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Field lines are used for explaining a wide variety of phenomenon. But is it really an accurate depiction of reality?

Is it more accurate to imagine a field in a different manner. For instance, using grey-scale colour to imagine intensity. For instance, for a positive point charge, instead of imagining infinite lines emanating from the point, we image concentric shells around the charge, of infinitesimally thin thickness and all of a different grey-scale colour. So if we imagine black to be the maximum strength(intensity) and white to be 0 intensity, you would effectively imagine an infinitely large sphere whose colour change from white at infinity to black at the centre. Rough approximation: enter image description here

The reason why I thought this might be more accurate was because we can do away with the whole problem (not sure if it is actually a problem!) of having gaps between lines that are filled with infinite other lines. It also seemed more natural to imagine this for inverse square law obeying phenomenon in 3d.

But I lost my confidence when I saw iron filings on a sheet above a bar magnet actually taking up shapes of lines. I think that this may be what inspired Faraday and other at the time. But I think that that could actually be because of some attraction that a magnetised filings have on each other. ie, if you were to move an entire field line of filings to a 'line' between itself and an adjacent filing, it wouldn't move back, would it?

So I'd like to know if this kind of thinking is a more accurate representation of reality?

EDIT: As Daniel Knapp points out: in case of a uniform field, one cannot determine the direction using this technique. It has to be explicitly mentioned. However, I think that for more complicated fields, would this be more accurate?

EDIT2: I think that using a similar diagram with 6 colours with suitable alpha values for up, down, 4 sides would be better. There will be atmost 3 colours blending together for any 2d slice, so it would represent it quite well, imho. I welcome comments regarding that, but the question has been answered and sadly, in the views of a few, this may not be the ideal place to have such discussion about improved field diagrams.

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    $\begingroup$ +1 for the bit on iron filings. I think you've more or less figured out the answer yourself. $\endgroup$
    – Kyle Oman
    Commented Jul 16, 2013 at 16:57
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    $\begingroup$ It is always true that you use different visualization aids for different purposes. Direction of the field is important in most basic pedagogical, so you need some kind of visualization that includes direction. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Jul 16, 2013 at 17:13
  • $\begingroup$ @dmckee: A few other people raised the question about direction. The change in intensity (dark to light) covers that, imho. But a uniform field would be its achilles' heel. $\endgroup$
    – mehfoos
    Commented Jul 16, 2013 at 18:23
  • $\begingroup$ And several people have already told you that not all fields can be represented as an intensity map and still get the direction right. In particular you can't even draw the field of a bar magnet or that around a current carrying wire that way because they have non-zero curl (equivalently they can not be represented as the diverges of a scalar field). I agree that the "field line" visualization has some deficiencies, but what you propose is used in other places (the search term you want is "heat map") and not used in E&M because it isn't the right one. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Jul 16, 2013 at 18:32
  • $\begingroup$ I see the limitation for non-zero curls. But I still think there is a better general purpose alternative. For instance: using more colours? 6 in total for all the direction issues, with a suitable alpha value so that the blending components can be figured out to determine the direction. Only three colours need to combine at once if you think about it, so I think that should be okay. It would be able to depict the field in much greater detail and avoid a lot of ambiguity imho. $\endgroup$
    – mehfoos
    Commented Jul 16, 2013 at 19:12

3 Answers 3

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Field line descriptions stand just for a pictorical description of vector fields. They are usually asumed to be smooth functions $\mathbb R^N\to \mathbb R^N$, so the problem you claim to solve is actually not a problem: you just fill the "missing vectors" with the information you get from your neighboors.

More important, the picture you upoloaded doesn't tell de direction of the field nowhere! So the only field it can represent, is a scalar field.

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    $\begingroup$ "More important, the picture you upoloaded doesn't tell de direction of the field nowhere! So the only field it can represent, is a scalar field." Is that necessarily true? I can take the gradient of this picture pretty easily in my head, if I picture dark as 'hilltops' and light as 'valleys'. Then I can easily define the direction 'downhill'. Not saying this is the best representation of a vector field, but I think it's a valid representation. $\endgroup$
    – Kyle Oman
    Commented Jul 16, 2013 at 16:56
  • $\begingroup$ Dark to light gives the direction. We can get rid off arrows too as you any two pixel can be used to convey the information. Makes better use of the image pixels available ;)T Also do you mean to say that the iron-filings assumption I made is right. The filing appear like lines because an inter-iron-filing attraction is involved? $\endgroup$
    – mehfoos
    Commented Jul 16, 2013 at 17:00
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    $\begingroup$ @Kyle: Not every field is the gradient of a scalar function. $\endgroup$
    – user4552
    Commented Jul 16, 2013 at 17:01
  • $\begingroup$ @Ben agreed, but some are, and this diagram could depict such a field. If you throw in a non-zero curl... better start drawing arrows I think. $\endgroup$
    – Kyle Oman
    Commented Jul 16, 2013 at 17:02
  • $\begingroup$ @user1218748 - my intuition is that you're right about the filings. There are no distinct "lines", the field of a point charge is spherically symmetric, not discrete. $\endgroup$
    – Kyle Oman
    Commented Jul 16, 2013 at 17:04
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I think the arrows are required for clarity in magnitude and direction of the field. Your picture illustrates your point in a lovely way, but what happens when the electric field is uniform (such as between parallel plates)?

