Can someone redpill me on classical E&M?

>Thanks for not being a twat.
Says the guy who can't form a sentence without using internet memes.

Maxwell's equations tell you that electric charge is the source of divergent electric fields, magnetic charge is the source of divergent magnetic fields.
They tell you that electric current and time varying electric fields are the source of rotational magnetic fields, and that magnetic current (i.e the flow of magnetic monopoles) and time varying magnetic fields are the source of rotational electric fields.
Magnetic monopoles don't exist (at low energy at least - they might be very massive particles not yet detected), so generally one takes the magnetic charge and hence magnetic current to be zero.

Some anons have already said the answer but in simple terms, Maxwell's equations tell us a few things:
i) [eqn]\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_{0}}[/eqn]
This tells you that a source charge distribution generates a divergent electric field. Invoking the divergence theorem leads to Gauss' law, which can be used to find either the electric field or the charge distribution.

ii) [eqn]\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}[/eqn]
This is the general form for Faraday's law of induction. This can be read as a rotating electric field induces a magnetic field OR a changing magnetic field generates an electric field.

iii) [eqn]\nabla \cdot \vec{B} = 0[/eqn]
This reflects the fact that there are no magnetic monopoles in classical EM. We have only observed multipoles (which can be broken up into dipoles in the multipole approximation).

iv) [eqn]\nabla \times \vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial \vec{E}}{\partial t}[/eqn]
This tells you that a rotating magnetic field is generated from a source current and a changing electric field. Invoking the curl theorem leads to Ampere's law.

This isn't the entire picture but hopefully it will be good enough.

Isnt the magnetic field just the opposing, orthogonal force of an electric field?

These equations combined with the continuity equation, which is essentially conservation of electric charge,
[eqn]\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}[/eqn]

and the Lorentz force law,
[eqn]\vec{F}_{EM} = q(\vec{E} + (\vec{v} \times \vec{B}))[/eqn]

tells us everything we would ever want to know from classical EM. I'm too lazy right now to do EM in a medium but that's the jist of it.

I bet OP looks like a sperg in his trench coat.

Obviously all valid replies and I appreciate those. For some reason I thought there was something simpler that explains the big picture? I understand maxwell but still feel like theres this giant gap in my brain that prevents me from tying everything together fundamentally.

I get this. This is obvious.

I don't mean it in a conspiracy sense. Im looking for 3-4 sentences that relate charge, current, and the fields, possibly without necessarily evoking maxwell. I know that doesn't make any sense, but its because I don't understand it what Im asking for well enough to produce a question. Im hoping somebody "gets it" beyond just
>> Hurr door much rotating time changing fields.

why wont you talk about this guy? He basically invented electromagnetism.

Time to wake up and understand what have been hidden from you.

No need to project. Im more a chad exterior with an autist interior.

Maxwell's equations model experimental results seen in classical EM. "Evoking" the equations is nothing magical, they are what they are.

>Experimentally, charge sources create electric fields.
>Currents , which are just moving charges, create magnetic fields.
>These electric fields happen to also create magnetic fields under certain conditions, visa versa.

>> hurr durr muh*

autocorrect.

That's not a question for classical Electrodynamics. They are components of the tensor that describes the curvature of the photon field. Since an electromagnetic force is really an exchange of photons, as a charged particle moves through regions of space where these photons are of different density or energy, the force it feels is different. Hence the variation of the photon field produces a force.

Is that QFT?

Can it be classically explained why a moving charge produces a magnetic field?

Not him but nothing classical can explain field theory. I may be completely wrong but the classical theories are naive in that they don't take relativity or quantum effects into account, meaning that vector bosons and whatnot are not explained.

Ok. I think that is what has been bothering me for years. Just taking Maxwell's equations on faith (almost every faggot in this thread) with no explanation for why the observed relations hold in the first place.

really basic QFT will tell you just why Maxwell's equations are true. The whole question of why electric fields and magnetic fields do different things but are highly connected is one that is answered in field theory.

In the modern language, Electrodynamics is a U(1) gauge theory, which just means that the electron has a complex phase, and the Lagrangian is locally invariant to changes in this phase. To make this possible, the theory must have a massless vector boson coupled to the electron. From this basic statement you can basically derive Maxwell's laws, if you have the right tools.

In a more complicated analysis of the rules that comes from this model (QED), you learn exactly how the electron behaves in the presence of the photon, and in terms of what we call electric and magnetic fields. You learn about the electron's spin and it's magnetic moment value and all of that.

OP, just know the answers are there. They're called quantum field theories

>thats not physical

Well, yes it it. They tell how those fields behave. That is physics.

Lets see the the really classical electromagnetism (after all, magnetism comes from special relativity):

[math]\nabla \cdot \varepsilon \mathbb{E}=\pho[/math]
[math]\nabla \times \mathbb{E}=\bar{0}[/math]

How would you describe this?

Lol I get ya. Remember, Maxwell's equations came before the development of quantum mechanics, so the "faith" physicists had on them were purely experimental. It is "good enough" of a model.

Thanks for coming back to explain further. I'm not versed yet in QFT or QED but hopefully one day after I finish going over basic topics first.

Much appreciated. Guess I have to start going through my QFT book.

>OP wants to be redpilled (whatever that means) about electrodynamics
>nobody shows him the relativistic formulation

>I may be completely wrong but the classical theories are naive in that they don't take relativity into account
You can do classical field theory and take relativity into account just fine. And you can also build a classical U(1) invariant Lagrangian for the electromagnetic field which is Lorentz invariant. That doesn't require QFT yet, but you will need it if you want to see photons.

