Essentially, the entire theory of electromagnetism can be found in the following four equations:

These are Maxwell’s Equations in differential form, not in integral form, which is the way they are often introduced. I will discuss them in this form, however, as I believe the differential equations convey more elegantly their physical meaning straight from the mathematics. Let’s get started.

Fields

If you ever did high school physics, you should have some idea of what electric and magnetic fields are. Below is an example of each (depicted using field lines):

What these field lines show is the direction of the respective fields at each location (indicated by arrows) as well as their relative strengths (indicated by density of field lines). So what are these fields actually representing? Well, for electric fields, this shows you the direction an object with a positive electric charge would be forced, while a negatively charged object would feel a force in the opposite direction.…

I came across the following YouTube video a while back which uses a strange trick to accurately approximate square roots. I suggest watching at least the first minute where the presenter explains how it’s done:

Let’s do an example. Approximate to 2 decimal places: 

First, YouTube tells us to find the nearest perfect square that’s less than 40, that’s 36, and take the root, giving us 6. So our answer is, obviously, 6 point something. That something is a fraction, where the numerator is the difference between 40 and 36, and the denominator is 2 times 6. So we have:

So what’s the actual answer to 2 decimals? It’s 6.32. That’s what I like to call: pretty darn close. Naturally, there are a few catches to this technique: you need to know your perfect squares, you need to know your fractions, and things tend to get hard with larger numbers.

But still, I thought this was surprisingly effective for such a simple piece of arcane trickery.…