Physics

I love dimensional analysis. It’s a good way to get a feeling of how different quantities and measurable features relate to each other. Helps you not to get lost in all those awful formulas that math or the real world can put you face to face with. Sometimes it even helps you to come up with formulas you didn’t know or reinvent the formulas you didn’t remember. In fact, this is how I’ve made it through my physics classes during the school years. The only time this approach didn’t help was the formula for kinetic energy, which isn’t $E=mV^2$ (despite what the famous $E=mc^2$ might make you believe), but in fact is $E=\frac{mV^2}{2}$ with a sneaky division by a dimensionless constant. Gross.

Cursed units

Anyways, as great as dimensional analysis is, sometimes it can leave you with very funky dimensions. This video has some great examples,

Logarithms

Introducing: a logarithm of a percent to base meter. $$ \log_{\text{meter}}\left(0;1\right] $$

As pretty much all the physics-related thoughts that come to my head, this one visited me during my time in a swimming pool¹. I was thinking about the fact that underwater the visibility falloff with distance is noticeably greater than in air, and even at about ~20 meters² the effects are visible.

The light intensity drops proportionately to the square of distance, obviously. But if we were to consider a single ray of light, the chances of it hitting a water molecule and getting off track are the same at every given span of the same length. So if it has a chance $p$ to not get diverted over 1 meter of distance, then over 2 meters of distance this chance becomes $p^2$. Yep, it’s just an exponential decay.

So, think about it, I take this probability $p$, raise it to a power of meters (or seconds if instead of a light ray in water I consider a particle decaying over time), and get a dimensionless value in range from 0 to 1. Or in other words, to get this $p$ I have to take a logarithm with base in meters (or seconds) of a dimensionless value.

Even more logarithms

Decibels are peculiar, they are essentially $10 \log_{10}{\frac{\text{Some unit}}{\text{The same unit}}}$.

It makes sense that for working with ratios you might want to use logarithmic scale. Base ten because ten fingers. Then multiply it by 10 to make the numbers you usually get to be more easily expressible with integers (and to justify deci- in the name).

Finally, add a suffix to the unit to clarify what ratio you were considering https://en.wikipedia.org/wiki/Decibel#List_of_suffixes

Electromagnetism VS Gravity

A few days ago, while having a dinner, I though about how some physical quantities are way easier for a human mind to grasp than others.

Take, for example, mass. You have a pretty good intuition about what it is. Electric charge? Not so much, but still comprehensible. Spin? Color charge? Good luck with that.

And even though your visualization for electric charge is probably electric current, magnets or a cat covered in foam peanuts, none of these are the most common way you experience electric charge. The most common way is touch, the repulsion of electron clouds in your body from those in the touchee.

You have to be thankful those clouds more or less cancel out their charge with the atomic nuclei, so you only experience this force at a short distance.

Anyways, this reminded me of the times in school when I would have to solve some problems involving interactions of charged particles and would always be told to just ignore gravity. At some point, I decided to check how significant gravitational interactions were in those cases and I remember getting a result that the gravitational force was way smaller that even just the Coulomb force.

But how much smaller?

Let’s crank some numbers.

Recall that the Newtonian force of gravity $F_g$ is equal to $G\frac{m_1 m_2}{r^2}$, where $G$ is the universal gravitational constant, $m_{1,2}$ are masses, and $r$ is the distance. The electrostatic force $F_e$ meanwhile is equal to $k_e\frac{q_1 q_2}{r^2}$, where once again $k_e$ is some constant (more on it later), $r$ is the distance, and $q_{1,2}$ are the charges.

So if we want $$F_g \approx F_e$$ we need $$G\frac{m_1 m_2}{r^2}\approx k_e\frac{q_1 q_2}{r^2}$$ We can get rid of the distance, neat! $$Gm_1 m_2\approx k_e q_1 q_2$$ For simplicity, let’s consider two identical particles. $$ Gm^2 \approx k_e q^2$$ Some rearranging… $$ \frac{q}{m} \approx \sqrt{\frac{G}{k_e}}$$

Wolfram Alpha tells me this is roughly $8.618 \times 10^{-11}$ coulombs per kilogram. The meaning of this value is the charge-to-mass ratio that a particle/body needs to have so that its electrostatic force is roughly equal to its gravitational pull.

Let’s see what this value is for different particles:

• Electron: $\sim 1.759 \times 10^{11}$ (21 orders of magnitude more charge per mass than needed).
• Proton: $\sim 9.579 \times 10^{7}$ (18 orders of magnitude more).
• Your body: zero. Well, I assume you are neutrally charged. Say you aren’t and there’s like $n$ electrons your body is missing or has in excess. Then, assuming you weigh 80 kg, this value works out for you to be $2n \times 10^{-21}$. (10 or fewer orders of magnitude less charge per mass than needed).

Make of this what you want.

Here’s a somewhat related video

(A great channel overall, consider this a recommendation.)

Ampere seconds

I lied to you a few paragraphs ago. Wolfram Alpha didn’t give me coulombs per kilogram, it gave me ampere-seconds per kilogram instead.

You see, back in the day electric charge wasn’t a fundamental dimension like mass, time, or length. Instead you had electric current, and coulombs would be defined using amperes. Amperes, meanwhile, would be defined as follows.

