Comprehensive Study Notes on Magnetic Materials and Bar Magnets

Fundamentals of Bar Magnets and Magnetic Dipole Moments

A bar magnet consists of two poles, the North (N) and South (S), with equal and opposite pole strengths denoted by $m$. The unit for pole strength is Ampere-meter ($Am$), often represented dimensionally as $[AL]$. The physical distance between the two poles is defined as the magnetic length ($2L$). It is noted that the magnetic length is approximately $5/6$ or $0.84$ times the geometric length of the magnet.

The magnetic dipole moment ($M$) is a vector quantity that characterizes the magnet's strength and orientation. It is defined as the product of the pole strength ($m$) and the magnetic length ($2L$):

$M = m \times 2L$

The unit for magnetic dipole moment is $Am^2$ ($AL^2$). The direction of the vector point is always from the South pole to the North pole ($S \rightarrow N$).

When multiple magnetic moments are present, the net magnetic moment ($M_{net}$) is calculated using vector addition:

M_{net} = √{M_1^2 + M_2^2 + 2M_1M_2os(̘)}

Special cases based on the angle (̘) between two equivalent moments ($M_1 = M_2 = M$) include:

• If ̘ = 60^∘, then $M_{net} = √{3}M$
• If ̘ = 90^∘, then $M_{net} = √{2}M$
• If ̘ = 120^∘, then $M_{net} = M$
• If magnets are placed in-line (̘ = 0^∘), then $M_{net} = M_1 + M_2 = 2M$
• If magnets are placed oppositely (̘ = 180^∘), then $M_{net} = M_1 - M_2 = 0$

Dipole Moments in Current-Carrying Wires

For current-carrying loops or wires, the magnetic dipole moment is given by the product of the current ($I$) and the cross-sectional area ($A$) of the loop:

$M = IA$

The unit for this expression is also $Am^2$. In the context of bending a steel wire of length $L$ and initial magnetic moment $M$, the new magnetic moment depends on the effective displacement ($L_{eff}$) between the ends:

1. For a wire bent into an L-shape (right angle): The effective length is the hypotenuse. If the wire is split into two equal halves of length rac{L}{2}, the new moment is M' = m \times rac{L}{√{2}} = rac{M}{√{2}}.
2. For a wire bent into a semicircle: The length $L$ equals ̐ R, so the radius R = rac{L}{̐}. The effective displacement between ends is the diameter ($2R$). Thus, M' = m \times rac{2L}{̐} = rac{2M}{̐}.
3. For a wire bent into a full circle: The displacement between the start and end point is zero ($L_{eff} = 0$), therefore $M = 0$.

When considering loops of different shapes formed from a wire of length $L$ carrying current $I$, the magnetic moment ($M = IA$) varies based on the area enclosed:

• For an equilateral triangle: The side length is rac{L}{3}. The area A = rac{√{3}}{4}(rac{L}{3})^2 = rac{√{3}L^2}{36}. Thus, M = rac{IL^2√{3}}{36}.
• For a square: The side length is rac{L}{4}. The area A = (rac{L}{4})^2 = rac{L^2}{16}. Thus, M = rac{IL^2}{16}.
• For a regular hexagon: The side length is rac{L}{6}. The area A = 6 \times rac{√{3}}{4}(rac{L}{6})^2 = rac{√{3}L^2}{24}. Thus, M = rac{IL^2√{3}}{24}.
• For a circle: The radius r = rac{L}{2̐}. The area A = ̐(rac{L}{2̐})^2 = rac{L^2}{4̐}. Thus, M = rac{IL^2}{4̐}.

Magnetic Induction due to a Bar Magnet

The magnetic induction ($B$) at a distance $r$ from the center of a bar magnet (where $r >> L$) can be calculated for two primary positions:

1. On the Axial Line (̘ = 0^∘): B_{axis} = rac{̑_0}{4̐} \times rac{2M}{r^3}

2. On the Equatorial Line (̘ = 90^∘): B_{eq} = rac{̑_0}{4̐} \times rac{M}{r^3}

The general formula for magnetic field at any point defined by angle ̘ is: B = rac{̑_0 M}{4̐ r^3} √{1 + 3os^2(̘)}

From these formulas, it is evident that for the same distance $r$, $B_{axis} = 2 \times B_{eq}$.

Dynamics of a Magnet in a Magnetic Field

When a bar magnet is placed in a uniform external magnetic field ($B$), it experiences a torque (̔) but no net translational force. The torque attempts to align the magnet with the field:

̔ = Μ \times B = MBin(̘)

Rotating a magnetic dipole within a field requires work ($W$). The work done in rotating the dipole from an initial angle ̘_1 to a final angle ̘_2 is calculated by integrating the torque:

W = ∫_{̘_1}^{̘_2} ̔ d̘ = MB[os(̘_1) - os(̘_2)]

If the magnet is allowed to oscillate in the field, it performs simple harmonic motion (provided ̘ is small). The time period ($T$) of these oscillations is given by:

T = 2̐ √{rac{I}{MB}}

Here, $I$ represents the moment of inertia of the magnet. The restoring torque equation is stated as Ȋ = -MB̘, where ̑ is the angular acceleration (rac{d^2̘}{dt^2}).

