The sentence, "No two magnetic field lines can intersect each other." has always confused me. The tangent of a magnetic field line gives us the direction of the magnetic force. When two forces act in two different directions at a specified point, we can simply find out the resultant force by vector addition. Why can't I do that in this case?
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General tip: Look in the right margin for related questions. – Qmechanic Aug 08 '20 at 08:01
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https://physics.stackexchange.com/q/529997/247580 might help you – imposter Aug 08 '20 at 08:46
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If the magnetic field could cross how would know in which direction the field would be (as you said: The tangent of a magnetic field line gives us the direction of the magnetic force. In the case of intersecting there is not a unique direction of the field.
Of course, you can always find the resultant of two magnetic field vectors, but this result will always be part of a magnetic field line.
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A magnetic field line is a line whose direction at every point along it is the direction of the (resultant) field at that point. But at any point the resultant field can be only in one direction – otherwise it wouldn't be the resultant!
If we trace (using, perhaps a plotting compass) the field lines of a magnet, remove the magnet and place another one near where the first one was and trace its field lines, the two patterns will intersect. But with both magnets in place, the pattern (different from either plotted before) will contain no intersections: it will show the resultant field direction at each point.
Philip Wood
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