How do magnets and electricity work?

Posts tagged "notes" represent my current knowledge and may be incorrect or misleading.

Electromagnetism feels like something I should have been taught before graduating high school. (My school didn't offer AP Physics E&M, but I'm not sure that would have covered everything I needed anyway.) Alas, as with many things, we can do nothing but fill in the gaps for ourselves. :)

Magnets and electricity superficially seem like two very different things. At least, I thought they were. They just... don't seem related. Magnets are, like, push-pull-y and electricity is, like, zappy or something. Everyone else also probably thought this until the 19th century, when electromagnetics developed into a successful field of study.

They're actually part of the same fundamental force, electromagnetism. Magnets in motion create electric fields; electric charges in motion create magnetic fields.

With these two things put together, electromagnetism is one of the four fundamental forces of nature, along with gravity and the two nuclear forces (strong and weak).

These fundamental forces seemed kind of mystifying to me, until I read the Advertisement section in David J. Griffiths' Introduction to Electrodynamics:

Mechanics tells us how a system will behave when subjected to a given force. There are just four basic forces known (presently) to physics: I list them in the order of decreasing strength:

1. Strong
2. Electromagnetic
3. Weak
4. Gravitational

He goes on to articulate that nearly all the forces you'll deal with in regular life are electromagnetic — friction, normal force, the forces that keep you from falling through the floor and that keep atoms together — except, of course, gravity, which is weak enough that it's only really noticeable with extremely large masses like planets or stars.

The strong force binds protons and neutrons together in nuclei and keeps quarks together to make hadrons (e.g. protons and neutrons, but notably not electrons, and including some other strange unfamiliar things); the weak force causes radioactive decay, idrk, it's complicated. Both of these nuclear forces operate on such a small scale that we do not notice them and don't ever deal with them in classical mechanics.

But, woah, nearly all the stuff you talked about in AP Physics C: Mechanics (this is my only reference point, sorry, it's the only useful physics class that I took in high school) is actually electromagnetic forces acting on a large scale.

Not only are electromagnetic forces overwhelmingly dominant in everyday life, they are also, at present, the only ones that are completely understood. There is, of course, a classical theory of gravity (Newton’s law of universal gravitation) and a relativistic one (Einstein’s general relativity), but no entirely satisfactory quantum mechanical theory of gravity has been constructed (though many people are working on it). At the present time there is a very successful (if cumbersome) theory for the weak interactions, and a strikingly attractive candidate (called chromodynamics) for the strong interactions. All these theories draw their inspiration from electrodynamics; none can claim conclusive experimental verification at this stage. So electrodynamics, a beautifully complete and successful theory, has become a kind of paradigm for physicists: an ideal model that other theories emulate.

(There are lots of other juicy things in the beginning of this textbook, I recommend checking it out! These quotes are from the fourth edition because that was easily accessible to me.)

Also, guess what? Light is just electromagnetic radiation! Magnets, electricity, and light are all the same fundamental thing manifesting in kinda different ways. It is here that I say WTF, this is awesome.

(section break here?)

Special relativity shows that electric and magnetic fields are different manifestations of the same thing, depending on the observer's frame of reference:

The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference.

(Wikipedia)

Which is weird, because I thought relativity was something Einstein came up with to figure out gravity-related things. Oh well, I guess this'll make sense eventually.

Along with Einstein's descriptions of special relativity, Maxwell's equations unify the theory of electricity and magnetism. They are "a set of four partial differential equations which provide a complete description of classical electromagnetic fields" (Wikipedia).

The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics.

(Wikipedia)

I have to say, it's disappointing that I wasn't exposed to this earlier in my life. Physics seems much juicier than I had previously perceived. It's probably good, though, because instead of setting some life-goal like "help create a unified theory of physics" (probably very high difficulty) I set the life-goal of like "help make aligned AGI" (????? difficulty) which at least seems... more materially impactful? In the short term?

Sorry, rant over.

These are the learning targets, I guess: an intuitive understanding of Maxwell's equations and special relativity. Idk, not that hard? Maybe hard? It's great, because I don't have enough grounding in the subject to know.

Actually, here are the learning targets.

questions to ask:

1. how do maxwell's equations unify electricity and magnetism?
2. what are the implications of the lorentz force in practical applications (e.g., particle accelerators)?
3. how does the concept of the electromagnetic field fit into the framework of quantum field theory?
4. in what ways do electric and magnetic phenomena manifest in everyday technologies (e.g., MRI machines, transformers[^1])?

