Author Topic: Questions for those who know electromagnetism better than I do  (Read 7173 times)

0 Members and 1 Guest are viewing this topic.

Offline cvancTopic starter

  • Frequent Contributor
  • **
  • Posts: 675
  • Country: us
(...which may very well be all of you  :-DD)

Hi all-

I'm scratching my head on a couple 'what ifs' regarding the shape of the magnetic field that gets set up around a wire carrying DC.

It's well known a thin round wire throws a circular field outside itself when carrying DC current. Right hand rule and all. My Googling shows it almost always worded exactly that way: 'thin round wire'.  It made wonder what would happen if you played with each of those words in turn. What shape does the magnetic field have then?

What happens if the wire isn't thin? What if it's say, inches in diameter?

What happens if the wire isn't round? What if the wire is triangular in cross section? Or square?  Or complex?

What happens if the wire isn't even a wire, but is instead a pipe (hollow)? Is there a field both inside and outside?

Thanks, I'm really curious about this (what software lets me play with stuff like this?)
 

Online rstofer

  • Super Contributor
  • ***
  • Posts: 9963
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #1 on: August 20, 2021, 03:27:51 am »
You wind up dealing with Maxwell's Equations and NOBODY wants to go there voluntarily.  Three people in history have understood Maxwell's Equations:  Maxwell, Feynman and Einstein.  The rest of us just think we know what is going on.  In the 48 years since I graduated, I have used Maxwell's Equations exactly never.

MATLAB can help!  Google for "MATLAB Maxwell's Equations".  There is some chance that what you find for MATLAB will also run on the free work-alike, Octave.

https://www.mathworks.com/matlabcentral/fileexchange/70394-biot-savart-law
https://www.mathworks.com/matlabcentral/fileexchange/48990-magnetic-field-simulator

https://en.wikipedia.org/wiki/Maxwell%27s_equations
« Last Edit: August 20, 2021, 03:45:18 am by rstofer »
 

Online ejeffrey

  • Super Contributor
  • ***
  • Posts: 4033
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #2 on: August 20, 2021, 03:50:19 am »
Thin doesn't matter at all.  A thick wire will do just fine. So will a pipe, and the field inside the pipe will be zero if everything is perfectly symmetric.

If the wire cross section is some other shape the result is qualitatively the same but quantitatively different.  More than a few wire diameters away it will basically look the same but near the wire will be different.
 
The following users thanked this post: RJSV

Offline CatalinaWOW

  • Super Contributor
  • ***
  • Posts: 5569
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #3 on: August 20, 2021, 04:15:17 am »
You can use Maxwell's equations for this, but for a DC current the magnetic field is given simply by Ampere's law.  The thin wire assumption avoids a problem that I have never seen discussed.  Many texts and online resources give solutions for a cylinder were the current density is uniform in the cross section of the cylinder.  But they never discuss whether the current density would be uniform.  Given the uniform current density assumption the solution is simple geometry, calculating the area inside a radius which then gives the field.

Presumably doing a full Maxwell's equation simulation including forces on the electrons starting from some initial distribution would give the actual current distribution.  I have never done this, and haven't thought of or seen a simplifying assumption that would justify the uniform density assumption.  But like the OP I am quite sure that I am not the sharpest tool in the shed.

The answers are mostly of academic interest.  As ejeffrey says, the answers converge to the same thing a few diameters outside of the wire.  Or in the pipe case, a few thicknesses outside the outer skin.
 

Offline T3sl4co1l

  • Super Contributor
  • ***
  • Posts: 22436
  • Country: us
  • Expert, Analog Electronics, PCB Layout, EMC
    • Seven Transistor Labs
Re: Questions for those who know electromagnetism better than I do
« Reply #4 on: August 20, 2021, 04:26:27 am »
Well, I wouldn't disregard theory out of hand... I certainly have a good grasp of it.  Granted, people call me smart, but I'm no Feynman either.

