Calculating the force on an electron from two positive point charges

[Antonis Hadjipittas]

So this is more of an intuitive question rather than a mathematical one. I present the problem.

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

 

[Nugatory]

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.

Field lines are just a visual aid that we draw to help us see which direction the electric field points at any given point. They're no more real than the lines of latitude and longitude that we draw on a map of the earth, so they cannot attract or repel or anything else.

 

[A.T.]

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.

Where do we assume that such a thing, whatever its supposed to mean. You compute the potential, and then can derive the field lines from that. Not the other way around.