I Visualizing fields around coils (statics and magnetostatics)

[SredniVashtar]

TL;DR Summary

Looking for examples of neat ways to visualize the magnetic field B, the vector potential A and their time derivatives of currents in curled paths in a quasi-static context. (Real conductor simulations welcome, as well).

I am looking for some neat way to visualize various fields in the space around the usual electronic components: a straight wire, a single turn circular coil, a multi-turn finite solenoid, and an infinitely long cylindrical solenoid.
I can get the expression, either in their analytical exact form (the infinitely long rectilinear wire and infinitely long solenoid are treated in all textbooks) or some approximation of the integrals involved. I have found publications with the A and B fields of single circular filamentary currents, and also of a finite solenoid (NASA Technical Note D-465, https://ntrs.nasa.gov/citations/19980227402).

But before embarking in the time consuming task of plotting a 3D entity in 3D space, I'd like to see some neat examples of the representation of these fields. By experience, my biggest problem with visualizing vector fields is with scaling: field lines can gets too crammed in one part and nonexistent in other parts, and when I try to plot arrow fields, I get either absurdly long arrows where the field is strong and points or empty space where it is weak.
How to plot - or even draw - these fields (the static A and B fields) in a way that gives an intuitive view of their direction (ok, circles around straight wires, deformed circles around the single loop, elongated football shapes around the finite solenoid, for the B field) and also magnitude?

I am looking for a didactic representation, more than a faithful numerical representation. If you look at the NASA technical node, the graphs are actually useful (smart people at NASA) but they are not exactly intuitive...

Also, do you agree that in the quasi-static approximation (the dynamic case is a sequence of static snapshots), the dA/dt (which - sign apart is the induced electric field) and dB/dt (which - sign apart - is the curl of E) fields should have the same 'shape' of the A and B fields? Can be in the same direction if the field is increasing or in the opposite direction when it is decreasing, but if the field is going in circles, then its time-derivative does too, right?

 

[Dale]

To plot vector fields I usually use a stream line plot rather than a vector plot. And I usually augment it with a color background for the magnitude, or sometimes contour lines for potentials.

 

[robphy]

Possibly useful:
https://www.glowscript.org/#/user/G...andinteractions/program/MatterAndInteractions

• 17-B-loop-with-r-dB
  
  

https://www.falstad.com/vector3dm/

• https://www.falstad.com/vector3dm/v...6&a1=40&a2=30&a3=36&rx=69&ry=-7&rz=-21&zm=1.2
  
  

• https://www.falstad.com/vector3dm/v...6&a1=40&a2=30&a3=36&rx=70&ry=-6&rz=-21&zm=1.2
  
  

https://web.mit.edu/8.02t/www/802TEAL3D/visualizations/guidedtour/Tour.htm#_Toc27302357

 

[SredniVashtar]

...and of course the winner had to be MIT.
I remember seeing those figures years ago. The page you linked is crammed with knowledge.

So, it seems they use a mixed 2D-3D representation: the object (the coils) is in 3D, while the streamlines - actually they call it 'grass seeds' and 'iron filings' lines - are shown in the saggital plane. This prevents the cramming of 3D field lines as seen in the Falstad simulations (which are good, don't get me wrong!)
In the MIT figures there is a mix of streamlines with their own alternating intensity (to show individual lines), and an overall intensity of the plot associated with the magnitude of the field (hence the disappearing into darkness - which is the part I am interested in).
A plot like this in the saggital plane and another one in the azimuthal plane should give a clear and intuitive view of the field.

Now, the problem is how to alternate the color of streamlines in order to get these 'grass seeds'/'iron filings' lines.

 

[robphy]

Possibly useful:
https://web.mit.edu/8.02t/www/802TEAL3D/visualizations/magnetostatics/index.htm

I don't recall Belcher's visualizations being real-time interactive (like Falstad's or Chabay+Sherwood's).
So, I think a lot of computation and rendering happened offline... then made into a video.

While these are all pretty to look at, I'm not sure if these representations as vector-fields
truly capture the physics and mathematics of the field configurations.

For a while, I've been interested in visualizing electromagnetic fields with differential-forms,
inspired by Bill Burke ( archived website: https://www.ucolick.org/~burke/home.html )
I have an ancient website that's in need of an update
https://www.visualrelativity.com/vrml/electromagnetism/
https://www.visualrelativity.com/vrml/electromagnetism/x_ite/
... probably to use Chabay+Sherwood's Glowscript/WebVPython ( https://www.glowscript.org/ )

 

[SredniVashtar]

I have played around with Falstad's simulations and it seems there's a problem with the toroidal solenoid. It seems it shows the A lines instead of the B lines.

https://www.falstad.com/vector3dm/v...&a1=46&a2=14&a3=100&rx=72&ry=7&rz=-4&zm=1.657

Strange because at one point I had the B lines inside the solenoid in white and nearly dark outside, as expected. But now I just can't get what I expect (and what is expected by theory). Does it happen to me alone?

The linear solenoid seems fine.