Magnetism and Matter: Exhaustive Physics Notes

Historical and Universal Context of Magnetism

Magnetism is a universal phenomenon observed across all scales of the universe, from the vast structures of distant galaxies to the microscopic architecture of atoms. It permeates living beings, including humans and beasts, through various magnetic fields stemming from diverse sources. The phenomenon of Earth's magnetism significantly predates human evolution. The etymology of the word "magnet" originates from Magnesia, an island in Greece where deposits of magnetic ore were discovered as early as 600 BC. In the early nineteenth century, the relationship between moving charges or electric currents and magnetic fields was established, with primary credit given to scientists such as Oersted, Ampere, Biot, and Savart.

Several fundamental observations characterize magnetism as an independent subject. First, the Earth acts as a giant magnet with its field pointing approximately from the geographic south to the geographic north. Second, a freely suspended bar magnet naturally aligns in a north-south direction; the end pointing toward the geographic north is termed the north pole, and the end pointing toward the geographic south is termed the south pole. Third, magnetic poles exhibit polarity-dependent forces: like poles (north-north or south-south) repel each other, while unlike poles (north-south) attract. Fourth, magnetic poles cannot be isolated. Breaking a bar magnet in half results in two smaller, complete bar magnets with their own sets of poles, albeit with weaker fields. This confirms that magnetic monopoles do not exist in nature, representing a significant departure from electric charges where isolated positive and negative charges are possible. Finally, magnets can be manufactured from iron and its various alloys.

The Bar Magnet and Field Line Characteristics

The physical manifestation of a magnetic field can be visualized by sprinkling iron filings on glass over a bar magnet. The filings arrange themselves in a pattern that suggests the magnet possesses two poles, mimicking the behavior of an electric dipole. One pole is the North pole and the other is the South pole. When suspended, these poles align with the Earth's geographic poles. A similar pattern of filings is observed around a current-carrying solenoid, reinforcing the connection between current and magnetism.

Magnetic field lines (sometimes historically referred to as magnetic lines of force) serve as a visual tool for understanding the magnetic field. Several properties define these lines. Unlike electric field lines, which start at a positive charge and terminate at a negative charge or infinity, magnetic field lines form continuous closed loops. The direction of the net magnetic field $\mathbf{B}$ at any given point is represented by the tangent to the field line at that point. The magnitude of the magnetic field is proportional to the density of the lines crossing a unit area; thus, a higher concentration of lines indicates a stronger field. Crucially, magnetic field lines never intersect, as an intersection would imply two different directions for the magnetic field at a single point, which is physically impossible. Field lines can be mapped experimentally by using a small magnetic compass needle and observing its orientation at various spatial positions.

Bar Magnet as an Equivalent Solenoid and Dipole Analogies

Based on Ampere's hypothesis that all magnetic phenomena result from circulating currents, a bar magnet can be viewed as an assembly of countless circulating currents, effectively making it equivalent to a solenoid. Cutting a bar magnet is conceptually identical to cutting a solenoid: each half becomes a new, smaller version of the original with continuous field lines emerging from one face and entering the other. This analogy is confirmed by the discovery that a small compass needle reacts identically when placed near either a bar magnet or a finite current-carrying solenoid.

Mathematically, the axial field of a finite solenoid at large distances resembles the field of a bar magnet. The magnitude of the axial magnetic field $\mathbf{B}$ at a distance $r$ is given by:

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

Where $m$ represents the magnetic moment. The magnetic moment of a bar magnet is defined as equal to the magnetic moment of an equivalent solenoid producing the same magnetic field. The analogy between electrostatics and magnetism allows for the conversion of electric field equations to magnetic field equations using specific replacements: the electric field $\mathbf{E}$ becomes the magnetic field $\mathbf{B}$, the electric dipole moment $\mathbf{p}$ becomes the magnetic moment $\mathbf{m}$, and the constant $\frac{1}{\epsilon_0}$ is replaced by $\mu_0$. Under these transformations, the equatorial field of a short bar magnet (r >> l) is $B_E = -\frac{\mu_0}{4\pi} \frac{m}{r^3}$ and the axial field is $B_A = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$.