Regarding wire diameter, the principle is Ampere's law, in same class as Gauss's law and so on, which state that the sum flux through a boundary only depends on what's enclosed within that boundary.  In the Amperian case, it's a closed loop curve, the enclosed area of which carries a current, and the flux is the magnetic field parallel to the path.  Gauss's law applies to electric charge contained within a shell, the flux of which (electric charge) extends through, perpendicular to the shell; but also to gravity for another example, where the acceleration depends on the mass contained within a shell.

As with the Earth: if we could stand at the same altitude (distance from center), and imagine all its mass compacted into an infinitesimal point, we would experience the same acceleration as with all that mass distributed as it is currently.  All that matters is that the mass is contained within a spherical shell beneath our feet.  (The exact shape of the shell, does and doesn't matter, due to more rigorous conditions that I won't go into detail about; suffice it to say, this works best when the shell shares the same symmetry as the problem, i.e. a spherical shell for a spherical mass distribution, and the shell is larger than said distribution.)

So, too, the magnetic field around a wire, at a given distance from center, is independent of the wire diameter, so long as the wire radius is smaller.

And as long as the wire is cylindrically symmetric, the law still applies within it; the magnetic field drops smoothly from its value at the surface of the wire, down to zero at the center.

(The story is different for AC, where self-induction within the wire causes an opposing magnetic field, cancelling out the internal field: skin effect.  Needless to say, this is more complicated, so I won't go into detail; just to note there's interesting things happening when wires or currents are changing over time.)


Which also gives us the tool to understand the other conditions.
Hollow: no field inside.  In effect, all the surrounding currents cancel out here, but we would need a lengthy calculation to show it that way!

Other (non-circular) cross sections: same outside the highest peak, and same inside the lowest valley.  The exact field now depends on angle as well as distance from center, so we can't say what the field is, at every point, so simply.  We might need to use a numerical solution in this case (e.g. Biot-Savart law, replacing the integral with a Riemann sum).

Tim
Seven Transistor Labs, LLC
Electronic design, from concept to prototype.
Bringing a project to life?  Send me a message!
 
The following users thanked this post: Gregg

Offline bob91343

  • Super Contributor
  • ***
  • Posts: 2675
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #5 on: August 20, 2021, 04:33:24 am »
Many of these questions can be answered by studying skin effect.  This deals with fields inside wires as well as outside, and shows what happens if a wire is hollow or not round.

A magnetic field can exist within a conductor, as will be learned in skin effect.  Almost any basic electronics text addresses it.
 

Online rstofer

  • Super Contributor
  • ***
  • Posts: 9963
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #6 on: August 20, 2021, 04:18:58 pm »
Well, I wouldn't disregard theory out of hand... I certainly have a good grasp of it.  Granted, people call me smart, but I'm no Feynman either.
I was never comfortable with the curl and divergence operators and I sure don't miss triple integrals.  It's clear you have kept up more skills in this area.
Quote
Other (non-circular) cross sections: same outside the highest peak, and same inside the lowest valley.  The exact field now depends on angle as well as distance from center, so we can't say what the field is, at every point, so simply.  We might need to use a numerical solution in this case (e.g. Biot-Savart law, replacing the integral with a Riemann sum).
Tim
I wrote some Fortran code for the left, center and right Riemann sums a while back.  The center sum compares nicely with the definite integral.  Trapezoidal integration is also workable because, in the limit, it is essentially the same as a center Riemann sum.  The code (attached) is a little pedantic, it was written for a tutorial.  The actual output is to the console but those lines are copied and pasted to the source as comments on lines 61..69.  Drop the .txt from the file name and compile with gfortran.

Skin effect is one of the reasons that copper or aluminum pipe is used for the high voltage conductors in some electric substations.  Pipe also has more useful structure than sagging wire.

A more interesting question is what does the field look like if the current is passing along an irregular shape.  I was thinking about an I beam but I think a piece of railroad track would be even more complex.  I'm wondering what it looks like at the 90 degree internal corners.  Something weird should be happening around the junction between the web and the flange(s).
 

