# 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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