Proof that e-field lines don't cross

[vortmax]

I understand that electric field lines do not cross and a simple explanation is that it violates existence-uniqueness. I'm thinking there is a more complete explanation out there though. I think the answer lies in the realm of dynamical systems and non-linear dynamics, but it's been a whlie since I've had that and can't piece together the theorems that would fully explain it.

 

[Gear300]

I'm not sure if this is the right answer, but classically, when analyzing electric fields, you take the vector sum...so technically, if field lines did cross, then you'd have a vector sum there; thus, altogether, you have a "unique" direction at each point.

 

[Xezlec]

I understand that electric field lines do not cross and a simple explanation is that it violates existence-uniqueness. I'm thinking there is a more complete explanation out there though. I think the answer lies in the realm of dynamical systems and non-linear dynamics, but it's been a whlie since I've had that and can't piece together the theorems that would fully explain it.

Easy. "Field lines" are just arbitrary lines drawn in a direction that is always perpendicular to every possible equipotential surface that they pass through. If the lines ever crossed, what would the equipotential surface look like at the point where they cross? A surface can only have one normal (subject to scalar multiplication), not two, so it can't have two lines coming out in different directions at the same place that are both perpendicular to it!

 

[uzair_ha91]

They simply can't cross because E can't have two values at a single time. Consider this, an electron can either push away a proton near it or the proton can push the electron, these two events cannot occur at the same time. Similarly, E can't have two values and hence electric field lines can't cross each other.