# Intersection of electric field lines where the two lines have the same slope at intersection

We know that the electric field lines cannot intersect in this way:

But what about this way?(where the two lines have the same slope at the point where they touch) :

Could the second situation ever occur? And why yes/no?

• In the second case if they have the same slope, then they have the same tangent. So they are not intersecting but touching. Is that what you mean? Oct 24, 2015 at 16:15
• @Aniket yes, sorry Oct 24, 2015 at 16:20

There are two properties one can read from a display of electric field lines:

• the direction of the field

• the strength of the field (up to a constant of proportionality)

The former is tangent to the field line, or some kind of weighted average of the tangents to nearby field lines. This is the property usually invoked to show that they can not cross (can't point in two directions at once, see?). You've more or less defeated that argument.

But let's consider the second property: the local strength of the electric field is inversely proportional to the distance between the lines in this region of space. As your proposed lines swoop towards each other that distance goes to zero with the consequence that the strength of the field increases without bound. Another un-physical conclusion.

You can arrange a physical situation that approximates your drawing by getting very high field strengths in a pinch, but not one in which the drawing is exactly correct.

• This argument is not convincing. The relation between the strength of the field and the distance between the field lines originates from the flux conservation, in regions of zero charge density, across two surfaces of a flux tube. Provided one of these surfaces shrinks in one direction and expands in another direction, the surface area may remain the same, thus allowing two field lines to become very close without implying a diverging field. Apr 28 at 6:42

Although based on a correct physical intuition, the original accepted answer is not mathematically sound.

The real problem with two field lines intersecting with the same slope is that, in such a situation, the uniqueness of the integral curves of the electric field would be violated: from the contact point, there would start two different field lines. The equation of the electric field lines $${\bf r}(s)$$ ($$s$$ is a one dimensional parameter) is: $$\frac{{\rm d} {\bf r}}{{\rm d}s} = {\bf E}({\bf r}(s)).$$ A sufficient condition for the local uniqueness of the solution of this set of differential equations is the local Lipschitz continuity of the electric field components. Therefore, in a point of non-uniqueness of the field lines, the electric field should not be Lipschitz-continuous. This would imply a locally unbounded variation of the field. Mathematically the non-Lipschitz possibility can be excluded for the electric field since the electric potential satisfies Laplace's equation and then is a harmonic function that is locally Lipschitz because it is real-analytic.

From the physical side, one could translate the mathematical properties into the fact that local unbounded variation of the electric field, in turn, would mean the large strength mentioned in the accepted answer. However, the reason is not directly connected to the relation between the density of field lines and the strength of the field. The real reason is in the way electric field can vary from point to point, as ruled by Maxwell equations.