Offline sandalcandal

  • Supporter
  • ****
  • Posts: 641
  • Country: au
  • MOAR POWA!
Re: Questions for those who know electromagnetism better than I do
« Reply #7 on: August 20, 2021, 05:37:39 pm »
I believe FEMM can let you simulate and play around with relatively simple 2D cross section questions like this.
https://www.femm.info/wiki/HomePage

Not exactly what you you were asking but a star cross section toroid. Change the .txt to .fem to open it.
Disclosure: Involved in electric vehicle and energy storage system technologies
 
The following users thanked this post: thm_w, Gregg

Online rstofer

  • Super Contributor
  • ***
  • Posts: 9963
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #8 on: August 20, 2021, 05:58:27 pm »
That's neat!  I sure could have used something like that back in the day.
 

Offline Slartibartfast

  • Regular Contributor
  • *
  • Posts: 63
  • Country: de
Re: Questions for those who know electromagnetism better than I do
« Reply #9 on: August 23, 2021, 11:08:37 pm »
You wind up dealing with Maxwell's Equations and NOBODY wants to go there voluntarily.  Three people in history have understood Maxwell's Equations:  Maxwell, Feynman and Einstein.  The rest of us just think we know what is going on.

You're trying hard to be silly here, aren't you?

Maxwell's equations really aren't that difficult to understand, particularly the local (i.e. differential) version. The global (integral) version is a bit harder, but more practical in some situations.
 
The following users thanked this post: Dave

Offline Slartibartfast

  • Regular Contributor
  • *
  • Posts: 63
  • Country: de
Re: Questions for those who know electromagnetism better than I do
« Reply #10 on: August 23, 2021, 11:14:25 pm »
You can use Maxwell's equations for this, but for a DC current the magnetic field is given simply by Ampere's law.

Do you realise that Ampére's Law is nothing else but the (in usual counting) fourth Maxwell's equation?
 

Offline CatalinaWOW

  • Super Contributor
  • ***
  • Posts: 5569
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #11 on: August 24, 2021, 04:19:00 am »
You can use Maxwell's equations for this, but for a DC current the magnetic field is given simply by Ampere's law.

Do you realise that Ampére's Law is nothing else but the (in usual counting) fourth Maxwell's equation?

Of course.  And you can the prove to yourself that the time derivative of a DC current is zero and write it looking like the original Ampere's Law which is accessible to those with less math background.

You realize that you can use Maxwell's equations to do circuit analysis.  But most folks find the simplified equations of Kirchoff are adequate and far easier to solve.
 

Online rstofer

  • Super Contributor
  • ***
  • Posts: 9963
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #12 on: August 24, 2021, 05:23:25 am »
You wind up dealing with Maxwell's Equations and NOBODY wants to go there voluntarily.  Three people in history have understood Maxwell's Equations:  Maxwell, Feynman and Einstein.  The rest of us just think we know what is going on.

You're trying hard to be silly here, aren't you?

Yes and no.  I remember Field Theory as the most difficult and least interesting course I had to take.  I will take the opportunity to avoid Maxwell's Equations at every opportunity.  Yes, intuitively, I understand them (with a few minutes review) but the math can get really ugly.

Quote

Maxwell's equations really aren't that difficult to understand, particularly the local (i.e. differential) version. The global (integral) version is a bit harder, but more practical in some situations.

 

Offline Slartibartfast

  • Regular Contributor
  • *
  • Posts: 63
  • Country: de
Re: Questions for those who know electromagnetism better than I do
« Reply #13 on: August 24, 2021, 08:32:04 am »
You can use Maxwell's equations for this, but for a DC current the magnetic field is given simply by Ampere's law.

Do you realise that Ampére's Law is nothing else but the (in usual counting) fourth Maxwell's equation?

Of course.  And you can the prove to yourself that the time derivative of a DC current is zero and write it looking like the original Ampere's Law which is accessible to those with less math background.

