Magnetic dipole moment of a bar magnet

Magnetic dipole moment of a bar magnet

Here, we will discuss the magnetic dipole moment of the bar magnet. Before going further, let’s discuss the bar magnet first.

What is a bar magnet?

A bar magnet is a rectangular parallelepiped body that produces a magnetic field. When a bar magnet is suspended freely by a thin inextensible thread then it rotates itself about the transverse axis passing through its center. It is found that it always aligns itself in the north and south directions. The endpoint of the magnet which is pointing towards the geographical north is called the north-seeking pole simply the north pole, while the end of the magnet which is pointing towards the geographical South pole is called south seeking pole or simply the South pole.

As we know that magnetic material is always attracted or repelled towards the other magnetic materials. Here, the bar magnet aligned itself in the geographical north and south poles, which means the earth is behaving like a large magnet. So the magnetic north pole of the earth is at the geographical South pole of the bar magnet and the magnetic South pole of the earth is at the geographical north of the bar magnet.

Does the magnetic monopoles exist?

It is said that there are two magnetic poles located at the endpoint of any magnet or any magnetic material. These two poles are equal and opposite in strength, it is exactly the same as the two equal and opposite charged particles.

Did you have break any magnet into two or more pieces? If yes! Then what did you see? Are they still attracting/repelling to each other? If they are still attracting or repelling each other, then it means they still have magnetic poles. It means by breaking any magnet or any magnetic materials we can’t separate their magnetic poles.

If you break a bar magnet by its center, then simultaneously another magnetic pole is created at the end, and each one starts behaving as a whole new magnet. If you can break a magnet into many simpler pieces, then you will find that each small piece of the magnet has a north and south pole. From this, we can conclude that we can’t separate their poles, and hence the magnetic monopoles can’t exist. [latexpage]

Fictitious poles

So, it is not possible to separate the two poles (north and south poles) of a magnet by breaking them into smaller pieces. Similarly, it is not possible to locate the position of the magnetic poles of a magnet. Can you guess that, where do magnetic poles lie within the magnet? It is very difficult to predict the position of the magnetic poles within any magnet. Hence, it is thought that imaginary or fictitious poles are located at the end of the magnet that forms a magnetic dipole. The position of these fictitious poles can be found by using the compass needle. And it is also thought that the position of magnetic poles is not exactly lying at the endpoint whereas, slightly inside the magnet.

• Geometric length of the magnet: It is the length of the edge parallel to the magnetic axis.
• Magnetic length of the magnet: It is the distance between the two poles of the bar magnet. The magnetic length of the bar magnet is $\frac{5}{6}$ times the geometric length of the bar magnet.
• Axis of a magnet: the line joining the poles of the bar magnet is called axis of the bar magnet.
• Equator of a magnet: perpendicular line drawn to the magnetic axis through the center of the magnet is called equator of a magnet.

Magnetic dipole moment of a bar magnet

Magnetic dipole moment is an analogy to the electric dipole moment. In electrostatic, when two equal and oppositely charged particles are separated by some distance then it is called an electric dipole. Same as the electric dipole, in magnetostatics, when two equal and opposite magnetic poles are separated by some distance, then it is called a magnetic dipole. Each pole of the magnet is the source of magnetic force which weakens with distance.

Since magnetic poles come in pairs, so they never are isolated. Forces exerted by the magnetic poles partially cancel each other because one pole pulls while the other repels. This cancellation of forces is greatest when the poles are closes to each other i.e. when the bar magnet is short. The magnetic force produced by a bar magnet at a given point in space is, therefore, depends on two factors, first is the strength p of its poles (magnetic pole strength), and the second is the vector ${\displaystyle \mathrm {\boldsymbol {\ell }}}$ separating distance of the poles. The magnetic dipole moment m of a bar magnet is given as-

What is the above expression? above expression is nothing but the product of the pole strength and the distance between their poles. It points in the direction from the South to the North pole within the magnet.

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