>redpill me on EM

>uses the phrase "redpill me"
>doesn't grasp basic physics

JJJJJUUUUUSSSSSTTTT

No you DON'T.

aw let him, worse thing would be that he can't understand it and will go back to lower level texts and build up his foundation

You have Coulomb's law.

Then you have Biot and Savart's law.

If you are really good in calculus, you should be able to derive Maxwell's equations from them.

quora.com/Do-Maxwells-equations-imply-the-Biot-Savart-law

>you should be able to derive Maxwell's equations from things which are direct consequences of Maxwell's equations
That's not a good way to learn.

Well, if you postulate Maxwell's equations, you can get both of these force laws. But that is not how it historically happened, is it?
Going the historical route (on some guided path so you don't end up with Weber's electrodynamics, though you could do that as an excursion after you understood Maxwell) may be worthwhile.

>A changing electric field induces an orthogal magnetic field, and vice versa.
>They tell you that electric current and time varying electric fields are the source of rotational magnetic fields, and that magnetic current (i.e the flow of magnetic monopoles) and time varying magnetic fields are the source of rotational electric fields.

I will just leave this here:
en.wikipedia.org/wiki/Jefimenko's_equations#Discussion

If you learn things in the historical context, you'll spend a lot of time struggling with incomplete models, which is what our contemporaries spent their lives doing. It's inefficient. You're trying to learn decades - perhaps a century's worth of work in a year or less. The only reason electrodynamics is taught this way is because most first-year students in a physics program haven't completed calculus or differential equations. At later stages though, it's probably more efficient to start with the model of the 'electromagnetic field' and to understand the connections through Maxwell's laws first.

For example, if a person with a BS in mathematics were to enter a physics graduate program, would it not be more helpful for them to begin with the governing equations (only learning later in the QFT class about gauge theories and where the governing equations come from)? I think it would be, since it's more important to understand the governing principle and the connection between ideas than it is to individually understand all the consequences of the governing principles.

I've had many long conversations with other students about more effective ways to teach this material. It's an important topic to me, so your input is welcomed.

>I feel like the relationship between the electric and magnetic field is much more basic than how I am perceiving it.
maybe this will help,
start with energy that can be stored in the form of an electric field or a magnetic field.
Then the curious facts;
A moving electric charge creates a magnetic field
A changing magnetic field creates a moving electric charge.
They are related in this moebis strip type relationship wherein they are not two sides of a coin they are the coin.
When the energy moves through ideal space as a electromagnetic wave it shows its true form and relationship.
Now at this point please pick up a couple of bar magnets and play with them. Notice the peculiarity of the interaction between the two magnetic fields. They can attract and replell, this is nothing new the electric fields can do the same. An electric positive attracts a minus and repells a positive in the same way a north magnetic attracts a south magnetic and repells a north. What is different here is the magnetic fields can induce torques upon each other. That is to say force vectors that are not linear (pushing / pulling) in effect but twisting. Also note the fact that these torques do not act on all axis equally. Two magnets aligned along their polar axis will have zero torque component between the fields, at all other alignments there is a force with a torque component between the fields as well as the attraction / repulsion due to polarity of poles. So the no torque situation is the special case of perfect alignment between the two magnetic fields.

I notice nonsense like
>This can be read as a rotating electric field induces a magnetic field OR a changing magnetic field generates an electric field.
mathematically it can be read either way, but in reality what is there in an electric field that can rotate?
Maxwell's equations were in quaternion for a reason and trying to make full sense of it otherwise is foolish.

>moebis strip type relationship wherein they are not two sides of a coin they are the coin
This line alone gave me cancer.

>A changing magnetic field creates a moving electric charge.

Well, maybe "learn things in the historical context" isn't exactly the best way to go about it.
What I mean is rather that there is value in starting with the force laws, since forces are what can be actually measured in experiment, and then obtaining Maxwell's equations from those. I believe Jackson does this, and it may serve well to reinforce the idea that Maxwell's equations don't "come from nowhere." The mathematician and even the physicist might be okay with just postulating Maxwell's equations. If he is not satisfied with this, you may tell him about U(1) symmetries and show him the relativistic formulation and how to derive Maxwell's equations and whatnot, but that might lead one into the trap of thinking you can do physics without physical intuition and reasoning. Just take some symmetry principle and write down your equations to axiomatize physics. Then go and see what happens.

The mathematician in particular might lose out on a lot of physical intuition. The physicist has probably seen Coulomb and Biot-Savart as starting point in his introductory E&M course, but probably got to Maxwell's equations from those involving some handwaving. So maybe repeating that using differential vector calculus might be a good idea, even if only as an exercise.

A stationary charge creates an electric field.

A moving charge creates a magnetic field.

The rest is relativity.

You have to take something as an axiom i.e based on measurements.

In classical electromagnetism, when nobody even considered moving charges, the only equation was Coulomb's force law. You could use it to show that equations

[math]\nabla \cdot \varepsilon \bar{E}=0[/math]

and
[math]\nabla \times \bar{E}=\bar{0}[/math]

are true. But actually they are equilevant, becouse curl and divergence describes a field.

But whe you have moving charges, you need to use also Biot-Savart law and you then you have an another field, and another curl and divergence of that field.