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.
