Trying to understand electromagnetism

[vasya]

Homework Statement

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Relevant Equations

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Hi! I'm trying to understand electromagnetism. Please help me.
Say we have an electron start moving a sine wave like pattern along z axis with amplitude of 1m up and down with frequency equal to 100Mhz. I want to determine an electric and magnetic field vectors in any arbitrary point in any time. But as far as I understand this requires knowledge about every other point in space in every previous time frame. I know what is divergence and curl (not fully, but somehow). I know meaning of maxwell equations. But all of these still is not enough for understanding em waves I want to be able to create em field simulations in something like MATLAB.

 

[hutchphd]

How many years do you wish to devote to this pursuit of knowledge?

 

[vasya]

I already fighting with this question for a very long time. And I will pusuit this till full and most deep understanding possible

 

[Delta2]

It's not clear to me what you want to do:
You want to calculate the EM wave emitted by an electron that is moving in a sine wave pattern
OR
You want to calculate the motion of an electron under an EM sine wave?

 

[vasya]

Ideally I want to be able to do both of these. But in my question I talked about EM wave, produced by an electron.

 

[tech99]

I imagine that you are applying a vertical electric field in order to move the electron. When it moves it is an electric current, so it will then have a magnetic field surrounding its path, like a wire. I think these are the two fields involved for low frequencies. For high frequencies, the acceleration of the electron is significant, so then we apply the Larmor formula to find the power lost to radiation. The radiated fields are in phase with each other and with the current. The applied electric field is leading by 90 degrees at low frequencies and moves closer to the magnetic phase at high frequencies.

 

[Delta2]

Ideally I want to be able to do both of these. But in my question I talked about EM wave, produced by an electron.

Both of these problems are hard but they have already been solved at the beginning of the 20th century (1900s)

The first problem is a special case of the problem of the EM field around a point particle that follows a specific trajectory. Hold on while I found the link to wikipedia.

The second problem can be simplified a lot if we make the simplifying assumption that the EM wave emitted by the accelerating (under the influence of the EM wave) electron is negligible. Then the electron simply does harmonic motion.

 

[vasya]

My main consern is how to calculate, or at least imagine all changes and their consequences in the field. Because when a charge moves, a field changes everywhere with a speed of light. And change in every point produces curl in magnetic field. And now we have to calculate resulting magnetic field in every point in space. For that we need to know curl of every another point. Every point change in magnetic field produces a change in all electric field and so on.
The result is insane computational complexity

 

[Delta2]

My main consern is how to calculate, or at least imagine all changes and their consequences in the field. Because when a charge moves, a field changes everywhere with a speed of light. And change in every point produces curl in magnetic field. And now we have to calculate resulting magnetic field in every point in space. For that we need to know curl of every another point. Every point change in magnetic field produces a change in all electric field and so on.
The result is insane computational complexity

Maxwell's equations are linear PDE's (Partial Differential Equations) with unknowns the Electric and Magnetic Field and sources the current and charge density. The are methods of solving PDEs either analytically or numerically. MATLAB has some tools towards this goal.

 

[PeroK]

The result is insane computational complexity

That's modern mathematical physics for you!

 

[Delta2]

This is the more generic problem of the waves emitted by a point particle which follows a given trajectory
https://en.wikipedia.org/wiki/Liénard–Wiechert_potential

You need to know calculus and vector calculus in order to understand it and Maxwell's equations in the potential formulation and the retarded potentials solutions.

 

[Delta2]

But my question still unsolved...

Nope it isn't unsolved , the first problem as I said is a special case of the Lienard-Wiechert potential.

 

[vasya]

sry I just didn't see that you wrote a new message)

 

[hutchphd]

This is an intermediate level standard problem in electrodynamics. Do you have access to beginning and intermediate textbooks on classical electrodynamics? And vector calculus?
You cannot do this by force of intellect. You must learn the tools (maths) and then the details. What have you done to acquire the necessary tools?

 

[vasya]

I tried to read a ton of books and videos on this topic. And now I know a tiny bit of vector calculus. But I'm more of a visual-thinking person, And just by loooking on equations its very hard to determine what is going on...

 

[hutchphd]

Yes it is difficult and there is no Royal road.

 

[tech99]

Homework Statement:: -
Relevant Equations:: -

Hi! I'm trying to understand electromagnetism. Please help me.
Say we have an electron start moving a sine wave like pattern along z axis with amplitude of 1m up and down with frequency equal to 100Mhz. I want to determine an electric and magnetic field vectors in any arbitrary point in any time. But as far as I understand this requires knowledge about every other point in space in every previous time frame. I know what is divergence and curl (not fully, but somehow). I know meaning of maxwell equations. But all of these still is not enough for understanding em waves I want to be able to create em field simulations in something like MATLAB.

Not sure you have to look at all space. For an alternating system at low frequencies, the energy in the fields comes back to the source the next half cycle, so can't go very far. For high frequencies, the radiated part of the energy is just gone forever.

 

[MidgetDwarf]

I tried to read a ton of books and videos on this topic. And now I know a tiny bit of vector calculus. But I'm more of a visual-thinking person, And just by loooking on equations its very hard to determine what is going on...

