For a lack of a better title and phrasing of my question, I am asking: do units have to make sense in physics?
Let me give an example of what I mean. Let's say that I have some arbitrary units (for argument's sake):
$$\frac{\mathrm{meters}^{2} \cdot \mathrm{seconds} \cdot \mathrm{henry}}{\mathrm{coulomb}^{3}}$$
And I want this to be equal to charge let's say,
$$C = \frac{\mathrm{meters}^{2} \cdot \mathrm{seconds} \cdot \mathrm{henry}}{\mathrm{coulomb}^{3}}$$
Which has units of Coulombs. But the units don't add up to equal one coulomb as you can see.
So my real question is that do you have to do some manipulation with the units in order for them to equal one coulomb? What are the rules and conventions in physics for this?
I have seen in EE (electrical engineering) that if you can cancel out all units, then you can give it any units you want in the end. It is the same for the physics field? (EDIT: I may have misunderstood this part and it is probably not true).
Or can you set the units to whatever you want even if they are not equal to one coulomb?
I have heard that the constant in the EFE's is a conversion factor to get the right units:
$$\frac{8\pi G}{c^4} T_{\mu \nu }$$
Where $\frac{8\pi G}{c^4}$ is the conversion factor. This is a real example that I found that I want to understand as well.
In summary:
- Can you set the units to whatever you want even if they are not equal to one coulomb?
- If you can cancel out all units, then can you give it any units you want in the end?
- Do you have to do some kind of manipulation of units to get the right units?
Best Answer
Units in physics are units of the real world. They must be consistent, because the real world is. After choosing a definition of necessary fundamental units (as defined by the SI system e.g. - I am not considering natural units here), all other units must be derived from those in a mathematically sensible manner.
Because, you move a distance in metres and carry an amount of Joules as kinetic energy and have inertia as an amount of kilograms. If you agree that you can't suddenly move a distance of "seconds" and have an inertia in "metres per second" and drink coffee that is "ten kilograms" hot, then you agree with this consistency of the world.
The fundamental properties like time, position, mass etc have fundamental units. All other properties are derived from those, and with units that are likewise derived from their units. Your speed is a change in position over time, so the distance-unit over the time-unit, metres-per-second. Speed can never have units that are not distance-units-over-time-units. This consistency must necessarily always be the case regardless of complexity. If not, then you have found a new fundamental property.
Note though: Now and then you might indeed meet equations with inconsistent units on either side of the equal sign. So that the unit of the left-hand-side isn't the unit of the right-hand-side. This is typically seen for emperical findings. It is mathematically sloppy but might work fine in applications. Nevertheless, it is mathematically sloppy.
As you also mention, a coefficient of sorts ought to be introduced to correct for unit mismatch. Typically, such unit correction is added to a constant or coefficient that is involved. For instance, in Hooke's Law for spring forces, we see proportionality between displacement from natural length and spring force - we thus invent a proportionality constant k:
$$F_{spr}=k\;\Delta x$$
What should the units of this constant $k$, which is called stiffness or simply the spring constant, be? If unitless, then this equation shows a number of Newtons equal to a number of metres. A mismatch. So, let's invent this $k$ to have units that make it all match: that would be Newton's-per-metre. And this definition of $k$ matches very well with a physical understanding of $k$ as stiffness - it is a material property that tells us something about how much the spring can be stretched before reaching a Newton of spring force; it does indeed tie distance with force, metres with Newton.
As another example, in Amontons' Law of kinetic friction at low normal pressures, we find proportionality between normal force and kinetic friction force - again we invent a coefficient to account for the proportionality constant:
$$f_k=\mu n$$
$\mu$ is called a friction coefficient. What unit should we give it? On the left-hand-side we have units of Newtons. On the right-hand-side we already have units of Newtons as well. By keeping $\mu$ unitless (or you can say "Newtonts-per-Newton" if you wish), this relationship fits. And so it went.