1. Introduction

2. Bar magnet

1. Fundamental Properties of Magnets

Before diving into the physics of fields, it is important to review the basic behavior of magnets:

Magnetic phenomena are universal in nature, with fields permeating everything from distant galaxies to tiny atoms.
When a magnet is freely suspended, its poles will point approximately toward the geographic north and south poles.
The pole pointing to the geographic north is designated as the North pole, while the opposite tip is the South pole.
Like poles repel each other, while unlike poles attract.
Unlike electric charges, isolated magnetic poles (magnetic monopoles) do not exist.
If you break a bar magnet in half, you do not isolate the poles; instead, you get two similar bar magnets with somewhat weaker properties.

2. Magnetic Field Lines

We can visualize the magnetic field of a bar magnet by looking at the patterns formed by iron filings sprinkled around it, which mimics the behavior of a magnetic dipole.

Magnetic field lines form continuous, closed loops.
This is a fundamental difference from electric field lines, which begin on a positive charge and end on a negative charge, or escape to infinity.
The tangent to a field line at any given point indicates the direction of the net magnetic field $\mathbf{B}$ at that specific location.
The density of the lines indicates field strength; a larger number of field lines crossing a unit area means the magnitude of the magnetic field $\mathbf{B}$ is stronger.
Magnetic field lines never intersect.
If they were to intersect, it would mean the direction of the magnetic field at that specific point is not unique, which is physically impossible.

3. The Bar Magnet as an Equivalent Solenoid

A bar magnet and a current-carrying solenoid share remarkably similar magnetic properties.

The resemblance in their magnetic field lines suggests that a bar magnet can be thought of as a large number of circulating currents.
Cutting a bar magnet in half is conceptually the same as cutting a solenoid; you simply get two smaller solenoids with weaker magnetic properties, but the field lines remain continuous.
The magnitude of the far axial magnetic field for both a finite solenoid and a bar magnet is identical:

$B = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$
Consequently, the magnetic moment of a bar magnet is exactly equal to the magnetic moment of an equivalent solenoid that produces the same magnetic field.

4. The Magnetic Dipole in a Uniform Field

When a magnetic dipole (like a compass needle or a bar magnet) with a known magnetic moment $m$ is placed in a uniform magnetic field, it experiences rotational forces.

The potential energy of the dipole depends on its orientation and is calculated as $-mB$ when aligned.
The system's potential energy is at its minimum ( $-mB$ ) when the angle $\theta = 0^{\circ}$ , representing the most stable equilibrium position.
Conversely, the energy is at its maximum ( $+mB$ ) when the angle $\theta = 180^{\circ}$ , which represents the most unstable position.

5. The Electrostatic Analog

Because the field equations at large distances are so similar, you can calculate the magnetic field of a bar magnet by using the equations for an electric dipole, simply by making a few substitutions:

Replace the electric field $\mathbf{E}$ with the magnetic field $\mathbf{B}$ .
Replace the electric dipole moment $\mathbf{p}$ with the magnetic moment $\mathbf{m}$ .
Replace the electrostatic constant $1/\epsilon_0$ with the magnetic permeability $\mu_0$ .

Using this analogy, for a bar magnet of size $l$ at a distance $r$ (where $r \gg l$ ):

The Equatorial Field ( $B_E$ ):

$B_E = -\frac{\mu_0}{4\pi} \frac{m}{r^3}$
The Axial Field ( $B_A$ ):

$B_A = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$

That is a very clever way to think about it, but it is actually the exact opposite! The poles on your hand-held magnet are named correctly based on the direction they seek out, which means it is actually the Earth's magnetic poles that are "backwards" relative to their geographic names.

Here is a breakdown of exactly how this works:

The Magnet's Poles: When a bar magnet is freely suspended, it naturally aligns itself in the north-south direction. The tip that points toward the Earth's geographic north is designated as the magnet's north pole, and the tip pointing toward the geographic south is the south pole.
The Law of Attraction: A fundamental rule of magnetism is that like poles repel each other, while unlike poles attract.
The Earth's Magnetic South: Because opposite poles attract, for the north pole of your magnet to be pulled toward the top of the globe, the magnetic pole located near Earth's geographic North must physically act as a magnetic south pole.
The Earth's Magnetic North: Similarly, the south pole of your magnet is attracted to the bottom of the globe because there is a magnetic north pole located near the Earth's geographic South.

So, your magnet's north pole really is a true north pole—it is just being attracted to the Earth's magnetic south pole, which happens to be hiding up near the geographic North.

Because a fundamental rule of magnetism is that opposite poles attract, the north pole of a compass or bar magnet is naturally pulled toward the top of the globe. This means that the magnetic pole located near Earth's geographic North must physically act as a magnetic south pole in order to attract it.

Therefore, the Arctic region is geographically north, but it behaves as a magnetic south pole, meaning Earth's magnetic poles are essentially "backwards" relative to their geographic names.

Location of the Earth's magnet:
- the Earth itself behaves as a magnet.
- This planetary magnetic field points approximately from the geographic south to the geographic north.
Size of the Earth's magnet:
- However, it does quantify the magnitude of the Earth's magnetic field, stating that it is about 3.6 × 10⁻⁵ T, and in another section, approximately 4 × 10⁻⁵ T.
Magnetic properties of other celestial bodies:
- It does confirm that magnetic phenomena are universal in nature.
- Furthermore, it states that vast, distant galaxies, tiny invisible atoms, humans, and beasts are all permeated through and through with a host of magnetic fields from a variety of sources.

