There is a considerable simplification in using quantities of mass, energy and momentum once relativity is taken into account. Really. You’ll see that by being careful with factors of we can make the equations simpler and the numbers more manageable.
First, let me remind you of the energy unit electron Volts. If I were to take a 9 V battery and use that voltage to accelerate an electron, it would gain “9 electron volts” of energy, or 9 eV. That’s just a standard that’s convenient, albeit with a conversion factor that’s sort of unwieldy:
That number might be familiar—by defining the electron volt this way, it brings in the fundamental charge unit and that’s the actual number .
Okay. So, we can convert from kg to a unit involving electron volts by remembering that the rest mass of an object is related to the rest energy of that object by the proportion factor of . So, the rest energy of an electron is calculated this way:
(2)The combination is the equivalent to J.
The conversion to electron volts is straightforward from here. Using Equation 2, we can calculate the rest energy in Joules to the rest energy in electron volts:
(3)Remember the names for the powers of 10? For numbers that we’ll care about, they are listed in Table~??. So, we can quickly convert the mass of the electron from a number with lots of zeros, to something simpler:
|power of 10||nickname||in energy units|
Of course, Equation 2 is just the defining relationship for rest energy and makes use of the mass of the electron in kilograms, . If you go to Wikipedia and search for “electron” you’ll find this mass listed this way, but also another way that seems to be an equation and not just a mass: 0.511 MeV/c. What’s that about?
From Equation 2, we can obviously see that mass could just as well have been written as:
and the trick here is to continue to use energy units, keeping the
'' as a symbol and not dividing it out all the way. In this fashion, we're done. We would quote the mass of the electron as . Yes, explicitly like that. You'd say that the mass of the electron is :0.511 MeV over c squared.”
This is how I solved some of the Tablet problems. You see, there are three sets of units treated in this way:
|energies?||keV, MeV, GeV, TeV||the energy of Fermilab’s accelerator was ~2 GeV while the energy of the LHC is 7 TeV and eventually will be 14 TeV|
|masses?||keV/, MeV/, GeV/||the mass of an electron is about half MeV/c and the mass of a proton is about 1 GeV/c|
|momenta?||keV/, MeV/, GeV/||GeV|
So, now let’s use it:
Example: Calculate the momentum of an electron whose velocity is . What is the total energy? Do the calculation in natural units. What is the kinetic energy of the electron? What is its relativistic mass?
Relativistic momentum is and for , is:
Now, keep the ‘s and insert what we know:
Notice that in the units part, that the in the denominator and the in the numerator can cancel, leaving the units of momentum . So, we never have to divide by , we just leave it with the ‘s.
and we say that the momentum is “zero point seven MeV over c.”
In fact, there’s a further simplification that’s sort of a shorthand notation…like a colloquial notation. We just refer to everything in MeV’s…and forget about the ‘s altogether. That’s equivalent to just defining . Then if we keep track of what the quantity is, we just add in the ‘s at the end to make the units be right. This latter, uber-simplified notation is called
Natural Units.'' I've invented my own name for keeping the 's as units.”
Continuing with the rest of the problem, now using “ units,” we need to calculate the total energy. We can actually do that two different ways, since we have two equivalent forms for the total energy. Using the momentum quantity:
The second way to characterize the total energy is:
See how the ‘s just don’t matter in this way of writing. In “Natural Units,” we would just go straight to the near-bottom line.
(11)The kinetic energy is:
Notice, that I’ve used Natural Units and just ignored ‘s. Sort of uncomfortable at first, but easy after you get used to it!
Finally, the relativistic mass of this electron is:
Remember, I told you that using Relativity was easier than working out the fundamental bases of it!