Lesson Video: Charge-to-Mass Ratio Physics

In this video, we will learn how to calculate the charge-to-mass ratio of different particles.

11:40

Video Transcript

In this video, we’re going to define a quantity called the charge-to-mass ratio, which is useful in describing how charged particles move in the presence of electromagnetic fields. As its name suggests, the charge-to-mass ratio is an object’s charge divided by its mass. We can represent this ratio symbolically by writing 𝑄 divided by 𝑚, where 𝑄 is the charge of the object and 𝑚 is its mass. Any unit of charge divided by any unit of mass is a valid unit for charge-to-mass ratio. Some common choices are coulombs per kilogram in SI units and elementary charge per unified atomic mass unit, which is commonly used in the field of chemistry. Because there are so many choices for these units, it’s always important to report both the numerical value and the units when working with charge-to-mass ratio.

To see these ideas concretely, let’s look at the charge-to-mass ratio for some common particles. In this table, we show the charge, mass, and charge-to-mass ratio of three particles: the proton, the muon, and the 𝛼 particle, which is a helium nucleus. Let’s focus on some of the qualitative ideas that we can learn from this information. Let’s first observe that the charge-to-mass ratio of the muon is negative because the charge of the muon is itself negative. In fact, the sign of a charge-to-mass ratio of any particle is always the same as the sign of its charge. So charge-to-mass ratio can be positive or negative, or, in the case of neutral particles with nonzero mass, charge-to-mass ratio can even be zero.

For particles and antiparticles then, since they have the same mass and charges with the same magnitude but opposite sign, their charge-to-mass ratios will also have the same magnitude but opposite sign. This is, of course, equivalent to saying that the charge-to-mass ratio of an antiparticle is negative one times the charge-to-mass ratio of its corresponding particle. Turning back to our table, we also see that the 𝛼 particle has half the charge-to-mass ratio of the proton. This is because even though the 𝛼 particle has twice the charge of the proton, it also has four times the mass. This comparison is only possible because we’ve reported the charge-to-mass ratio of the proton and the 𝛼 particle in the same units of coulombs per kilogram.

However, to compare these to the charge-to-mass ratio of the muon, we need to do some unit conversion since we’ve reported the charge-to-mass ratio of the muon in units of elementary charge per unified atomic mass unit, since those are the units of charge and mass that we reported in the table. This illustrates the importance of always reporting the numerical value as well as the units when working with charge-to-mass ratio. Alright, let’s now see how charge-to-mass ratio can help describe the motion of charged particles in electromagnetic fields. Here we have an object of charge 𝑄 and a mass 𝑚 moving with velocity 𝑣 through an electric field 𝐸 and a magnetic field 𝐵. The force on such an object can be expressed as the charge of the object times the sum of the electric field vector and the vector cross product of the velocity vector and the magnetic field vector. Note that this equation is the vector combination; that is, it includes both size and the direction of the electric force and the magnetic force on the object.

In any case, Newton’s second law tells us that the force on any object can also be expressed as the mass of the object times a vector representing its acceleration. Equating these two expressions for force gives us an equation that, at least from a classical mechanics perspective, is enough to completely describe how the object will move. Let’s look carefully at what this equation depends on. We have acceleration and velocity, which describe the motion of the object. We have electric field and magnetic field, which describe the environment in which the object is moving. And we have mass and charge, which are properties of the object itself. If we divide both sides of this equation by 𝑚, on the left-hand side, we’re left with just the acceleration. And on the right-hand side, we have the charge of the particle divided by its mass times the contribution from the electric and magnetic fields.

But look what we’ve done. Now, the only term in this equation that depends on properties of the object itself is 𝑄 divided by 𝑚, which is just the charge-to-mass ratio of the object. Since charge and mass appear nowhere else in this equation, we can see that it is the ratio of an object’s charge to its mass that determines its motion in electromagnetic fields, not the particular value of its charge or the particular value of its mass. For example, a deuterium nucleus with one proton and one neutron has the same charge-to-mass ratio as a helium nucleus with two protons and two neutrons. And according with our general principle, if these two particles were in the same electric and magnetic fields with the same initial motion, they would follow the same path. This is because even though the helium nucleus has twice the charge of the deuterium nucleus, it also has twice the mass and so the same charge-to-mass ratio.

There is one specific and experimentally useful way that the motion of a charged particle depends on its charge-to-mass ratio in a particular case where there is only a magnetic field. Charged particles in the presence of only magnetic fields tend to move in circles or helixes, and this motion is called cyclotron motion. The radius of these circles is called the cyclotron radius, and it depends on the charge-to-mass ratio of a particle, not on its particular charge or mass. This radius can be directly measured, which gives us an experimental way to determine the charge-to-mass ratio of a particle. This is quite useful if, for example, we know the charge of a particle but not its mass because we can then use the charge-to-mass ratio to determine its mass. Alright, let’s apply what we’ve learned about charge-to-mass ratio to some examples.

An electron has a charge of negative 1.60 times 10 to the negative 19th coulombs and a mass of 9.11 times 10 to the negative 31 kilograms. What is the charge-to-mass ratio of an electron? Give your answer to three significant figures.

This question is asking us to find a charge-to-mass ratio, which exactly as its name suggests is an object’s charge divided by its mass. In this case, the object we’re interested in is an electron, and we’re given a value for its charge and also a value for its mass. So all we need to do is plug in negative 1.60 times 10 to the negative 19th coulombs for the charge and 9.11 times 10 to the negative 31 kilograms for the mass. To start this calculation, note that the units will be coulombs per kilogram, which is a valid unit of charge-to-mass ratio since it’s a unit of charge divided by a unit of mass. For the numerical part of our answer, we plug in negative 1.60 times 10 to the negative 19th divided by 9.11 times 10 to the negative 31 into a calculator. This gives us negative 1.75631 et cetera times 10 to the 11th with units of coulombs per kilogram.

