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# 1. Magnetism and the B Field

## Magnets, North and South Poles

Imagine we are given a set of magnetics, and we begin playing with them to explore magnetic phenomena. One of the first things we find is that we can get a pair of magnets to stick when some ends face each other, but turning one of the magnets around causes the magnets to start repelling. We find that bar magnets can attract or repel other magnets depending on which ends are brought together. We need a way to label the two ends.

The convention is that these two distinct ends of a magnet are called the north pole and the south pole. This convention is related to the use of magnets in navigation. The development of the navigational compass (around the 12th century) makes use of the interaction between bar magnets and the Earth (which, for the purposes of magnetism, is one large bar magnet). The poles are defined thus: If we suspend any magnet freely on a table (by a string or sticking it to a cork which is free to float on a liquid, to name some examples) the end of the magnet that points towards the north geographic pole of the Earth is referred to as the “north-seeking pole” or just the north pole. The opposite end is the “south-seeking pole” or just the south pole.

With this definition, we can go back and explore the behavior of magnets (just like one can explore the behavior of electricity once the charge-sign convention has been established). If we bring together two magnets and want to make their north poles touch, we notice that there is a repulsion between the two magnets. If we flip one of the magnets so that now we make a south pole approach a north pole, we notice that the magnets will attract, even stick together. Therefore, we can conclude from these observation that like poles repel and opposite poles attract. This behavior is illustrated below, and is something you should also verify in your D/L.

We suspend two bar magnets from a pair of strings, such that the north poles of both magnets are in close proximity. The two bar magnets will exert a repulsive force on each other, represented by the arrows in the picture. The repulsive force that each bar magnet exerts on the other will be equal in magnitude and opposite in direction (Newton’s Third Law). In (b), we do the same thing but flip one of the magnets. Now, the north pole on the left magnet is close to the south pole on the right magnet, so the two bar magnets experience an attractive force.

## Magnets Exist as Dipoles

One interesting aspect of magnets, a feature that is distinct from electric phenomena, is that magnets always have two poles. Imagine that you have a long bar magnet with the north pole on the right and the south pole on the left end. Let’s say you want to break it into two pieces. One might think that you end up with only a north pole in the right piece and only a south pole in the left piece. It turns out that upon breaking it, each new piece still acts like it has two poles, a north an south.

This behavior has no analog in electricity, where one can physically separate positive and negative charges, and each charge can exist separately in other objects. In nature, as far as we know, there are no north poles without a south pole, nor south poles without a north pole. In other words, magnetic monopoles do not exist as far as we know; it's one of the most important aspects of magnetism.

## The Magnetic Vector Field

Once we have defined a convention for the two types of magnetic poles, we are ready to define the magnetic vector field $$\mathbf{B}$$. Imagine placing several round magnetic compasses on a plain sheet of paper on a flat table. In the absence of any other magnets, the compasses will all align themselves in the same direction: their north poles will be pointing towards geographic north by our definition. So we rotate the compasses such that the label “N” points to the top of the page. Assume that we now place a large bar magnet horizontally on the sheet of paper as shown below, with the North pole on the right of the page (as displayed below).

What happens? The magnetic needles of the compasses will reorient themselves in the presence of the bar magnet (like "test" magnets responding to a "source" magnet). The compass body, which is not magnetic, remains with the “N” labels pointing to the top. The compass needles now do not point to the top; they will be pointing in different directions, as shown in the figure below:

The bar magnet produces a magnetic field in its surrounding space, and the magnetic needles of the compass reorient themselves because the field exerts a force on them. For example, the compass needle that is just right of the bar magnet is closest to the North pole of the bar magnet. Therefore the north pole of the compass needle will be repelled by the north pole of the bar magnet, so it will now point to the right. Examining the other compass needles, we can use this situation to define the direction of the magnetic field vector $$\mathbf{B}$$. At a given location in space, $$\mathbf{B}$$ will point in the same direction that the north pole of a compass would point if placed in the field.

Another way to say this is to use similar language from the $$\mathbf{E}$$ field definition. Recall that we defined the direction of the $$\mathbf{E}$$ field as the direction of the force that a positive charge would feel at a given location in space. Similarly, the $$\mathbf{B}$$ field direction is defined as the direction of the force that a north-seeking pole would feel at a given location in space. So if we place a compass near the north pole of the bar magnet, the bar magnet repels the compass's north pole (and the south pole will be attracted to it) causing our compass to point as shown in the figure above.

Imagine now placing compasses one after the other, such that their compass needles are touching head to tail and you could make a path out of them. If you follow this procedure starting near the north pole of the bar magnet, you will find that the compasses eventually point to the south pole (as shown by the progression of arrows below the magnet above). This schematic illustrates the idea that we can follow a line from the north to the south pole of a magnet. Just as we did with the $$\mathbf{E}$$ field, we can represent the magnetic field of a magnet by drawing the magnetic field lines, which will follow the path of the head-to-tail compasses in our example.

Note that in the figure, there is one north pole and one south pole present, so we have two opposite type poles. This configuration is very similar to the electric dipole, where we had one positive and one negative charge present. You will notice that the field lines of magnetic dipoles and electric dipoles have very similar configurations.