Why do two field lines never intersect?

[sanalsprasad]

Why can't two field lines intersect? The answer that i have often heard is that if they were to intersect, then there would be two different directions for force. But then two field lines can interesct in such a way that their tangetes at the point of intersection point in the same direction. ( Example: Consider the curves y = x^3 and y = -x^3. They intersect at (0,0) and the tangets are same )

I searched a bit on google and came across something called existence uniqueness theorem. Could someone explain in a little detail, maybe with some examples. Thanks.

 

[Philip Wood]

Two magnets a long way away from each other will have their own sets of field lines. Move the magnets closer together and one might want to say that their field lines intersect. But that would be to forget that, by definition, the tangent to a field line gives you the direction of the resultant field at that point. The resultant field will have a single direction, that of the vector sum of the fields due to each magnet, at the point in question.

'Resultant' is the key word.

 

[sanalsprasad]

I agree that the tangent to the electric field line points in the direction of the force. But then two field lines can have the same tangent at the point of intersection and the resultant will be in the direction of the tangent.

 

[Philip Wood]

The lines must surely diverge at some point, in which case we have don't have a clearcut direction for the field at that point, yet the field is not zero. Seems unphysical doesn't it?

 

[mfb]

You can consider field strength as "density of field lines" (when drawn properly). If two lines meet each other, all lines in between would do the same - your field strength would be infinite, which is unphysical and does not happen without point charges (which are unphysical, too, but a good approximation in many cases).

 

[Khashishi]

Field lines are allowed to bunch up infinitely close inside a true dipole. I suppose this could be considered to be something like intersecting.

 

[omega_minus]

I agree with Phillip, if you draw two intersecting field lines, they vectorially add and would end up not intersecting. Picture two electric field lines pointing diagonally downward and crossing in an "X". the left and right components add to zero and you're left with a "||" shape. Whether one can consider this crossing or not is beyond me.

 

[vanhees71]

The field lines are non-intersecting only if there are no singularities around. As a counter example take the Coulomb-force Law for the electrostatic field of a point charge (which provides the singularity here!): All field lines are radial and begin (or end) at the charge.

If there is no singularity of the vector field $\vec{A}(\vec{x})$ within an open region of space, then by the uniqueness theorem for the solutions of differential equations the equation for the field lines (tangent to the field)
$$\frac{\mathrm{d} \vec{x}}{\mathrm{d} \lambda}=\vec{A}[\vec{x}(\lambda)]$$
with the initial condition $\vec{x}(0)=\vec{x}_0$ with $\vec{x}_0$ within the neighborhood of the has a unique solution within this neighborhood. This would be not the case if there would exist two such field lines that cross at a point.

 

[mrspeedybob]

magnetic field lines are not physical. They are not real. They are a human invention used to make a graphical representation of a magnetic field. The field itself in continuous.

 

[Emilyjoint]

if 2 field lines truly intersected then a charged particle could make a choice as to which way to go at the point of intersection... don't think this has ever been observed or is considered inphysics

 

[raopeng]

Agree with Emilyjoint, cannot think of a practical example in real life. And to think it in mathematical terms, if it does not compromise the physical image, I think field is a one-one mapping of coordinates to physical quantities, except for singular points(the source of the field, point charge etc). So I guess here it cannot be called field because it is not a one-one mapping even if the two vectors have the same direction.

 

[vanhees71]

Again: Take the electrostatic Coulomb field which has a singularity at the place of the point charge. There the (radial) field lines intersect, but as this example shows this can happen only if there are (singular) sources of the field.

 

[TrickyDicky]

Again: Take the electrostatic Coulomb field which has a singularity at the place of the point charge. There the (radial) field lines intersect, but as this example shows this can happen only if there are (singular) sources of the field.

I know point charges are idealized constructions to some extent, however it seems as the whole physics edifice is based on their existence. I have certain confusion about how can one base physics on singularities.
On the other hand, within QED it is hard to claim that "electric monopoles" really exist, due to the whole issue about bare vs dressed charges and vacuum polarization.

 

[mikeph]

I would suggest the field lines don't intersect at the location of a point charge because they don't exist at this point; the charge density is discontinuous here so it would be a mistake to rely on the differential form of Gauss' law to derive a meaningful value of the electric field.

It makes no more sense than trying to determine the gradient of y = |x| at x = 0.

IMO it is a mathematical hiccup, not a physical paradox.