All that’s left now is to give this answer to three significant figures. Starting on the left of the number, we have one, seven, and then five. Since five is the last significant figure we’re looking for, we round it on the basis of the next digit. The next digit is six, which is greater than five, so five rounds up to six. So to three significant figures, the charge-to-mass ratio of an electron is negative 1.76 times 10 to the 11th coulombs per kilogram. Note that this answer is negative because the charge of an electron is negative. Also, if we’d used different units, we would’ve gotten a very different numerical answer because, for example, in units of the elementary charge, the charge of an electron is exactly negative one.

Okay, let’s see another example.

A neutron has a charge of zero coulombs and a mass of 1.67 times 10 to the negative 27th kilograms. What is the charge-to-mass ratio of a neutron?

Given the charge and mass of some object, in this case a neutron, its charge-to-mass ratio is simply its charge divided by its mass. In symbols, we’d write the charge-to-mass ratio as capital 𝑄 divided by 𝑚, where capital 𝑄 is the charge and 𝑚 is the mass. The units of this quantity are whatever units we use for charge divided by whatever units we use for mass. For this particular question, the calculation is quite easy because the neutron has a charge of zero and a mass that is not zero. Therefore, when we divide charge by mass, we have zero divided by a number that isn’t zero, the result of which is just zero. Carrying over the units, this gives us a charge-to-mass ratio of zero coulombs per kilogram. A charge-to-mass ratio with a numerical value of zero is characteristic of all neutral particles with nonzero mass, like the neutron. Particles like the photon that are neutral but also have zero mass do not have a well-defined charge-to-mass ratio.

Now that we’ve seen some examples of calculating charge-to-mass ratio, let’s see an example where we use charge-to-mass ratio to calculate another quantity.

A new particle has been discovered with a charge of 3.2 times 10 to the negative 19th coulombs and a charge-to-mass ratio of 4.45 times 10 to the seventh coulombs per kilogram. What is the mass of the particle? Give your answer to three significant figures.

The question is asking us to find the mass of a particle given its charge and its charge-to-mass ratio. This is possible because the charge-to-mass ratio provides a relationship between an object’s charge and its mass. Specifically, the charge-to-mass ratio is a particle’s charge divided by its mass. If we plug in values from the question, we have 4.45 times 10 to the seventh coulombs per kilogram is equal to 3.2 times 10 to the negative 19th coulombs divided by the unknown mass. To solve for mass, we multiply both sides by mass and divide by 4.45 times 10 to the seventh coulombs per kilogram.

On the left-hand side, we have 4.45 times 10 to the seventh coulombs per kilogram in both the numerator and the denominator, so the division leaves us with just mass. On the right-hand side, mass in the numerator divided by mass in the denominator is just one, and we’re left with 3.2 times 10 to the negative 19th coulombs divided by 4.45 times 10 to the seventh coulombs per kilogram. All that’s left is to calculate. Let’s start off by dividing 3.2 times 10 to the negative 19th by 4.45 times 10 to the seventh. Plugging into a calculator gives us 7.19101 and several more decimal places times 10 to the negative 27th. As for the units, we have coulombs divided by coulombs per kilogram. Coulombs in the numerator divided by coulombs in the denominator is just one. And having per kilograms in the denominator is equivalent to having just kilograms in the numerator. So the overall units of our answer will be kilograms, which is good because we’re looking for a mass.

Now we just need to round our answer so it has three significant figures. The first three significant digits of our answer are seven, one, and nine. Looking at the next digit, one is less than five, so nine doesn’t change when we round to three significant figures. So to three significant figures, the mass of our particle is 7.19 times 10 to the negative 27th kilograms. As it happens, this mass that we’ve calculated and the provided charge match up very closely with the mass and charge of an 𝛼 particle.

Alright, now that we’ve worked through several examples, let’s review what we’ve learned about the charge-to-mass ratio. In this video, we learned about the aptly named charge-to-mass ratio, which is an object’s charge divided by its mass. In symbols, we write 𝑄 divided by 𝑚, where 𝑄 is the object’s charge and 𝑚 is the object’s mass. Any unit of charge and any unit of mass can be combined into a valid unit for charge-to-mass ratio, some common choices being coulombs per kilogram and elementary charge per unified atomic mass unit. Charge-to-mass ratio is also a property inherent to an object or particle. That is, two identical objects or particles will always have the same charge-to-mass ratio.

Considering the definition of particles and antiparticles, the charge-to-mass ratios of a particle and its corresponding antiparticle always have the same magnitude but opposite sign. We also learned by way of example that the charge-to-mass ratio of any neutral particle with nonzero mass is zero. Except in this special case, it’s always important to report both the units and the numerical value when working with charge-to-mass ratio. This is because two quantities can only be directly compared if they have the same units. We also saw three ways that the charge-to-mass ratio is a useful quantity. For charged particles moving in electromagnetic fields, the property that affects their motion is the charge-to-mass ratio rather than the particular values of their charge or mass.

In the special case of cyclotron motion, where there’s only a magnetic field and no electric field, if the velocity and the magnetic field are known, then the charge-to-mass ratio sets the cyclotron radius. This actually allows for the direct measurement of the charge-to-mass ratio of a particle moving in a magnetic field. As a result, if we measure the charge-to-mass ratio of a particle and also its charge, we can determine its mass. And if we measure its mass, we can use a measure of the charge-to-mass ratio to determine its charge.

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