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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 $O$.

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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?

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    $\begingroup$ What about the field lines perpendicular to this from the charges? Looks like they terminate at O, Yes? $\endgroup$
    – BowlOfRed
    Commented Apr 8, 2022 at 19:43

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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. Field from two identical point charges

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  • $\begingroup$ 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? $\endgroup$
    – Asher2211
    Commented Apr 8, 2022 at 14:10
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    $\begingroup$ The crux is that points where lines 'originate' are mathematically well-defined as points where the divergence of the vector field is non-zero. Because the field is a function of (x,y,z) you cannot draw conclusions from viewing it along one axis. $\endgroup$
    – Cyrus Tirband
    Commented Apr 8, 2022 at 14:20
  • $\begingroup$ The field line between the charges cannot be attributed to a single charge, since it is the superposition of both charges that creates the total field. Along the bisector they both contributed half. $\endgroup$
    – Cyrus Tirband
    Commented Apr 8, 2022 at 14:26
  • $\begingroup$ But doesn't a field line originate from only one charged particle? $\endgroup$
    – Asher2211
    Commented Apr 8, 2022 at 14:33
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    $\begingroup$ Take a look in this image. $\endgroup$
    – Frobenius
    Commented Apr 8, 2022 at 17:10
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Electric field lines do originate only at charges, but they can cancel out.

  • 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.

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  • $\begingroup$ In my book it mentions that two field lines cannot intersect and field lines originate from a positive charge so if we say that field lines in the perpendicular bisector originate from both charges then doesn't it contradict the above facts? $\endgroup$
    – Asher2211
    Commented Apr 8, 2022 at 15:55
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    $\begingroup$ @Asher2211 Your book is referring to a field created by one single source. In your scenario here you have two fields from two different sources merging. $\endgroup$
    – Steeven
    Commented Apr 9, 2022 at 7:30
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Electrical field lines have no physical meaning. The rules describing them are just rules to make illustrations. They therefore do not require rigor.

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  • $\begingroup$ This is not true at all. Field lines are not even specific to electrostatics, they are mathematical constructs, useful whenever you have vector fields, and their properties are in fact rigorously defined. See e.g. Smythe, Static and Dynamic Electricity, chapter 1. $\endgroup$
    – Sam Gallagher
    Commented Jan 6, 2023 at 18:33

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