Dipoles in Uniform Magnetic Fields

When a small compass needle with magnetic moment $\mathbf{m}$ is placed in a uniform magnetic field $\mathbf{B}$, it experiences a torque $\tau = \mathbf{m} \times \mathbf{B}$. The magnitude of this torque is $\tau = mB \sin(\theta)$, where $\theta$ is the angle between $\mathbf{m}$ and $\mathbf{B}$. This torque acts as a restoring force, aiming to align the dipole with the field. The magnetic potential energy $U_m$ associated with this configuration is derived by integrating the torque over the angle:

$U_m = \int \tau(\theta) \,d\theta = \int mB \sin(\theta) \,d\theta = -mB \cos(\theta) = -\mathbf{m} \cdot \mathbf{B}$

By convention, the zero of potential energy is set at $\theta = 90^\circ$. Consequently, the potential energy is at its minimum value of $-mB$ when $\theta = 0^\circ$ (representing the most stable equilibrium) and reaches its maximum value of $+mB$ when $\theta = 180^\circ$ (representing the most unstable equilibrium).

Gauss's Law for Magnetism

Gauss's law for magnetism states that the net magnetic flux $\phi_B$ through any closed surface is zero. This is expressed mathematically by dividing a surface $S$ into small area elements $\Delta \mathbf{S}$ and summing the flux:

$\phi_B = \sum_{\text{all}} \mathbf{B} \cdot \Delta \mathbf{S} = 0$

This law reflects the fundamental physical reality that isolated magnetic poles or monopoles do not exist. In electrostatics, the flux through a closed surface is $\frac{q}{\epsilon_0}$, where $q$ is the enclosed charge. Because there are no magnetic "charges" (sources or sinks), every field line that enters a closed volume must also exit it. All magnetic phenomena are thus explained through current loops or dipoles. If magnetic monopoles were to exist, Gauss's law would be modified such that the integral of $\mathbf{B} \cdot \Delta \mathbf{S}$ would equal $\mu_0 q_m$, where $q_m$ is the enclosed magnetic charge.

Magnetisation, Intensity, and Susceptibility

To categorize the magnetic properties of materials, several vector fields and constants are defined. Magnetisation $\mathbf{M}$ is defined as the net magnetic moment per unit volume of a sample:

$\mathbf{M} = \frac{\mathbf{m}_{\text{net}}}{V}$

$\mathbf{M}$ has dimensions $[L^{-1} A]$ and is measured in $A \, m^{-1}$. In a long solenoid with $n$ turns per unit length carrying current $I$, the magnetic field in a vacuum is $B_0 = \mu_0 nI$. When a material core is inserted, the total field $\mathbf{B}$ becomes the sum of the vacuum field and the field contributed by the material $\mathbf{B}_m$. Since $\mathbf{B}_m = \mu_0 \mathbf{M}$, the total field is expressed as:

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

Here, $\mathbf{H}$ is the magnetic intensity, defined as $\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}$. The influence of the external field on the material is quantified by the magnetic susceptibility $\chi$, a dimensionless quantity representing the ratio of magnetisation to magnetic intensity:

$\mathbf{M} = \chi \mathbf{H}$

The total field can also be written in terms of relative magnetic permeability $\mu_r$ and the magnetic permeability of the substance $\mu$:

$\mathbf{B} = \mu_0 (1 + \chi) \mathbf{H} = \mu_0 \mu_r \mathbf{H} = \mu \mathbf{H}$

Where $\mu_r = 1 + \chi$ and $\mu = \mu_0 \mu_r$. These constants are interrelated; if one is known, the others can be calculated.

Classification of Magnetic Materials

Materials are classified into three primary categories based on their response to external magnetic fields and their susceptibility $\chi$.

Diamagnetism: Diamagnetic substances have a small, negative susceptibility (-1 \leq \chi < 0). They are weakly repelled by magnets and tend to move from stronger to weaker parts of a non-uniform magnetic field. This occurs because orbiting electrons in atoms act as current loops; an external field induces a change in their motion (Lenz's law), creating a net magnetic moment opposite to the applied field. Examples include bismuth, copper, lead, silicon, water, and sodium chloride. Superconductors are "perfect" diamagnets with $\chi = -1$ and $\mu_r = 0$, exhibiting the Meissner effect where magnetic field lines are completely expelled from the material.

Paramagnetism: Paramagnetic substances have a small, positive susceptibility (0 < \chi < \epsilon). They are weakly attracted to magnets and move from weak to strong field regions. Individual atoms possess permanent magnetic dipole moments, but random thermal motion prevents bulk magnetisation until an external field aligns them. Magnetisation increases as temperature decreases or field strength increases. Examples include aluminium, sodium, calcium, oxygen (at STP), and copper chloride.

Ferromagnetism: Ferromagnetic substances have large, positive susceptibility (\chi >> 1). They are strongly attracted to magnets. Atoms possess permanent dipole moments that spontaneously align in macroscopic regions called domains (typically $1 \, mm$ in size containing $10^{11}$ atoms). In an unmagnetized state, domain orientations are random. An external field causes domains to align and grow, leading to high field concentration. Hard ferromagnetic materials (like Alnico) retain magnetisation after the external field is removed, forming permanent magnets. Soft ferromagnetic materials (like soft iron) lose magnetisation when the field is removed. Ferromagnetism is temperature-dependent; above a certain temperature, the domain structure disintegrates, and the material becomes paramagnetic.