(Generated by GPT-4 as part of a larger response.)

I'll add on "how does special relativity interact with, support, and depend on, the theory of electromagnetism?" because I kind of want to learn about special relativity too.

These all seem like insanely interesting questions. Let's get to it!

Starting with the first question, how do Maxwell's equations unify electricity and magnetism?

Well, I mean, looking at them, they're describing characteristics of electric and magnetic fields in terms of one another along with some extra variables. If they're (sort of) a complete description of classical electrical and magnetic fields, that means that together they can describe any behavior of those fields and things that are subject to those fields, at least at a level of abstraction above or equal to those variables (i.e. the equations don't describe quantum phenomena, since they're below the level of abstraction of the equations).

I will be able to answer this question better once I have actually understood the equations.

Each of them have corresponding "conceptual descriptions" on Wikipedia, which give intuitive natural-language descriptions of the laws. Here's what I understand from an initial pass. There's more nuance to these laws, but I want to see what I can get the first time through.

• Gauss's Law tells you about the relationship between electric charges and electric fields. The field "points away from" (is repelled by?) positive charges, and points towards (is attracted to?) negative charges. Then there's something about flux or outflow or something for, like, a certain part of space (a gaussian surface?), being proportional to the amount of charge that's in it times some constant $\epsilon_{0}$ ("vacuum permitivity")
• Gauss's Law for Magnetism says that, while electric fields can start from single points/particles, magnetic fields can only originate from paired poles — “dipoles,” the north and south ends of a magnet — and the net outflow is zero. Seems like these are describing generally, like, what creates magnetic and electric fields, and defining a key distinction (outflow) between them? I’ll be interested to see how outflow becomes important, and what flux means.
• Faraday's Law tells you about how a magnetic field will respond over time to changes in the circulation of an electric field, and some other stuff I don’t really get relating the work needed to move a charge around a loop to the rate of change in the flux. Uh, yay!
• Ampère's law with Maxwell's addition states that magnetic fields are related to electric current, and vice-versa. Not really clear how this is different from Faraday’s law.

All these equations have multiple alternate forms, which express the same key ideas in a different way; above they’re differential equations, but there’s an integral form and such.

They all seem relatively simple equation-wise. It’s cool that that is the case. Let’s see if we can actually figure them out.

First, maybe clarify a couple of concepts that came up.

First, flux. Flux is actually pretty easy. Khan Academy’s 10-minute video probably does a better job of building intuition than I can, so I'll just assume you've watched it and continue.

Flux basically considers the strength of a 3-d field, at a given (subsection of a) cross-section. Magnetic flux measures the strength of the magnetic field passing through an area; electric flux measures the strength of an electric field passing through an area.

Sticking with the banger Khan Academy videos, Khan Academy india has a pretty simple explanation of how to calculate electric flux:

$$\Phi = \int \text{E} \cdot d\bf{A_{\perp}}$$

or

$$\Phi = \int \text{E} \cdot d\bf{A} \cdot cos(\theta)$$ to make that extra $\perp$ notation more mathematically explicit.

I.e., "to get the magnetic flux $\phi$, integrate along a surface $A$ (hence $dA$), measuring the magnitude of the surface-perpendicular (hence the $\perp$ in $dA_{\perp}$) component of the vector given by the electric field $\text{E}$ evaluated at that point." That's a lot of words, but it's not a difficult explanation; skip through the video above (it's very heavy on intuition-building diagrams, so you can mostly just figure it out from those and reading subtitles every couple seconds) if that didn't click, because I am unsure it would if I didn't already get the idea here.

Wow, though, that's actually pretty easy. Looking back at Maxwell's first equation, Gauss's law:

"The net electric flux through any hypothetical closed surface is equal to $1/ε_{0}$ times the net electric charge enclosed within that closed surface".

This almost makes sense now, except for the fact that it's a closed surface and not just an, idk, "regular" surface? It's like we're looking at the boundaries of a defined 3d space, rather than a 2d surface.

Luckily, Khan Academy has yet another video, on Gauss' law itself! We don't even need to figure it out ourselves.

And, once again, it's actually mad easy! Obviously this is a simplification for an intuition pump, which probably removes depth/beauty from the equation, but it's a great start.