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  • $\begingroup$ I think that magnitude is represented by the colour shade, isn't it? Do conventional field line diagrams represent magnitude at all? As for direction: dark to light. You make use of the entire picture resolution too, as every pixel is used. When it's uniform!! Good catch! $\endgroup$
    – mehfoos
    Commented Jul 16, 2013 at 17:08
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    $\begingroup$ @user1218748: Do conventional field line diagrams represent magnitude at all? Yes, it's represented by the density of field lines. $\endgroup$
    – user4552
    Commented Jul 16, 2013 at 17:10
  • $\begingroup$ Of course! Got carried away there. $\endgroup$
    – mehfoos
    Commented Jul 16, 2013 at 17:12
  • $\begingroup$ With parallel plates, one plate is white, one plate is black, and the space in between is a smooth gradient. $\endgroup$
    – user20574
    Commented May 1, 2017 at 4:01
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Field lines generally depict the direction of the field vector at that point. So if you have a set of field lines, and if you draw a tangent to any one of those lines at some point, the tangent will give the direction of the field vector at that point.This is precisely why the iron fillings align themselves into lines in a magnetic field. They are in fact aligning themselves parallel to the field at that point.

The concentration or density of the field lines depicts the magnitude of the field at that point. Although, I think this is a very loose definition, and it has lots of potential problems with it. For eg. you can have infinite number of lines near that point, so how exactly would you define density? Personally, I have never been satisfied with this part of the depiction, and I believe your way presents this better.

This answers your first question, that yes field lines are an accurate depiction of reality, they tell you the direction of the field at that point, which your diagram doesn't do so efficiently.(The diagram for a point charge looks easy to understand, try making one for complicated field like the one for a dipole, and you'll realize that field lines are better in understanding how the field is acting in the space around it.)

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  • $\begingroup$ Just to clarify, so do you mean that the assumption I made about moving the line in my question is inaccurate? If no, then I guess we're on the same page and please ignore the following: I did not get the explanation for iron filings aligning themselves as lines. The field is continuous, so, unless it is due to some inter-iron-filing interactions, why would they align themselves into lines rather than a continuous pattern such as the image I uploaded (modified for dipole)? $\endgroup$
    – mehfoos
    Commented Jul 16, 2013 at 17:58
  • $\begingroup$ About the density of field lines being loose definition: I think it helps to look at whether they are diverging or converging. If they are diverging, then the density has to decrease as the same number of lines passing through an earlier section is passing through a larger volume. About a similar diagram for dipole: it would take more time to generate, but I have a feeling it'd be easier to understand without much ambiguity unless it's a simple uniform field. In fact, I think it gets more difficult to generate and more easier to understand as the the complexity of the field increases. $\endgroup$
    – mehfoos
    Commented Jul 16, 2013 at 18:03
  • $\begingroup$ @user1218748 Maybe its a geometrical constraint. The iron fillings are long and thin, so they tend to look like lines. Perhaps iron powder will give you different results. And, pardon me but I really can't understand what you mean by "moving the line" $\endgroup$
    – udiboy1209
    Commented Jul 16, 2013 at 18:03
  • $\begingroup$ Oh wait, I got it... You mean to say that if you align all the iron fillings on one line along another, they shouldn't move, right? Well, you also have to consider that the iron fillings are magnetized and have fields of their own, so they will repel each other and try to attain an equilibrium, which would explain the equal spacing between lines. $\endgroup$
    – udiboy1209
    Commented Jul 16, 2013 at 18:07
  • $\begingroup$ Good point; I think we are on the same page, then. I did think that it would not be possible to move the field line, but I couldn't think of a better way to convey the message that I think the field lines are not justified by looking at the pattern you get with the iron filings alone. $\endgroup$
    – mehfoos
    Commented Jul 16, 2013 at 18:11

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