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We know that the electric field lines cannot intersect in this way:
enter image description here

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

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

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  • $\begingroup$ 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? $\endgroup$ – SchrodingersCat Oct 24 '15 at 16:15
  • $\begingroup$ @Aniket yes, sorry $\endgroup$ – TheQuantumMan Oct 24 '15 at 16:20
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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.

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