So Gauss's law just says that, if you stick an electric charge inside a closed surface (say, a sphere), the electric flux through that surface will be independent of the size of the surface; all that matters is $q$, the charge of the particle, and the electric constant $\epsilon_{0}$ which I don't understand yet — it has something to do with, like, how charge creates a field in a vacuum or something, I think, but don't take me on that because I'm not trying to define $\epsilon_{0}$ here.

So taking a look at the wikipedia notation, from page on Electric flux:

$$\Phi = {{\subset\!\supset}\llap{\iint}_{S}} \: \text{E} \cdot d\mathbf{A} = \frac{Q}{\epsilon_{0}}$$

(The fancy double integral with the circle (that I had to get from stack overflow because mathJax doesn't render closed surface integrals natively) is just telling you that it's an integral across a closed surface. :)

In other words, this is the exact thing that the khan academy video just derived lol

→don't know how to calculate $Q$, but that's something I can figure out probably pretty easily. Apparently it's derived from Coulomb's law.)

We see an equivalent of $\frac{Q}{\epsilon_{0}}$ reappear in the canonical page we started with, the one for on Maxwell's Equations:

$$\nabla \cdot \text{E} = \frac{\rho}{\epsilon_{0}}$$

We just replaced $Q$ with $\rho$ (rho) which is I guess a different way to label it that has some semantic utility I don't understand, or maybe it's just a different convention or something.

Another part I don't understand even more is the $\nabla$ part. We're not talking about flux anymore, we're just talking about some property of the electric field. It has something to do with divergence of a vector field, which is something I don't know yet. But 3blue1Brown has a video on it :D

To calculate divergence at a point, you just look at a very small region around the point. We'll talk in terms of water for now. If it tends to flow away from the point (like a fountain at the top of a hill) divergence is positive. If water tends to flow into the point net more than it flows out (like the drain at the bottom of a sink), divergence is negative. This is still true if you just grab a random section — if water flows out moving faster than it flows in, divergence is positive there too. It's sort of like a derivative; you're measuring "how much the point tends to act like a source or a sink."

I... sort of understand this intuition, enough to vaguely grasp its place in this equation. Rather than integrate across the whole closed surface, we're just looking at the thing that makes flux (???) at a point: the divergence of the electric field $\text{E}$ at that given point. And it's proportional to the charge, $Q$ or $\rho$ or whatever. This makes sense; if the charge is net negative, at the point there will be a pulling in of... stuff? (Still haven't built the intuition of, like, what an electric field actually is, sorry.) And if the charge is positive, that means the field will be repulsive, and it'll act more like a source — positive divergence.

Ok. I asked a friend to help me with the intuition of what "the divergence of the vector field at a point" means in terms of particles.

Divergence requires a direction to be "in/negative" and a direction to be "out/positive." If we think about a particle moving right, we can set "moving right" as the positive direction. If the force acting on the particle is increasing the magnitude of its velocity, the divergence is positive. If the force acting on the particle is reducing its velocity, the flux would be negative.

We can also think about it — going back to the charge-inside-a-sphere example — as setting the positive direction at any given point on our sphere as the direction perpendicular to the surface of the sphere at that point; positive divergence means the particle is pushing things away from it, negative divergence means the particle is attracting things to it.

For those who have taken multivariable, the notation hints at how to calculate divergence: find the gradient (the 3d vector containing each of the field's partial derivatives) and then sum each element in the vector, as you would with a dot product! :D

OK, so Maxwell's first equation basically just tells you some stuff about how if you stick some electric charge in a box, then the measure of the field flowing through the surface of the box will be proportional to the amount of electric charge you put in the box. Lmao, it's really funny to see it reduced like this.

But I think I get how this is fundamental — it's establishing the fundamental point that electric fields are, at their core, just defined by the charges that create them — they seem to kinda just transcend other factors. Maybe? Well, if that's what it's saying, cool! If not, tragic L for me in the couple hours that I spent rummaging around the internet for this.

I'll leave the first equation there for now. I'm just trying to get a very vague sense of what's going on here, to anchor myself before I move on, and I think i've done that.

Now we get to look at its magnetic equivalent!

...

What is a field? What is a field, actually? - have you ever done differential equations? - Each point has a corresponding vector defining the force the field will exert - Exert on what? - gravity has a field, so does magnetism. - It kind of seems to break our idea of causality that there can be just, idk, an invisible force field that can act on stuff, but alas. The notion of “things need to touch each other to interact” is not really true once you go down much deeper.

• fields don't actually flow. things within them flow — electrons or something. we just pretend they do i guess