Introduction

This chapter delves into the fascinating world of magnetism, exploring its properties and the behavior of magnetic materials. It revisits the fact that moving charges and electric currents are sources of magnetic fields, as discovered by Oersted and others. The chapter introduces the concept of a magnetic field, its visualization through field lines, and the behavior of magnetic dipoles (like bar magnets) in external magnetic fields. It also discusses Gauss's Law for magnetism, the classification of materials based on their magnetic properties (diamagnetism, paramagnetism, ferromagnetism), and the crucial concept of magnetisation.

The Bar Magnet

The Magnetic Field Lines

Magnetic field lines are used to visualize the magnetic field of a magnet. They form continuous closed loops, starting from the north pole and entering the south pole, both outside and inside the magnet. The tangent to a field line at any point gives the direction of the magnetic field, and the density of field lines indicates the strength of the field. Unlike electric field lines, magnetic field lines never intersect.

Figure 5.1 shows the pattern of iron filings around a bar magnet, mimicking magnetic field lines. Figure 5.2 compares the field lines of a bar magnet, a solenoid, and an electric dipole, highlighting similarities at large distances.

Bar Magnet As An Equivalent Solenoid

A bar magnet can be understood as an equivalent solenoid, where the magnetic field produced by circulating currents within the atomic structure of the magnet is analogous to the field of a solenoid. Cutting a bar magnet in half results in two smaller magnets, similar to cutting a solenoid.

The magnetic field on the axis of a finite solenoid at large distances resembles that of a bar magnet. The magnetic moment of a bar magnet is equivalent to the magnetic moment of a solenoid that produces a similar field.

The Dipole In A Uniform Magnetic Field

When a magnetic dipole (like a bar magnet or a current loop) with magnetic moment $\vec{m}$ is placed in a uniform external magnetic field $\vec{B}$, it experiences a torque $\vec{\tau} = \vec{m} \times \vec{B}$. This torque tends to align the magnetic moment with the magnetic field.

The potential energy ($U$) of the magnetic dipole in the external field is given by $U = -\vec{m} \cdot \vec{B}$. The potential energy is minimum (stable equilibrium) when $\vec{m}$ is aligned with $\vec{B}$ ($\theta = 0^\circ$) and maximum (unstable equilibrium) when $\vec{m}$ is aligned opposite to $\vec{B}$ ($\theta = 180^\circ$).

Example 5.1 discusses phenomena related to cutting magnets and the forces/torques experienced by magnetic materials in fields.

Example 5.2 analyzes the equilibrium configurations of two magnetic dipoles.

The Electrostatic Analog

There is a strong analogy between electric dipoles in electric fields and magnetic dipoles (like bar magnets or current loops) in magnetic fields. This analogy is summarized in Table 5.1, showing corresponding quantities like $E \leftrightarrow B$, $p \leftrightarrow m$, and $\epsilon_0 \leftrightarrow \mu_0$.

Magnetism And Gauss’s Law

Gauss's Law for magnetism states that the net magnetic flux ($\phi_B$) through any closed surface is always zero:

$\oint_S \vec{B} \cdot d\vec{S} = 0$

This is a fundamental law reflecting the fact that isolated magnetic monopoles (sources or sinks of magnetic field) do not exist. Magnetic field lines always form closed loops, meaning the number of field lines entering any closed surface must equal the number leaving it.

Example 5.3 identifies incorrect representations of magnetic and electrostatic field lines based on their properties and Gauss's Law.

Example 5.4 discusses the interpretation of magnetic field lines and the implications if magnetic monopoles existed.

Magnetisation And Magnetic Intensity

Magnetisation ($\vec{M}$) of a material is defined as its net magnetic dipole moment per unit volume. It's a vector quantity measured in A/m.

When a material is placed in an external magnetic field $\vec{H}$ (magnetic intensity), it develops a magnetisation $\vec{M}$. The total magnetic field $\vec{B}$ inside the material is given by:

$\vec{B} = \mu_0 (\vec{H} + \vec{M})$

where $\mu_0$ is the permeability of free space. The magnetic intensity $\vec{H}$ represents the contribution from external factors (like solenoid windings).

The relationship between $\vec{M}$ and $\vec{H}$ for linear materials is $\vec{M} = \chi_m \vec{H}$, where $\chi_m$ is the magnetic susceptibility (a dimensionless quantity). This leads to $\vec{B} = \mu_0 (1 + \chi_m) \vec{H} = \mu_0 \mu_r \vec{H} = \mu \vec{H}$, where $\mu_r = 1 + \chi_m$ is the relative permeability and $\mu = \mu_0 \mu_r$ is the permeability of the material.

Example 5.5: Calculates magnetic intensity ($H$), magnetisation ($M$), magnetic field ($B$), and magnetizing current ($I_m$) for a solenoid with a magnetic core.

Magnetic Properties Of Materials

Materials are classified based on their magnetic susceptibility ($\chi_m$) and relative permeability ($\mu_r$):

Diamagnetism

Diamagnetic materials have a small, negative magnetic susceptibility ($\chi_m < 0$) and $\mu_r < 1$. They are weakly repelled by a magnetic field and tend to move from stronger to weaker field regions. This effect arises from induced orbital magnetic moments in atoms due to Lenz's Law, opposing the applied field. Examples include bismuth, copper, water, and superconductors (which exhibit perfect diamagnetism, $\chi_m = -1$, known as the Meissner effect).

Paramagnetism

Paramagnetic materials have small, positive magnetic susceptibility ($\chi_m > 0$) and $\mu_r > 1$. Their atoms possess permanent magnetic dipole moments which tend to align with an external magnetic field, causing a weak attraction and a slight enhancement of the field inside. Examples include aluminum, sodium, and oxygen.

Ferromagnetism

Ferromagnetic materials have very large, positive magnetic susceptibility ($\chi_m \gg 1$) and $\mu_r \gg 1$. They are strongly magnetized in an external field and exhibit strong attraction towards magnets. This arises from cooperative alignment of atomic magnetic moments within regions called domains. Ferromagnetic materials can retain their magnetism even after the external field is removed (permanent magnets, like iron, nickel, cobalt) or lose it easily (soft ferromagnets, like soft iron).

Table 5.2 summarizes the magnetic properties of these material classes.

Figure 5.7 illustrates the behavior of field lines near diamagnetic and paramagnetic substances. Figure 5.8 shows the domain structure in ferromagnetic materials.

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