these two particles and, with an additional piece of information, even managed to give a likely identification for both particles in the detector.

Great! Now that we’ve seen how to use our equation and qualitative information from particle detectors, let’s work through a quantitative example.

A charged particle moves through a uniform magnetic field. It moves along a circular path with a radius of 0.0200 meters. Its speed is 3.40 times 10 to the sixth meters per second. The strength of the magnetic field is 0.200 teslas. What is the charge-to-mass ratio of the particle? Give your answer in coulombs per kilogram to three significant figures.

Since this question is asking us to find the charge-to-mass ratio of a particle moving through a uniform magnetic field, it makes sense to consider the equation speed divided by magnetic field strength times radius is equal to charge-to-mass ratio. This gives us the charge-to-mass ratio of a particle moving in a uniform magnetic field with motion described by the speed and the radius of the path. Recall that we derived this equation by equating the magnetic force on a charged particle moving in a magnetic field to the centripetal force on any object moving in a circle.

Now that we have this equation, to calculate charge-to-mass ratio, all we need to do is find appropriate values for the speed, magnetic field, and radius. In this question, the values are simply given to us. We have 0.0200 meters for the radius, 3.40 times 10 to the sixth meters per second for the speed, and 0.200 teslas for the strength of the magnetic field. Plugging those values into our formula gives us an expression for the charge-to-mass ratio. Let’s start by evaluating the numerical portion, 3.40 times 10 to the sixth divided by 0.200 times 0.0200. Plugging into a calculator, this gives us 850 times 10 to the sixth.

We’re asked to report our answer to three significant figures. Since this number is written in the form of three digits, 850 times a power of 10, the three significant figures for this number are 850. We could either leave the number in this form, or to be absolutely clear with the significant figures, we could write it as 8.50 times 10 to the eighth. For the units, meters in the numerator divided by meters in the denominator is just one. And per seconds in the numerator divided by teslas in the denominator is just one divided by tesla seconds.

Aside from the fact that these are somewhat odd units, the question wants us to have our answer in coulombs per kilogram. So we need to convert one per tesla seconds to coulombs per kilogram. The most direct way to do this is to recall that one tesla is exactly one kilogram per coulomb seconds. So one tesla second is one kilogram second per coulomb second, which is just one kilogram per coulomb. Taking the reciprocal of both sides, we see that despite one per tesla second looking rather odd, it’s exactly equivalent to one coulomb per kilogram, which is precisely the units we want for our answer.

This one-to-one relationship is actually guaranteed from the definition of SI units and the fact that the tesla is the basic unit of magnetic field, the second is the basic unit of time, the coulomb is the basic unit of charge, and the kilogram is the basic unit of mass. Because one per tesla seconds and coulombs per kilogram are both valid units for charge-to-mass ratio and both only contain basic SI units, by definition they must be equivalent. So to three significant figures, the charge-to-mass ratio of our particle is 8.50 times 10 to the eighth coulombs per kilogram. As it happens, this is almost exactly the magnitude of the charge-to-mass ratio of both the muon and the antimuon.

Alright, now that we’ve seen how to apply this formula both qualitatively and quantitatively, let’s review what we’ve learned about identifying particles and their properties. We started by deriving the equation that the charge-to-mass ratio of a particle is equal to its speed divided by the radius of its path and the strength of the magnetic field in which it is moving. In the context of a particle detector, this strength is typically fixed so this equation effectively relates the charge-to-mass ratio of a particle to its motion.

We saw by way of example that this equation can be used to calculate the charge-to-mass ratio of a particle from knowledge of its speed, the radius of its path, and the strength of the magnetic field. And in fact, we can calculate any quantity appearing in this equation from knowledge of the other quantities. We also saw that particle detectors can show us the path of a charged particle as it moves through a magnetic field. In this case, the paths are spirals not circles because as the charged particle changes direction, it loses energy, which causes its speed to decrease with a corresponding decrease in radius.

Nevertheless, using the output of such detectors, we can still find the radius at any particular time. We can also determine if two particles have the same or opposite sign for their charge by looking at whether their paths curve in the same or opposite directions. Finally, if we include information like charge and initial speed that are available from other types of measurement, we can qualitatively compare the charge-to-mass ratio and the mass of two particles in a detector and sometimes even come up with quantitative values for those quantities. Sometimes it is even possible to use all this information to identify a particular particle by its path in the detector.