This means that you do not have the background to learn the material needed for your question. So work on math.

What is your current math level, and what formal math and physics courses have you taken?

 

[vasya]

This means that you do not have the background to learn the material needed for your question. So work on math.

What is your current math level, and what formal math and physics courses have you taken?

I finished a high school. There wasn't any calculus or vector algebra in the curriculum in my country, so any further knowledge I acquired myself.

 

[Delta2]

I finished a high school. There wasn't any calculus or vector algebra in the curriculum in my country, so any further knowledge I acquired myself.

The math behind the solution of what you want to learn is quite more complicated than high school math. However I have to say that you did a fairly good job on describing the problem and its solution in qualitative/intuitive terms.

I enjoyed the term "Insane computational complexity" in one of your posts :D.

 

[vasya]

the question is: do fields really interact with each other in that complicated manner or it's just my misconception?

 

[PeroK]

the question is: do fields really interact with each other in that complicated manner or it's just my misconception?

The fields obey Maxwell's equations.

 

[Delta2]

the question is: do fields really interact with each other in that complicated manner or it's just my misconception?

Maxwell's equations in integral form seem to imply something like your intuitive understanding , that the E-field in a point at a time, depends on the values of the B-field in all the space around.

However this is not exactly true. Maxwell's equations in differential form tell us what exactly is happening point by point in space. So if we take faraday's law in differential form $\nabla\times \mathbf{E}=-\frac{d\mathbf{B}}{dt}$ tell us that the curl of the electric field at a specific point depends only on the time derivative of the magnetic field at the same point

But yeah you need to understand calculus and vector calculus before you fully understand Maxwell's equation in integral and differential form.

 

[vasya]

So does curl of B creates B in area around after C/distance_to_point_of_our_interest

 

[Delta2]

Curl of B creates E in the same point at the same time instant. Maxwell's Ampere's law in differential form and in free space is $curl(\mathbf{B})=\frac{d\mathbf{E}}{dt}$

 

[vasya]

Sorry. But does curl of B creates B in area around after distance_to_point_of_our_interest/C seconds?

 

[Delta2]

Don't want to spoil your intuitive understanding but In my opinion Curl B creates E and curl E creates B at the same point, at the same instant. It is wrong in my opinion to introduce the retarded time in your intuitive thinking like this.

The retarded time comes into play only after you have write the wave equation for B (or E) with source term the current and/or charge densities.

It is the current and/or charge density that creates B or E in distance x after time= x/c.

 

[PeroK]

Sorry. But does curl of B creates B in area around after distance_to_point_of_our_interest/C seconds?

Maxwell's equations encapsulate two things:

1) How the E and B fields depend on charge and current densities.

2) The relationship between the E and B fields.

These are expressed in terms of the divergence and curl of the fields. The transmission of EM information at the speed of light is encapsulated implicity in these equations. There is no retarded time in Maxwell's equations.

In order to analyse, study and solve these equations, however, it is useful to have the concept of electromagnetic potentials. This in turn leads to the idea of gauge transformations and retarded potentials. And, in particular, to use the heuristic idea that EM information travels at the speed of light. This is where retarded time enters the equations.

This allows the intuitive idea to generalise the Coulomb and Biot-Savart laws from electro- and magneto-statics to the general case of electrodynamics. But, the mathematics required to work out this simple idea is quite advanced, even for a single moving point charge (which leads to the Lienard-Wiechert potentials).

 

[vasya]

...I want to believe that there is the simple rules that govern all of electromagnetic phenomena. such rules, that by following/simulating them it is posible to derive everything like waves, their dispersion, refraction, etc. I'm starting to feel hopeless

 

[PeroK]

...I want to believe that there is the simple rules that govern all of electromagnetic phenomena.

They are called Maxwell's equations. Deal with it!

 

[Delta2]

Let me ask you something, you study Maxwell's equation in integral form or in differential form or both?

 

[vasya]

so how to simulate maxwell equations that way so I can get EM waves shown on screen of my computer? To simulate them in such a way so they will travel at a speed of light. I'm software developer myself, so I want to be able to simulate EM field in my head or at least on my pc. I call this an understanding.

 

[PeroK]

so how to simulate them that way so I can get EM waves shown on screen of my computer? I'm software developer myself, so I want to be able to simulate EM field in my head or at least on my pc. I call this an understanding

You are then talking about a specific solution to Maxwell's equations. Note that an EM wave is actually a vacuum solution (where there are no charges or currents). For example:
$$\vec E(z, t) = E_0 \cos(kz - \omega t) \hat x + \frac 1 c E_0 \cos(kz - \omega t) \hat y$$You can check that is a wave moving at speed $c$ in the $z$ direction and confirm that it satisfies Maxwell's equations in a vacuum. Note that $\omega = ck$.

You'd need 3D graphics of some description to simulate that on a screen.

 

[vasya]

maybe numerical methods would be more suitable for my problem... because for me your formula seems like completely unconnected with maxwell equations... I'm more like a geometry/visual thinking person. I hope to get understanding of em waves through geometry of maxwell equations one day

 

[Delta2]

Just keep studying, never give up. Maybe your thinking is correct and I just fail to understand it.