Consider two parallel infinitely thin, infinitely long wires, 1 meter apart from each other in vacuum. If each meter of the said wires experiences a magnetic force of $2 \times 10^{-7}$ newtons, then you have a current of 1 ampere.

And 1 coulomb is a charge that passes each second through a cross-section of a wire with a 1 ampere current.

Majestic, isn’t it?

However, in 2019 things got redefined. And now 1 coulomb is some constant times elementary charge, while an ampere is just a coulomb per second. Boring.

Sadly, I had to take my physics exam before the new definition.

Coulomb’s constant

While on topic of coulombs and constants, let’s get back to that $k_e$ I promised we would revisit.

It is defined as $k_e=\frac{1}{4 \pi \varepsilon_0}$, where $\varepsilon_0$ is some other well-known constant, and $4\pi$ is a thing that sparks my interest.

Math

(Two) Pi

When was the last time you saw a $\pi$ that was not multiplied by $2$? Aside from the fractional angles, such as $\frac{\pi}{3}$.

I can tell you with great certainty that the only two place where I’ve seen it without the coefficient are the area of a circle $\pi r^2$ and Euler’s identity $e^{i\pi}+1=0$.

Everywhere else it appears with a two attached to it.

Normal distribution? $$\frac{1}{\sigma \sqrt{2\pi}}e^{\frac{-1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

Planck constant? Comes up divided by two pi so often they made a special “h-bar” for it: $$\hbar=\frac{h}{2\pi}$$

An approximation for factorials? $$n!\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$$

Full turn? $2\pi$ radians.

The solution of $sin(x)=a$? $$x=2\pi k + \arcsin a$$

The solution of $z^n=1$? $$z=e^\frac{2\pi ki}{n}$$

Circle circumference? $$2\pi r$$

Recurrent formulas for $n$-dimensional ball/sphere ’s volume/area?

$$ V_0=1 $$ $$ S_0=2 $$ $$ S_n=2\pi r V_{n-1} $$ $$ V_n=\frac{r}{n}S_{n-1} $$

I would argue that even fractional things like 30° in radians make more sense to be expressed as fractions of a full turn rather than fractions of a half turn.

Tau

Say hello to $\tau$, a relatively young symbol (first proposed in 2010), and set to be exactly $2\pi$. Twice as big, but with only half as many legs. (Although a three-legged pi was proposed even back in 2001.)

It has somehow found its way into some programming languages (Python, Rust, and Java, to name a few) as well as in the minds of people.

I honestly don’t have that much to say about it. I never really understood why someone would need to break the status quo and switch to a different constant, until I actually started noticing the oddly close friendship of $\pi$ & $2$.

Just the other day, I was reading a book about noise-resistant coding and there was a formula

$$ f(y|0)=\frac{1}{\sqrt{\pi N_0}} e^{-\frac{(y+\sqrt{E})^2}{N_0}} $$

Essentially we have a binary value $b$ and we turn it into a rational-valued signal $(-1)^b \sqrt{E}$, where $E$ is some constant of our choice. Then it passes through a noisy channel where a random variable $\eta \sim N(0, \sigma^2)$ gets added to it.

So the formula above is the probability density for observing a given value $y$ on the receiving end of the channel given that $b=0$.

What immediately caught my eye was the pi, without a two and inside a square root, which means it was unlikely that this pi’s two got cancelled out with something else in the fraction. Were the tau-evangelists wrong? Are there actual real-world examples of half a tau appearing in formulas? Nope, turns out that several pages above they defined $\eta$’s variance to be $N_0/2$. So really this $\sqrt{\pi N_0}$ is just the $\sigma\sqrt{2\pi}$ from the normal distribution’s PDF with a shorthand of $2\sigma^2=N_0$ applied. To paraphrase a popular Haskell motto, “Avoid tau at all costs”.

Another thing I wanted to mention is that I’ve been under an impression that tau’s popularity in USA is higher than in other parts of the world (but I may be wrong). Does it say something about the society? Not sure.

Well, that’s all I’ve got for today, go away now.

Addendum

Since decibels can be used to express any ratio, here’s some cool facts:

• The Eiffel Tower is $7.78$ decibels higher than The Leaning Tower of Pisa.
• Russia is $2.493$ decibels larger than USA in terms of area.
• USA population is $3.72$ decibels larger than Russia’s.
• This means that USA is $2.493 + 3.72 \approx 6.21$ decibels more densely populated than Russia.
• You are $105$ decibels taller than an atom.
• Alcohol contents of vodka is $-3.9790$ decibels.
• You spend $-4.77$ decibels of your life asleep.
• A magnitude 5 earthquake releases $15$ decibels more energy than a magnitude 4 one.
• You are $69$ decibels shorter than The Earth and $229$ decibels lighter.
• A hydrogen atom’s electron cloud is $43.2$ decibels wider than its nucleus. This means that pretty much the majority of anything you see around is empty, consisting mostly of the nothingness in-between the atom nuclei & the electrons floating around.

Initially I was going to mention the last fact in this post, in the section about electromagnetism, but genuinely forgot. Alongside with yet another sensory input that gets cut off when you are swimming in a swimming pool: haptic perception. Or, in other words, you feel especially touch-starved while swimming.

• Updates
  • 2024-04-28: an addition of the addendum to add some things I forgot to add initially.