I What is wrong with crossing electric fields? Why can't you sum them?

[annamal]

I do not understand why electric fields cannot cross. Can't you just sum the two electric fields vectors to get a net electric field?

 

[Hornbein]

Yes you can sum. The net field contains no crossing lines.

 

[annamal]

Yes you can sum. The net field contains no crossing lines.

https://physics.stackexchange.com/q...-two-or-more-electric-field-lines-never-cross
says they cannot cross, which means you cannot sum

 

[Drakkith]

https://physics.stackexchange.com/q...-two-or-more-electric-field-lines-never-cross
says they cannot cross, which means you cannot sum

That is not what the explanations on that page say. Summing multiple field vectors is literally what you do if you want to find the net electric field from multiple electric charges. If you do this at each point in a field and then draw field lines you will find that no lines ever cross.

 

[annamal]

That is not what the explanations on that page say. Summing multiple field vectors is literally what you do if you want to find the net electric field from multiple electric charges. If you do this at each point in a field and then draw field lines you will find that no lines ever cross.

Ok, so electric field lines are always drawn from the net electric field

 

[Hornbein]

Ok, so electric field lines are always drawn from the net electric field

Right, from the field you would get if you went out and measured it. All the "field lines can't cross" means is that you can't measure two different field values in one place and time.

 

[DaveE]

I've always liked the "test charge" analogy. Suppose you could magically put an electron at a point in space, the electrostatic force that electron experiences is proportional to the e-field at that location. Yes, due to everything (except it's own charge, which is a rabbit hole I won't go into).

 

[vanhees71]

I do not understand why electric fields cannot cross. Can't you just sum the two electric fields vectors to get a net electric field?

Where did you get this from? It's obviously wrong, because the electric field has sources, which are charges. Consequently the electric-field lines cross at the position of the charge. The most simple example is the Coulomb field of a point charge: All field lines cross at the position of this point charge.

 

[Drakkith]

The most simple example is the Coulomb field of a point charge: All field lines cross at the position of this point charge.

I'd say they terminate/originate at a point charge, not cross.

 

[vanhees71]

Sure, but the point charge is the most simple example of a field singularity, where the argument concerning the uniqueness of the direction of the field that field lines never can cross, goes wrong, and it's in fact a very important example, because it's nothing else than the Greens' function of the Laplace operator (in free space), which naturally is very important for electrostatics.

 

[annamal]

Where did you get this from? It's obviously wrong, because the electric field has sources, which are charges. Consequently the electric-field lines cross at the position of the charge. The most simple example is the Coulomb field of a point charge: All field lines cross at the position of this point charge.

https://physics.stackexchange.com/q...-two-or-more-electric-field-lines-never-cross

 

[Drakkith]

Sure, but the point charge is the most simple example of a field singularity, where the argument concerning the uniqueness of the direction of the field that field lines never can cross, goes wrong, and it's in fact a very important example, because it's nothing else than the Greens' function of the Laplace operator (in free space), which naturally is very important for electrostatics.

Forgive me, I'm not familiar with Green's function or the Laplace operator. Can you elaborate on how field lines leaving or converging on a point charge is the same as field lines crossing? Or have I misunderstood you?

 

[vanhees71]

The field lines of the Coulomb field all intersect (or rather start) at the location of the point charge.

 

[Drakkith]

The field lines of the Coulomb field all intersect (or rather start) at the location of the point charge.

But they don't appear to cross each other at the location of a point charge.

 

[DaveE]

But they don't appear to cross each other at the location of a point charge.

I think his point was more that the question of "crossing or not" makes no sense at the singularity. But I don't really see how any of that relates to the OP's post. They don't cross; you know, where you can ask that question.

 

[DaveE]

says they cannot cross, which means you cannot sum

You can not sum because there aren't really two (or more) things to sum. At each point in space there is only one e-field vector. It only points in one direction*.

The field lines you speak of are just a conceptual or graphical tool to visualize a vector field. They are a way of showing the direction of a whole bunch of vectors, each unique and at one location.

It can sometimes be easier to deal with the solution to the e-field by splitting the problem up into a sum of other simpler problems. This is a really great thing about EM, the equations are linear, which means that you can just add up separate solutions to get the solution of the combined problem. I'm not sure I described that too well, but you can look into "superposition" to learn more. So yes your OP was correct, if you have a field solution from one source and a field solution from another source, the field for the combination of those two sources is simply the vector sum of the individual solutions. It's sort of based on this concept: $f(a) + f(b) = f(a+b)$. This is what linearity means, and EM has that property.

* except at point charges, a singularity where either they point in every direction, or (more correctly, IMO) it just isn't well defined. However, arbitrarily close to the singularity it's OK, then they have a single well defined direction.

 

[Vanadium 50]

If filed lines crossed, in which direction would a test charge move? How would it pick which line to follow?

 

[vanhees71]

Maybe it's a language problem. English is not my mother tongue. For me field lines cross at a point charge, and of course at such a singularity the force on another point charge due to the presence of this point charge is simply undefined.

It's also true that for a proper charge distribution no such singularities occur, and then the force on a point charge is everywhere well defined and the lines of the electrostatic field don't "cross" anywhere.

 

[Ibix]

For me field lines cross at a point charge

I would say they meet, rather than cross. I'd reserve "cross" for the (unphysical) case where you have one field line that satisfies $x=0\ \forall\ y$ and another that satisfies $y=0\ \forall\ x$ - they cross at the origin because they carry on through (or would, if that were physically plausible). On the other hand I'd say field lines from a point charge at the origin meet there, because they don't pass through the meeting place - the lines in the $\pm x$ directions are distinct field lines.

 

[Drakkith]

Maybe it's a language problem. English is not my mother tongue.

Ah, that's probably it. The lines in the letter 'X' cross, but the lines making up the letter 'V' do not, they simply meet each other.