Question

Why do magnetic field lines not intersect each other?

Answer 1

We can get the answer of the question by first understanding the magnetic field lines and their property. The property of magnets that an electric field line shows will help us to define the reason for which the two lines never cross each other.

Complete step by step answer:
Magnetic field lines are the pictorial representations used to visualize the direction and strength of magnetic fields which in turn tells us about the magnetic force. When we place a magnetic compass in a magnetic field, then the direction in which its north end points defines the direction of the magnetic field line at that point.

Suppose around a bar magnet, we want to map a magnetic field line using a small compass. Then the north end will always direct away from the north pole of magnet and towards the south pole of the magnet. Then if we connect all the points at which we measured the direction of field lines, we will get a closed loop emerging from the north pole of the magnet to the south pole.

As magnetic field lines are not any physical entity and just a pictorial representation. There are certain rules that governs magnetic field lines,
1.)If we draw a tangent at any point of a magnetic field line along its direction, then it will give us the direction of the magnetic field.
2.)The number of magnetic lines present per unit area determines the strength of a magnetic field. More the number, higher is the magnetic field strength and vice versa.
3.)Magnetic field lines form closed and continuous loops having no beginning or end and goes from north pole to south pole.
4.)The direction of the magnetic field at any point on the magnetic field line is unique as if we put a compass at that point its north end will point only in one direction and thus these lines never cross each other.
Now, suppose we have two magnetic lines of forces $a$ and $b$, which intersects at point P, so, if we draw tangents at this point for both the lines, we will get two different directions of the magnetic field, which is impossible. This can be illustrated in the diagram below.

Hence, magnetic field lines never cross each other.

Note:
If we consider a bar magnet, then inside the magnet, the magnetic field lines move from south pole to north pole and are almost parallel to each other. At places with very high magnetic strength, the magnetic field lines are very closely spaced, but never intersect each other.