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Antonis Hadjipittas

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Assume I have 2 charges of charge +q at a distance r from each other on the z axis. Position of two charges is (0,0,r/2) and (0,0,-r/2). Assume now that I want to calculate the force these two charges exert on an electron which is at a point (x,0,0) or (0,y,0). Basically any point on the plane which lies on z=0, i.e. exactly between the two charges.

Mathematically the way to go about this, you calculate the electric potential for a system of charges and take both positive charges into account. Then you calculate the gradient of the electric potential which gives you the electric field and then find the force on the electron. If this is done correctly, you will find that there is a net force on the electron, however this is not intuitive to grasp. You see, since both charges are positive, the electric field lines should repel each other leaving the plane (x,y,0) in the middle with no field lines, therefore the force on the electron should be zero (Also assume that the electron's effect is minimal).

So my issue is the following: When we calculate the electric potential of a system of charges, we assume that the electric field lines do not interact with each other. If we assume that they do interact with each other, in this case we get a different answer since there should be no field lines in the plane lying exactly in the middle of the two charges. What am I missing?

Thanks in advance