Questions and Discussion

Example 5.1 Analysis: (a) Cutting a bar magnet either transverse or along its length results in two smaller magnets, each with its own North and South pole. (b) A magnetised needle in a uniform field feels torque but no net force because the poles experience equal and opposite forces. However, an iron nail near a bar magnet is in a non-uniform field and has an induced magnetic moment. Because the induced pole closer to the magnet (e.g., the south pole) is in a stronger part of the field than the further pole (the north pole), there is a net attractive force. (c) Not every configuration has N/S poles; a toroid or infinite straight conductor has a field but no net magnetic moment/poles. (d) To identify which of two identical bars is magnetised: if they repel in any orientation, both are magnetised. If they only attract, one is a simple iron bar. To find which is which, touch the end of bar A to the middle of bar B. If there is no force, B is the magnet (since the field is weakest at the center of a magnet). If the force remains constant from end to middle, A is the magnet.

Example 5.3 Analysis of Field Lines: Field lines cannot emanate from a single point (violates Gauss's law). They cannot cross (direction must be unique). Static magnetic field lines in empty space cannot form closed loops; they must enclose a current. Field lines at the ends of a solenoid must curve to form loops, not remain straight. Fringing of lines at pole pieces is inevitable due to Ampere's law.

Example 5.4 Discussion: Magnetic field lines do not represent lines of force for moving charges because the magnetic force ($F = qv \times B$) is always perpendicular to the field. If monopoles existed, Gauss's law for magnetism would resemble Gauss's law for electrostatics. Elements of a current-carrying wire do not exert force on themselves, but they do exert force on other elements of the same wire. A system with zero net charge (like an atom) can still have a magnetic moment due to individual current loops (electron orbits).

Summary of Physical Quantities and Units

Permeability of free space ($\mu_0$): Scalar, units $T \, m \, A^{-1}$. Value $\frac{\mu_0}{4\pi} = 10^{-7}$.

Magnetic field ($\mathbf{B}$): Vector, units Tesla ($T$). $1 \, T = 10^4 \, Gauss$.

Magnetic moment ($\mathbf{m}$): Vector, units $A \, m^2$.

Magnetic flux ($\phi_B$): Scalar, units Weber ($W$). $W = T \, m^2$.

Magnetisation ($\mathbf{M}$): Vector, units $A \, m^{-1}$. Defined as $\frac{\text{Magnetic moment}}{\text{Volume}}$.

Magnetic intensity ($\mathbf{H}$): Vector, units $A \, m^{-1}$. Relation: $\mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M})$.

Magnetic susceptibility ($\chi$): Scalar, dimensionless. Relation: $\mathbf{M} = \chi \mathbf{H}$.

Relative magnetic permeability ($\mu_r$): Scalar, dimensionless. Relation: $\mathbf{B} = \mu_0 \mu_r \mathbf{H}$.

Selected Exercise Illustrations

Exercise 5.1: A short bar magnet at $30^\circ$ to a $0.25 \, T$ field experiences a torque of $4.5 \times 10^{-2} \, J$. The magnetic moment $m$ is calculated using $\tau = mB \sin(\theta)$.

Exercise 5.2: For a magnet with $m = 0.32 \, J T^{-1}$ in a $0.15 \, T$ field, stable equilibrium is at $\theta = 0^\circ$ (potential energy $U = -mB = -0.048 \, J$) and unstable equilibrium is at $\theta = 180^\circ$ (potential energy $U = +mB = +0.048 \, J$).

Exercise 5.3: A solenoid with 800 turns, area $2.5 \times 10^{-4} \, m^2$, and current $3.0 \, A$ acts as a magnet. The magnetic moment is calculated as $m = NIA = 800 \times 3.0 \times 2.5 \times 10^{-4} = 0.6 \, A \, m^2$.

Exercise 5.7: A short bar magnet ($m = 0.48 \, J T^{-1}$) at a distance of $10 \, cm$ ($0.1 \, m$) has an axial field of $B_A = \frac{\mu_0}{4\pi} \frac{2m}{r^3} = 10^{-7} \times \frac{2 \times 0.48}{(0.1)^3} = 0.96 \times 10^{-4} \, T$ and an equatorial field of $B_E = \frac{\mu_0}{4\pi} \frac{m}{r^3} = 0.48 \times 10^{-4} \, T$.