# Can the magnetic lines of force of two or more magnets intersect each other?

Basic property of magnetic lines of force is that, they can never intersect each other. Among the two points given below, which one is correct?

• Magnetic lines of force of same magnet can't intersect each other, but magnetic lines of force of different magnets can intersect each other.
• Magnetic lines of force can't intersect each other irrespective of their origin (i.e whether the lines were of same magnet or different magnets, they can't intersect)

Suppose I have two bar magnets. If I tend to make the repelling poles join together by exerting force (i.e, if I tend to make north pole of two magnets, or south pole of two magnets join together). Would the magnetic lines of force intersect each other?

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Suppose if we assume magnetic lines of force of two different magnets intersect each other.There will be two directions for the magnetic field at a single point,i.e we can draw two tangents at a single point.One tangent will be giving direction of magnetic field of one magnet at that point,and other tangent will be giving direction of magnetic field of another magnet at the same point. – Immortal Player Oct 21 '13 at 5:00
Yes, they can (in a quadrupole magnet). But that means that the field is zero at that point. – jinawee Oct 21 '13 at 9:06

Here are from wikipedia drawings of the field lines of two magnets in two orientations, like-like, like-unlike .

North pole to north pole

North pole to south pole.

The lines distort but do not intersect.

These field lines are solutions of the formal Maxwell differential equations. Differential equations do not give discontinuous solutions, as would be the case if two lines crossed. Discontinuities in the solutions when smooth boundary conditions exist are not possible, as in the drawings. Discontinuities can exist as singularities, which can only exist at the source of the field. The field lines themselves follow smooth functions away from sources.

Mawell's equations have been continually validated by an enormous number of experiments and applications and thus we trust the descriptions of nature that the solutions give.

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I think the important part here is that it's a vector field. A vector field associates one vector (with one direction and one length) with each point in space. The magnetic field lines follow the vectors of the magnetic field at each point. If field lines would cross, that would imply that there are two vectors with two directions at one point. – MSalters Oct 21 '13 at 10:33
@MSalters well, that is what I would call "discontinuity". – anna v Oct 21 '13 at 10:35
I know, but a scalar field can have discontinuities as well, and a scalar field could also be described by differential equations too. It's rather common knowledge that the magnetic field is not scalar, but this wasn't exactly a PhD level question. – MSalters Oct 21 '13 at 10:38
@MSalters.If we would had considered a single magnet,with its magnetic lines intersecting, like as you said there would be two vectors with two directions at one point.It would be incorrect.If you consider magnetic lines of two different magnets intersecting eachother,they would also create two vectors with two directions at a point.But here one vector will be indicating the direction of magnetic line of one magnet and the other vector will be indicating the direction of magnetic line another magnet.It would not be a contradiction. – Immortal Player Oct 21 '13 at 18:26
You are forgetting that The great in practice solver of differential equations is Nature itself. We as physicists have the task of finding which differential equations nature is using so that we can predict behavior in future situations. We know from the way the data fit the Maxwell equations that those are the equations of nature as far as macroscopic magnetic fields go, which is the situation in your problem. Once more, differential equations do not lead to double values, in vectors or scalars or tensors. They are single valued functions of the coordinates of the coordnates in the problem. – anna v Oct 22 '13 at 7:04

Both fields will distort each other. The topology will look like compressed field lines, but they will never intersect. If you flipped the magnets so that opposite poles are in proximity, then the field lines will combine together

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The other answers are quite complete, but magnetic (and electric) lines DO cross in some special cases.

An example would be the magnetic quadrupole configuration:

This doesn't contradict the other answers, but it means that at the centre there is no field and there are several directions of getting there (it would be an equilibrium point).

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See also magnetic reconnection – Kyle Kanos Oct 23 '13 at 10:59
In any physical instance of a quadrupole field, wouldn't there be an object at the equilibrium point? And wouldn't the field inside the object no longer be accurately modeled by the quadrupole field? – Flavin Oct 28 '13 at 18:53
Quadrupole solution doesn't work at the origin. Multipole decompositions are done at the distances which are large comparing to the size of the source. – Alexey Bobrick Oct 29 '13 at 21:41
@AlexeyBobrick Why not? I didn't talk about multipole decomposition (which is an aproximation, correct me if I'm wrong) but cuadrupole configuration. Or are you saying that you can't calculate the field of a cuadrupole? – jinawee Oct 29 '13 at 22:08
Quadrupole field is the approximation to the field of a magnetic quadrupole at large distances. At small distances it does not approximate the field of a magnetic quadrupole anymore. – Alexey Bobrick Oct 29 '13 at 22:24

When you have two magnets there are 3 magnetic fields it makes sense to talk about:

i) The magnetic field of the first magnet (described by the magnetic lines of the first magnet),

ii) The magnetic field of the second magnet (described by the magnetic lines of the second magnet),

iii) The total magnetic field which is the sum of the two above and is described by magnetic lines that are effectively the sum of the previous two.

It is true that if you draw them on a diagram, the magnetic lines in i) and in ii) intersect BUT: The magnetic lines of the TOTAL magnetic field never intersect for the reasons mentioned in the other answers.

In your example, you can still make the same distinction. The answer is that even if you try to bring together the north poles of two magnets the magnetic lines of the TOTAL field will NOT intersect. Instead, they will look like the pictures posted in the other answers.

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In quadrupoles and the picture jinawee posted, the lines don't intersect either. As jinawee pointed out, the field at the center (where the lines "seem" to intersect) is zero. The field strength "continuously" decreases as you move from a pole towards that point, and is zero at that specific point. Mathematically you can continue the field lines "as close as you want" to this point, but since the field itself is zero there, you have no field lines and therefore there is no intersection. As you can guess, two of the lines in the figure are pointed towards this point, and the other two are pointed outwards. The reason is that the potential in this particular configuration is saddle shaped, and the "seemingly" intersection point is the saddle point. So you can guess again that it isn't an equilibrium point. This means if you fix a very very tiny piece of iron or any magnetizable metal in the middle of this configuration (the saddle point), a small perturbation can send it going towards the poles on the path of the field lines.

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Field lines of both electric and magnetic origin do not intersect ever. But you have to remember when we say this we are talking about the resultant field lines.

Obviously the individual fields of two charged particles or magnets if drawn separately could be shown to intersect. But wherever these 2 fields would intersect their components along some directions would add up and along some would cancel out, which will leave you with a single field line, since this will happen at all intersections the field lines if plotted as done in previous answers would be seen as compressing or expanding in certain places rather than several intersections.

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In addition to the answers above, electromagnetic field equations are linear, meaning the fields obey the superposition principle. So if you separately calculate the field of different sources at a point, the field at that point will be a superposition of your calculated fields, leaving only one vector at each point (and keeping you're vector field description of the electromagnetic field safe).

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anna v ' s answer is complete and satisfactory, but she misses a little detail I thought I'll mention.

The field lines created in space due to a magnet or any combination of magnets at any point depict the direction of the net magnetic field at that point in space. The tangent to the field line curve depicts direction, and the density of field lines depicts magnitude.

If field lines ever intersect at any point, there will be two tangents to the field line curve at that point, which is contradictory, as there can only be one net magnetic field vector having only one direction at any point in space.

This also proves that the field lines have to be continuous and diffrentiable, meaning they can't have breaks or sharp corners.

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