I'm not sure what you're referring to. The time derivative of current does not enter anywhere. Are you talking about so-called "displacement current"? That name is misleading, it is not a current, but a varying electric field.

AFAIK the only difference between the "original" and later version of Ampére's Law is the inclusion of the time derived electric field in the latter. That the time derived electric field is zero, is trivial in the given situation, and does not need proof.

Quote
You realize that you can use Maxwell's equations to do circuit analysis.  But most folks find the simplified equations of Kirchoff are adequate and far easier to solve.

That includes me, of course. But the analogy is a poor one: Kirchhoff's are derived from Maxwell's, but Ampère's IS one of Maxwell's.

I'm always amused when guys declare Maxwell's are too difficult to deal with, but then go and happily use Gauss' Law (1st and 2nd of Maxwell's), or Faraday's Law (3rd Maxwell's), or even Ampére's Law (4th), and claim that's a simplification. As simplification goes, the situation given by the topposter in fact calls for using the Biot-Savart law, which is the general solution of Ampére's Law (which by itself is a differential equation) in the given situation.

Wikipedia has a quite decent article on Maxwell's. It's well written IMHO, and very accessible, at least when excluding the advanced parts (e.g. relativistic, i.e. manifest Lorentz invariant formulation, and the classical limit of quantumelectrodynamics).
 

Offline Slartibartfast

  • Regular Contributor
  • *
  • Posts: 63
  • Country: de
Re: Questions for those who know electromagnetism better than I do
« Reply #14 on: August 24, 2021, 08:47:16 am »
Many texts and online resources give solutions for a cylinder were the current density is uniform in the cross section of the cylinder.  But they never discuss whether the current density would be uniform.

That's because there isn't much to discuss. In the general case it is not uniform, of course. The case that is usually assumed if talking about a uniform current density distribution needs:
0.) a DC current. With AC, effects related to skin effect destroy the uniformity.
1.) the conductor to be the shape of a generalised cylinder (i.e. size, shape, and orientation of crosssection constant along it's length).
2.) uniform conductor's material (more specifically, constant conductivity across the conductor's volume).
3.) the electric potential to be constant across the end surfaces.

That's it. If any of these are violated, the current densitiy in the best case is only approximately uniform. Violation of more than one can be made such that their effects on the current uniformity cancel, but that's a very tricky business.
 

Offline Slartibartfast

  • Regular Contributor
  • *
  • Posts: 63
  • Country: de
Re: Questions for those who know electromagnetism better than I do
« Reply #15 on: August 24, 2021, 09:00:13 am »
I will take the opportunity to avoid Maxwell's Equations at every opportunity.

That won't be difficult, as pretty much all cases where they are analytically solvable are covered in literature, and all other cases need numerical methods (simulation) to solve.

Quote
but the math can get really ugly.

I guess you just haven't seen ugly math. ;) Dealing with Maxwell's equations is a treat compared to some stuff in General Relativity, or even worse, Quantum Field Theory.

But, yes, high school math is not sufficient to solve Maxwell's equations.
 
The following users thanked this post: Dave

Offline bsdphk

  • Regular Contributor
  • *
  • Posts: 212
  • Country: dk
Re: Questions for those who know electromagnetism better than I do
« Reply #16 on: August 24, 2021, 12:57:30 pm »
Three people in history have understood Maxwell's Equations:  Maxwell, Feynman and Einstein.

Just because you do not grok them, does not mean that only card-carrying geniuses can.

There are lots of non-nobel-laurates who have done amazing things with Maxwells Equations, for one thing Bell Labs where knee-deep in the stuff for 60 years.

Here is a random example: https://archive.org/details/bstj40-1-233/mode/2up
 
The following users thanked this post: Dave

Online rstofer

  • Super Contributor
  • ***
  • Posts: 9963
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #17 on: August 24, 2021, 05:46:33 pm »
I will take the opportunity to avoid Maxwell's Equations at every opportunity.

