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I see a lot of images, including one in my textbook, like this one, where at the ends of a uniform field, field lines curve.

enter image description here

However, I know that field lines are perpendicular to the surface. The only case I see them curving is when drawing field lines to connect two points which aren't collinear (like with charged sphere or opposite charges) and each point of the rod is collinear to its opposite pair, so why are they curved here?

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  • $\begingroup$ see also the closely related physics.stackexchange.com/q/406837/36194 $\endgroup$
    – ZeroTheHero
    Sep 8, 2020 at 1:55
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    $\begingroup$ Each positive charge in the left plate creates an electric field radially outward away from it, and the total field produced by the plate is the vector sum of each of these individual fields (plus those of the negative charges, but let's focus on the positive ones). At points near the middle of the plate, the charges above it and charges below it produce fields with cancelling vertical components. At points near the top of the plate, most of the charges are below it, so the majority have a field with an upward vertical component. $\endgroup$
    – jawheele
    Sep 8, 2020 at 19:27
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    $\begingroup$ Does this answer your question? What is the reason for the edge effect in capacitors? $\endgroup$
    – Tim
    Sep 9, 2020 at 7:19
  • $\begingroup$ @Tim It's the same question but it's really beyond my OL phyiscs level. For me the answers here are much more easy. $\endgroup$
    – Manar
    Sep 9, 2020 at 22:54

4 Answers 4

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I have taken your image and created a few additional field lines at one end of the plates in the first diagram below.

When you come to the ends of the plates, the field starts to resemble that associated with two point charges instead of a sheet of charge. The second diagram below shows the field lines between two point charges. Note that as you move away from the two point charges an equal distance apart, the lines look like those at the ends of your parallel plate capacitor (curved lines). Towards the center between the charges, the field lines start to look straight and evenly spaced (parallel lines).

Hope this helps.

enter image description here

enter image description here

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  • $\begingroup$ Do field lines also exist that emerge from one of the edges of the plates and end up on the opposite edge of the opposite plate? Say, a field line that emerges from the upper edge of the left plate ending up on the lower edge of the right plate? Can such an emerging line end up perpendicularly on the right plate? Or do all field lines lay within the confines of the (imaginary) infinite extensions of the parallel plates? Because otherwise, field lines would intersect? Which is not possible. $\endgroup$
    – Deschele Schilder
    Sep 28, 2020 at 13:17
  • $\begingroup$ "Say, a field line that emerges from the upper edge of the left plate ending up on the lower edge of the right plate? " The field lines represent the direction of the force that a positive charge would experience if placed in the field. Where do you think a free positive charge placed in the space just above the upper left edge of the left plate would go? To the upper edge of the right plate or the lower edge of the right plate? $\endgroup$
    – Bob D
    Sep 28, 2020 at 15:00
  • $\begingroup$ That's indeed pretty clear. To the edge of the right plate. But what happens to a positive charge somewhere on the left (right) of the left (right) plate? $\endgroup$
    – Deschele Schilder
    Sep 28, 2020 at 15:18
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These are so-called edge effects. The straight electric field lines connecting two surfaces is a solution for the infinite charged plates. In practice, no plates are infinite: they have edges. Far from the edges (close to the center of the plates) one can still think of the plates as infinite, but at the edges this is clearly not true.

Note that the same is true for an infinite charged wire or cylinder: in practice one always has a finite one, but far enough from the edges, one can assume that it is infinite and thus simplify the math.

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    $\begingroup$ Thanks a lot for saying this phenomenon's name! I found a whole lot more of information searching the name than when I googled my question directly and here's another question which adds even more details. physics.stackexchange.com/questions/389766/… $\endgroup$
    – Manar
    Sep 7, 2020 at 16:10
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    $\begingroup$ Another name is "fringe field." $\endgroup$
    – d_b
    Sep 8, 2020 at 0:15
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This is one of those questions where you just have to see it. Here is fieldline drawing of two charges. Red is a positive charge and blue is negative.

Field lines 1 charge

Now for 6 charges:

Field lines 3 charges

and finally for 40 charges:

enter image description here

Here is the Mathematica code for anyone interested

range = 1.4;
nCharges = 20;
xSeparation = .5;
e[r_, r0_] := (r - r0)/Norm[r - r0]^3
chargeY[n_] := If[nCharges == 1, 0, (n - 1)/(nCharges - 1) - .5];
Show[
  StreamPlot[
    Sum[e[{x, y}, {-xSeparation, chargeY[n]}], {n, 1, nCharges}] - 
    Sum[e[{x, y}, {xSeparation, chargeY[n]}], {n, 1, nCharges}], 
    {x, -range, range}, {y, -range, range}],
  ListPlot[Table[{-xSeparation, chargeY[n]}, {n, 1, nCharges}], 
    PlotStyle -> {Red, PointSize[.03]}],
  ListPlot[Table[{xSeparation, chargeY[n]}, {n, 1, nCharges}], 
    PlotStyle -> {Blue, PointSize[.03]}]
 ]
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    $\begingroup$ Beautiful answer. Most important for me, it reveals that none of the lines are straight; the edge effect is just that they get more curved near the edge. $\endgroup$
    – Don Hatch
    Sep 9, 2020 at 19:08
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    $\begingroup$ @Don Hatch, even the line in the exact middle between two charges won't be straight? $\endgroup$
    – Manar
    Sep 9, 2020 at 22:47
  • $\begingroup$ @Manar doh! Missed that one. Well spotted :-) $\endgroup$
    – Don Hatch
    Sep 9, 2020 at 22:59
  • $\begingroup$ I love answers with Mathematica code. $\endgroup$
    – Display Name
    Sep 10, 2020 at 13:06
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Rather than thinking of the plates as solid line charges, think of them as lines of infinitely many point charges.

2 point charges will have a straight field-line directly between them, and weaker curved field-lines outside of that. If you place 2 pairs of point charges next to each other — positive with positive, and negative with negative — then their field lines can overlap. However, much like with waves, electric fields can interfere with each other, both constructively and destructively.

This means that the overlapping curved field lines will average out as a straight field line, through the middle of the point-charge pairs. The outermost edges of the electric field, on the other hand, will have nothing to interfere with, and remain curved.

Diagrams showing: 1 pair of point charges; 2 adjacent pairs, with overlapping field-lines; 2 adjacent pairs, with interfering field-lines; 5 adjacent pairs, as extended plates

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    $\begingroup$ This is a beautiful answer because it helps bridge the gap between microscopic and macroscopic views to some extent, and explains rather than invokes calculus; I like it very much! I wish it used a different word than "interference" though, because that term is so often associated with waves. If it were me I'd say that it's the perpendicular components adding to zero, or "cancelling". $\endgroup$
    – uhoh
    Sep 9, 2020 at 12:12
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    $\begingroup$ @uhoh I will admit, I hesitated before using 'interference' for much the same reason — this is Flux rather than Waves — but the underlying principle is close enough. I can't remember what the specific language used for discussing the equivalent magnetic phenomenon is in Halbach Arrays, but it serves to demonstrate that the field lines can add together rather than cancel out (depending on the circumstances), and keeping the language sign-neutral struck me as more important. $\endgroup$
    – Chronocidal
    Sep 9, 2020 at 15:27
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    $\begingroup$ It is interference in the limit as $\omega \to 0$ :-) $\endgroup$
    – uhoh
    Sep 9, 2020 at 15:54

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