5.2.2. Bar magnet as an equivalent solenoid

In the previous chapter, we have explained how a current loop acts as a magnetic dipole (Section 4.9). The resemblance of magnetic field lines for a bar magnet and a solenoid suggest that a bar magnet may be thought of as a large number of circulating currents in analogy with a solenoid.

The concept here is that both bar magnets and solenoids can create a magnetic field, and they do this in similar ways. A bar magnet generates a magnetic field due to the alignment of its internal atomic magnets, while a solenoid generates a magnetic field through the flow of electric current. This comparison allows us to understand the behavior of a bar magnet as if it were made up of many small loops of current, each contributing to the overall magnetic field.

Think of a bar magnet as a group of friends holding hands in a circle (the circulating currents). Each friend represents a small current loop, and together they create a strong magnetic force just like the bar magnet does. Similarly, when the group organizes into a line and starts moving in sync (like current in a solenoid), they produce a magnetic effect that can be seen when near a compass.

Cutting a bar magnet in half is like cutting a solenoid. We get two smaller solenoids with weaker magnetic properties. The field lines remain continuous, emerging from one face of the solenoid and entering into the other face.

When either a bar magnet or a solenoid is cut, each resulting piece retains both a north and a south pole. This means that no matter how many times you cut the original magnet or solenoid, you will always end up with smaller versions of the original, each capable of creating a magnetic field. The continuity of the magnetic field lines is important because it signifies that the magnetic field is always present and never isolated.

Imagine a chocolate bar that you break into pieces. No matter how small the pieces become, each piece still tastes like chocolate. Similarly, when you break a bar magnet into parts, each part still acts as a complete magnet, just smaller in size.

One can test this analogy by moving a small compass needle in the neighbourhood of a bar magnet and a current-carrying finite solenoid and noting that the deflections of the needle are similar in both cases.

By placing a compass needle near a bar magnet or a solenoid with current running through it, you can observe how the compass needle reacts. The direction the needle points reflects the direction of the magnetic field in that area. If the deflections are similar, it confirms that both a bar magnet and a solenoid create comparable magnetic environments, reinforcing our understanding of their equivalent nature.

Consider how your phone's compass works. When you're near your fridge magnet or a wire carrying electricity, the compass tells you where north is by pointing in the direction of the magnetic field. This is similar to how both the fridge magnet and electrical wire create forces that can influence the compass.

To make this analogy more firm we may calculate the axial field of a finite solenoid depicted in Fig. 5.3 (a). We can demonstrate that at large distances this axial field resembles that of a bar magnet.

By performing calculations using the appropriate equations, we can demonstrate that the magnetic field produced by a solenoid at a distance resembles that created by a bar magnet. This means that theoretically and empirically, a solenoid can be treated as the equivalent of a magnet when analyzing their properties at a distance. Hence, if we know the properties of solenoids, we can predict behaviors of bar magnets and vice versa.

Think of using a flashlight. When you shine it close, you see the beam clearly; move it far enough away, and it looks like a glow – similar to how the detailed structure of the solenoid fades into a simple resemblance with the bar magnet as distance increases.

The magnetic moment of a bar magnet is thus equal to the magnetic moment of an equivalent solenoid that produces the same magnetic field.

This statement emphasizes a key point: regardless of whether we're dealing with a bar magnet or a solenoid, if both have the same magnetic moment, they will create the same effect on the surrounding magnetic field. The understanding of magnetic moment is crucial for comparing these two types of magnets, as it allows us to establish a direct relationship between them.

Imagine two cars moving at the same speed; you can't tell which one is faster just by looking at them from a distance. The speed makes them comparable, just like having the same magnetic moment makes both the bar magnet and solenoid behave similarly.

--