That won't be difficult, as pretty much all cases where they are analytically solvable are covered in literature, and all other cases need numerical methods (simulation) to solve.

Quote
but the math can get really ugly.

I guess you just haven't seen ugly math. ;) Dealing with Maxwell's equations is a treat compared to some stuff in General Relativity, or even worse, Quantum Field Theory.

But, yes, high school math is not sufficient to solve Maxwell's equations.

I took that Field Theory class in '73.  No computers, no plotters, no calculators, if numbers were used so were slide rules.  Everything was a handwave and all that was presented were endless equations.  I still have the book but it hasn't gotten a lot of reading time over the decades.  I noticed the Bell Labs paper had not a single graph, chart or plot.  Just endless equations.  Not my cup of tea but it was written in 1961.  Tools just weren't as sophisticated.

Breaking up the Bell system was a huge mistake.  Bell Labs was a national treasure.  Now owned by Nokia...  Eleven Nobel Prize winners worked for Bell Labs.

There's a reason I concentrated on digital in grad school.
 

Offline mawyatt

  • Super Contributor
  • ***
  • Posts: 4117
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #18 on: August 25, 2021, 12:51:57 am »
Breaking up the Bell system was a huge mistake.  Bell Labs was a national treasure.  Now owned by Nokia...  Eleven Nobel Prize winners worked for Bell Labs.

I've always considered breaking up of Bell Labs as one of the greatest blunders of all time. Recall a comment by the USG about this, saying "don't worry our nationals labs will continue with all the research!!", yeah right :P

Best,
Curiosity killed the cat, also depleted my wallet!
~Wyatt Labs by Mike~
 

Offline mawyatt

  • Super Contributor
  • ***
  • Posts: 4117
  • Country: us
Re: Questions for those who know electromagnetism better than I do
« Reply #19 on: August 25, 2021, 01:02:49 am »
Isn't Oliver Heavyside credited with consolidating Maxwell's works into what today we refer to as "Maxwell's Equations"?

I took Fields and Waves long ago and recall we used this conductive paper which we painted conductors on. This was basically a type of analog simulation where you could measure and plot the equipotential field lines using a VOM and power supply. This was the same time when we were designing hybrids using Rubylith and an Exacto knife ::)

Best,
Curiosity killed the cat, also depleted my wallet!
~Wyatt Labs by Mike~
 

Offline T3sl4co1l

  • Super Contributor
  • ***
  • Posts: 22436
  • Country: us
  • Expert, Analog Electronics, PCB Layout, EMC
    • Seven Transistor Labs
Re: Questions for those who know electromagnetism better than I do
« Reply #20 on: August 25, 2021, 08:24:37 am »
Yes, Heaviside invented the form of vector calculus as we know it today, simplifying E&M down to the four field equations we use most commonly.  He also brought complex numbers to electronics!

Maxwell himself only collected the known laws/relationships, IIRC.  We could also credit Einstein for inventing the tensor shorthand, in which the whole law is merely a single equation.  The expressiveness is no accident, as relativity is built upon E&M, an extrapolation of the curious form of Lorentz invariance.

Tim
Seven Transistor Labs, LLC
Electronic design, from concept to prototype.
Bringing a project to life?  Send me a message!
 

Online iMo

  • Super Contributor
  • ***
  • Posts: 5570
  • Country: va
Re: Questions for those who know electromagnetism better than I do
« Reply #21 on: August 25, 2021, 09:28:56 am »
My first lecture on "EM Field Theory" started with an intro given by the professor - "..do you really think the electrons transfer the energy from the power station to your household?? Have you ever seen an electron?? Electron is small and lightweight, almost nothing.. Moreover, the electrons travel a few centimeters in a second in a copper wire.. You would never have lit a lamp on your desk if it had worked that way.."
And he followed with Maxwell's equations, messing with calculus integrating EM fields over the entire spacetime of Universe, as that is the way the EM energy lights up my lamp.. A positive part of this EM Field Theory nightmare - the professor was a passionate Sinclair ZX Spectrum user and he was encouraging us to write some code solving his EM stuff numerically (as the Basic computers were pretty common among students in that time already)..
 :D
« Last Edit: August 25, 2021, 09:32:58 am by imo »
Readers discretion is advised..
 