Magnetic field and Electric Field

Electric field lines and magnetic field lines share several fundamental properties, but they also have distinct physical differences primarily related to their sources and geometry.

Similarities

Both electric and magnetic fields are long-range, meaning they both depend inversely on the square of the distance from their source.
The principle of superposition applies to both fields.
Neither electric nor magnetic field lines can ever cross each other. If they did, it would mean the direction of the field at that point of intersection is ambiguous.
For both fields, the tangent to a field line at any given point represents the direction of the net field at that location.
The relative density of the lines indicates the strength of the field; the field lines crowd together in regions where the field is strong and are spaced further apart where the field is weak.

Differences

Closed Loops vs. Terminating Lines: Magnetic field lines form continuous, closed loops. In contrast, electrostatic field lines must start at positive charges and end at negative charges (or escape to infinity), and they can never form closed loops.
Sources: An electrostatic field is produced by a scalar source, namely an electric charge. A magnetic field, however, is produced by a vector source, such as a current element ( $I dl$ ).
Gauss's Law and Monopoles: Because isolated magnetic poles (monopoles) do not exist, the net magnetic flux through any closed surface is always zero. For electric fields, the flux through a closed surface is determined by the total net electric charge enclosed by that surface.
Direction of Force: Electrostatic field lines indicate the direction of the force that would act on a positive test charge. Unlike electrostatics, magnetic field lines do not indicate the direction of the force on a moving charge.

3. Magnetism and Gauss's law

4. Magnetisation and Magnetic intensity

5. Magnetic properties of materials

Imagine taking a magnet and holding it up to various things—a copper pipe, an iron nail, or even a glass of water. In physics, we classify materials based on exactly these varied reactions, which can be measured by a property called magnetic susceptibility ( $\chi$ ). Think of $\chi$ as a measure of how a material responds to an external magnetic field.

Based on this mathematical metric and their physical behaviors, the Earth's bewitching variety of elements and compounds fall into three main categories:

1. Diamagnetic Materials (The Rebels)

Behavior: These substances have a tendency to move from a stronger to a weaker part of an external magnetic field. In simple terms, they are repelled by magnets.
The "Why": The resultant magnetic moment in a single atom of these materials is naturally zero. When you apply a magnetic field, the orbiting electrons change their speeds—those with moments in the same direction slow down, and those in the opposite direction speed up due to an induced current (Lenz's law). This develops a net magnetic moment in the exact opposite direction to your applied field, causing a slight repulsion.
Susceptibility ( $\chi$ ): Small and negative ( $-1 \le \chi < 0$ ).
Practical Examples: Copper, bismuth, lead, silicon, nitrogen, water, and sodium chloride (table salt).
Superconductors (The Extreme Example): When certain metals are cooled to extremely low temperatures, they exhibit perfect diamagnetism ( $\chi = -1$ ) and become perfect conductors. They completely expel all magnetic field lines in a phenomenon known as the Meissner effect. This is the exact science exploited to run magnetically levitated (Maglev) superfast trains!

2. Paramagnetic Materials (The Weak Conformists)

Behavior: These materials get weakly magnetised in an external magnetic field and have a tendency to move from a weak to a strong field, meaning they are weakly attracted to magnets.
The "Why": Their individual atoms or molecules possess their own permanent magnetic dipole moments. However, because of ceaseless, random thermal motion, these atoms point in all directions, meaning no net magnetisation is seen initially. When a strong external field is applied, it forces these atomic dipoles to align in the same direction, which slightly enhances the field inside the material.
Susceptibility ( $\chi$ ): Positive and small ( $0 < \chi < \epsilon$ ).
Practical Examples: Aluminium, sodium, calcium, oxygen, and copper chloride.
Temperature Dependence: As the temperature drops or the external field increases, more of these dipoles are able to align. This continues until they reach a "saturation value" where all dipoles are perfectly aligned with the field.

3. Ferromagnetic Materials (The Enthusiastic Followers)

Behavior: These substances get strongly magnetised in an external magnetic field and are strongly attracted to magnets.
The "Why": Like paramagnets, their atoms have dipole moments, but they interact cooperatively to spontaneously align themselves in massive groupings called domains. A typical domain is only about 1mm in size, but it holds roughly $10^{11}$ atoms all pointing in the exact same direction! Initially, these domains point randomly and cancel each other out. However, when an external field is applied, the domains oriented with the field rapidly grow in size, creating a highly concentrated net field.
Susceptibility ( $\chi$ ): Extremely large and positive ( $\chi \gg 1$ ).
Practical Examples: Iron, cobalt, nickel, and gadolinium.
Hard vs. Soft Ferromagnets:
- Some materials, like Alnico (an alloy) or naturally occurring lodestone, keep this magnetisation even after the external field is removed. These are "hard ferromagnets" used to make permanent magnets like compass needles.
- Others, like soft iron, lose their magnetisation when the field disappears, making them "soft ferromagnets".
Temperature Limits: This incredible ferromagnetic property depends heavily on temperature. If heated high enough, the structured domains completely disintegrate, and the ferromagnet gradually becomes just a regular paramagnet.