Why do the electric field lines not originate from a positive charge in the following situation? Consider two fixed positive point charges, each of magnitude $Q$ placed at a finite distance apart. Let point $O$ be the midpoint of the two charges. We can see that the electric field at $O$ is zero, but for any other point in the perpendicular bisector of the two charges, the electric field is nonzero and the direction of the electric field is along the perpendicular bisector and opposite to that of point $O$. So it appears that there are electric field lines originating from point $O$.

I have learnt that electric field lines originate only from positive charges, but there aren’t any positive charges at point $O$, so where did I go wrong in my reasoning?
The field lines can only originate from only one of the charges. But if we consider the field line through the perpendicular bisector, the field line is symmetric to both the charges, so how can we say that this field line originates from one of the charges?
 A: Electric field lines do originate only at charges, but they can cancel out.

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*Consider point O. From the top charge, a field line points downwards in this point. From the bottom charge, a field line points upwards in this point. Their effects cancel out and there is no net field at this point O.


*Now look slightly to the side, horizontally from point O. In such a point the top charge causes a field line with an angle, so downwards-and-a-bit-sideways. The bottom charge also causes a field line with an angle, but upwards-and-a-bit-sideways.
Now, note how the sideways components of the field from either charge is in the same direction. But the vertical components are opposite. The vertical components thus cancel out whereas the sideways components add up.
The final result is a net field directed perfectly horizontally from point O. This does not mean that the field originates at point O - it is just the merged result of two fields originating at the charges that just happen to perfectly cancel out vertically due to ideal symmetry in this setup.
A: Electrical field lines have no physical meaning. The rules describing them are just rules to make illustrations. They therefore do not require rigor.
A: I drew the field associated with two identical point charges below as a visual aid.
As you can see, the origin is not a place where the 'electric field originates', the charges are.
A more mathematical description of 'originate' would be that it is not enough that $\frac{\partial}{\partial x}E_x$ is non-zero. Rather, it is the divergence, $\frac{\partial}{\partial x}E_x + \frac{\partial}{\partial y}E_y +  \frac{\partial}{\partial z}E_z$ that is non-zero when field lines 'originate' from a charge. At the origin, the divergence is zero so there's nothing strange going on.