Offline T3sl4co1l

  • Super Contributor
  • ***
  • Posts: 22436
  • Country: us
  • Expert, Analog Electronics, PCB Layout, EMC
    • Seven Transistor Labs
Seven Transistor Labs, LLC
Electronic design, from concept to prototype.
Bringing a project to life?  Send me a message!
 
The following users thanked this post: duckduck

Offline Slartibartfast

  • Regular Contributor
  • *
  • Posts: 63
  • Country: de
Re: Questions for those who know electromagnetism better than I do
« Reply #23 on: August 27, 2021, 06:11:42 pm »
Yes, Heaviside invented the form of vector calculus as we know it today, simplifying E&M down to the four field equations we use most commonly.  He also brought complex numbers to electronics!

Maxwell himself only collected the known laws/relationships, IIRC.  We could also credit Einstein for inventing the tensor shorthand, in which the whole law is merely a single equation.  The expressiveness is no accident, as relativity is built upon E&M, an extrapolation of the curious form of Lorentz invariance.

With that, you have the causal relationships all upside down. You'd be right if you only wanted to describe the historic sequence and linkage of discoveries, but as a causal chain of nature's features, it does not make sense.

You cannot consider electromagnetism more fundamental than space and time. Lorentz invariance is a fundamental property of spacetime, it is a subset of the full set of symmetries of spacetime described by the Poincaré group. It is this fact, where Lorentz invariance of electromagnetism comes from, not the other way round. Being a symmetry of spacetime, every physical law must be rewritable in a manifestly Lorentz invariant form; this has been an important guidance principle for physics research since it's discovery in the early 1900s. Maxwell's equations is just the first instance where Lorentz invariance had been discovered, and it took until around 1900 because the Maxwell equations in the forms known before are not manifestly Lorentz invariant. It could be argued that Heaviside helped obscuring it, by introducing a form of vector algebra and calculus that only works in three dimensions and cannot be simply extended to spacetime's four dimensions.

Tensors are not shorthand, they are mathematical (geometric, in fact) objects, just like vectors. They are not an invention of Einstein either. Riemann and Christoffel used them many decades earlier to built the then new field of differential geometry, which later turned out to be an essential ingredient for General Relativity. When Einstein had his main insights about the principles that would allow him to generalise Special Relativity, he was searching for ways to express them in mathematical ways. He had his mathematician friends teach him about matrices (not at all commonly known at the time) and differential geometry, and tensors.
 
The following users thanked this post: Rod

Offline Slartibartfast

  • Regular Contributor
  • *
  • Posts: 63
  • Country: de
Re: Questions for those who know electromagnetism better than I do
« Reply #24 on: August 27, 2021, 06:41:05 pm »
My first lecture on "EM Field Theory" started with an intro given by the professor - "..do you really think the electrons transfer the energy from the power station to your household?? Have you ever seen an electron?? Electron is small and lightweight, almost nothing.. Moreover, the electrons travel a few centimeters in a second in a copper wire.. You would never have lit a lamp on your desk if it had worked that way.."

An exceedingly silly rant, if it really happened. Of course the marching electrons (a.k.a. current) need the electric field (a.k.a.) voltage, to, as the multiplicative product, deliver the power. Makes you wonder, if the uphill bicyling professor cannot find out where his efforts go, surely not into the movement (a.k.a. velocity)? Completely analogous, the velocity needs the backwards pulling force, to, as the product, become the power sink that makes him pant.

I'd consider this a way to confuse students, a pedagogical desaster.
 
The following users thanked this post: Rod


Share me

Digg  Facebook  SlashDot  Delicious  Technorati  Twitter  Google  Yahoo
Smf