# Explanation on the resulting forces of two positive point charges

Why will the resulting force lines of two positive point charges be like this:

I would expect this:

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Essentially for two reasons - Lines of force always have to be smooth, there can't be a sharp bend in them like in your second diagram. Also because they can never intersect, they'll always be separate, however small the separation. –  Kitchi Feb 27 '13 at 9:45
Google(Asymptote)=Understanding. –  Asphir Dom Feb 27 '13 at 10:29

First a comment about the following statements made by Kitchi and Wouter:

Lines of force always have to be smooth, there can't be a sharp bend in them like in your second diagram

and

The comment by @Kitchi basically says it all: lines of force should be continuous and differentiable and they should never intersect.

These statements are incorrect without qualification, and here are two reasons why:

1. The electric field lines of a point charge consist of rays all of whom intersect at the location of the point charge. Moreover, the field is not smooth at the location of the point charge (it is not even continuous there since there is a singularity).

2. There is a discontinuity in the electric field in passing through a surface charge with charge density $\sigma$, in fact the discontinuity is proportional to the surface charge density; $$(\mathbf{E}_2 - \mathbf E_1)\cdot \mathbf n = \frac{\sigma}{\epsilon_0}$$

If you want to make some statement about smoothness of electric field lines, (which you should try to avoid calling "lines of force"), then you really need to make some other qualifications that exclude such cases.

Second, these smoothness justifications are, to some extent, missing the forest for the trees so to speak. The crux of the issue is really addressed by richard. The way one obtains the field due to a charge distribution is by invoking the principle of superposition. This immediately rules out the second diagram you drew because if you simply add the fields of the two point charges vectorially, then you will see that the field lines look like you draw them in the first diagram.

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It is easy to see that the second plot is wrong! For example consider the line at $45^o$ from the left charge. It says that the electric field direction is $45^o$. But you have to add electric field vectors of both charges at that point (at any point) which can not be in $45^o$ from the horizontal line!

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The comment by @Kitchi basically says it all: lines of force should be continuous and differentiable and they should never intersect.
!The above is in general not true! The answer from @joshphysics addresses this and states some very blatant counterexamples. He also rightly goes into the reason for the actual shape of the lines (which you could mathematically find using Classical Field Theory). My answer (the part below) is mostly an attempt to show why the second image is wrong using rather basic physical arguments.

From a different, less mathematical point of view: the laws of physics are presumed to be deterministic (at least for electromagnetism, I imagine there are areas of study where this presumption is not made). So let's look at your picture and follow the path of a particle, say along the green line pointing to the upper left. After a while you get onto the orange line. Now, if you reverse time (a way to check for determinism), you follow the orange line down until you get to the splitting point. There, you have a problem, because you could go either way and you cannot deterministically determine (no pun intended) which way to go. So determinisim fails. This is another way to see why your picture cannot be the case. The first picture doesn't suffer from this problem.

In fact, I just realized you don't even have to reverse time: you can just consider an oppositely charged particle starting on the orange line. Then there is a huge issue when this particle reaches the splitting point. This problem does not occur in the first picture, where the force lines get infinitely close together after a while, but they never overlap.
The cool thing then is that the behaviour of the tracked particle will have a hypersensitivity to the initial conditions. For example, if you have the particle in an initial position high up on the vertical axis and just a little bit to the right so that it will go spinning off to the right, you could move its initial position ever so slightly to the left and cause its trajectory to turn to the other side completely. But I'm getting off topic.

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You say, "lines of force should be continuous and differentiable"; this is not true. Consider, for example, the discontinuity in the electric field across a surface charge. (see my response below). –  joshphysics Feb 27 '13 at 17:43
@joshphysics You're quite right. I didn't really think that through since I mostly wanted to look at it from another perspective (the larger portion of my answer). I'll edit the first paragraph. –  Wouter Feb 27 '13 at 17:50