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I wanted to ask a question. My question is that are the properties of field lines ( number of field lines leaving/entering a point charge is proportional to the charge, field strength between points can be compared using relative field line density etc.) true for any symmetrically drawn diagram. Like if I had charges $-q$ and $2q$, then will out of all the field lines drawn symmetrically from $2q$ only half of them enter $-q$? Or are the field lines drawn in such a way that they they follow the mentioned properties?

Also what is the mathematical idea associated with field lines? For example divergence of a vector field is intuitively 'how much' of a source/sink a point is. Rigorously, it is limit of the flux of vector field per area of surface as the surface approaches zero. I am asking this because I feel that this may help me derive results related to field lines in a well-drawn diagram. So, could somebody tell me what to associate a 'field line' with? Flux? Electric field strength? Something else?

Emilio Pisanty
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user194517
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3 Answers3

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Field lines are a graphical representation of a vector field (you can draw as many lines as you like). In a well drawn three dimensional sketch, the line density will be proportional to the field strength in each region. Electric field lines should start and end on charges with the number of lines proportional to the charge.

R.W. Bird
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I can do my best to answer the first half of your question. Although in mind that field lines are a less formal depiction (simply a visual aid) of electrostatic fields than an actual vector field might be, they still need to follow a few rules.

The first is, as you said, that their density in a specific region corresponds to the strength of the field there. This can actually be connected to flux: if you trace a surface in space, count the field lines passing through it, and multiply by the area of the surface, the result is proportional to the electrostatic flux (taking into account, of course, angles, the accuracy of the image, and other details).

The second property is simple. The field lines are tangent to the electrostatic field vectors at each point; simply put, they point in the same direction as the field itself.

Now, let us use these properties to answer your question. In the region surrounding the $2q$ charge, the field will be twice as strong as and pointing in the opposite direction from (in terms of divergence) the field in the region surrounding the $-q$. Field strength corresponds to line density, so $2q$ has twice as many field lines diverging from it as $-q$ has converging. Since lines can't combine or cross, only half of the lines starting at $2q$ make it to $-q$. The rest do not end.

This should make sense. Remember, field lines can be compared to flux. If we create a closed (Gaussian) surface around our two particle system, we get that the half of the lines not ending at $-q$ leave the closed surface. Tallying up these lines and multiplying by the area of the surface gives us the total flux which, if we recall Gauss' law, must be proportional to the net charge ($q$) of our system. Indeed, this is half of the flux we would calculate if our surface only enclosed $2q$.

If I may be allowed to answer the second half, field lines may represent a few different mathematical concepts. I personally prefer to use them, with an appropriate surface, to calculate relative flux, but they may just as easily be used to represent divergence: whatever the situation necessitates.

BillyTheSquid
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Field lines are a cartoon. I once saw a standardized test question with picture of a current loop, and single arrow going through it labeled a magnetic field, with the caption:

NOT TO SCALE

Face Palm emoji was not an answer option.

So drawing them is up to the artist, but they should come out of + and go into -, and in your case, half the $+2q$ lines should terminate in $-q$, and the remaining ones should fizzle out at infinity.

JEB
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