Modification and Cutting of Bar Magnets

The properties of a bar magnet change depending on how it is sectioned:

1. Cutting Perpendicular to the Length: If a magnet of moment $M$ and pole strength $m$ is cut into two equal halves perpendicular to its axis, the pole strength remains $m$, but the length becomes $L$. The new magnetic moment is M' = m \times L = rac{M}{2}.
2. Cutting Parallel to the Length: If cut longitudinally into two equal halves, the magnetic length remains $2L$, but the pole strength of each piece becomes rac{m}{2}. The new magnetic moment is M' = rac{m}{2} \times 2L = rac{M}{2}.

Atomic Magnetism and Revolving Electrons

Magnetism at the atomic level arises from the motion of electrons. An electron revolving in a circular orbit of radius $r$ with speed $v$ constitutes a current loop.

The current is I = rac{e}{T} = rac{ev}{2̐ r}. The resulting magnetic moment is:

M = IA = rac{ev}{2̐ r} \times ̐ r^2 = rac{evr}{2}

The relationship between the magnetic moment ($M$) and angular momentum ($L = m_e vr$) is expressed via the gyromagnetic ratio:

rac{M}{L} = rac{e}{2m_e}

The gyromagnetic ratio for an electron is approximately 8.8 \times 10^{10}\text{/kg}. The Bohr Magneton (̑_B) is the base unit of magnetic moment at the atomic scale, corresponding to the first Rydberg orbit, and is calculated using Planck's constant (h = 6.63 \times 10^{-34}\text{s}):

̑_B = rac{eh}{4̐ m_e} = 9.27 \times 10^{-24}\text{Am}^2

Magnetic Properties and Classification of Materials

Key parameters for defining magnetic behavior include:

1. Magnetization ($M_z$): Divided as the total net dipole moment per unit volume. M_z = rac{M_{net}}{Volume}. Units are $Am^{-1}$ ($A/m$). Dimension: $[A^1L^{-1}]$.
2. Magnetic Intensity ($H$): For a solenoid, $H = nI$. Units are $Am^{-1}$.
3. Magnetic Susceptibility (̗): Relates magnetization to intensity, M_z = ̗ H.
4. Permeability (̑): The ability of a material to allow magnetic field lines to pass through it. ̑ = ̑_0 ̑_r, where ̑_0 is vacuum permeability and ̑_r is relative permeability.
5. Relation: B = ̑_0(H + M_z) = ̑_0 H(1 + ̗) = ̑ H. Thus, ̑_r = 1 + ̗.

Materials are classified into three categories based on their behavior in an external field:

• Diamagnetic: Weakly repelled by magnets. Examples: Bismuth, Copper, Gold, Mercury, Quartz, Alcohol, Hydrogen. Field lines are less dense inside. They move from stronger to weaker parts of a non-uniform field. Properties are independent of temperature (̗ is small and negative; ̑_r < 1).
• Paramagnetic: Weakly attracted to magnets. Examples: Aluminum, Platinum, Manganese, Oxygen, Copper Sulphate. Field lines are slightly more dense inside. They move from weaker to stronger parts of a field. These follow Curie's Law: M_z ∝ rac{B}{T} or ̗ = rac{C}{T}, where $C$ is the Curie constant (̑_r > 1; ̗ is small and positive).
• Ferromagnetic: Strongly attracted to magnets. Examples: Iron, Cobalt, Nickel, Gadolinium, and alloys like Alnico. They move quickly toward stronger field regions. They are temperature dependent; above the Curie Temperature ($T_c$), they become paramagnetic (̑_r >> 1; ̗ is large and positive).

Domain Theory and Hysteresis

Ferromagnetism is explained by Domain Theory. A domain is a region of spontaneous magnetization where atomic dipoles are aligned parallel. In an unmagnetized state, these domains are randomly oriented, resulting in a net moment of zero. Applying an external field causes domains to align and merge, resulting in strong magnetization.

The Hysteresis Loop describes the lag of magnetic induction ($B$) behind the magnetizing field ($H$):

• Point O: Initial unmagnetized state.
• Point A: Saturation point, where all domains are aligned.
• Point B (Retentivity): The residual magnetism left in the material when the external field $H$ is reduced to zero ($H=0, B ≠ 0$).
• Point C (Coercivity): The magnitude of the reverse magnetic field required to reduce the residual induction to zero ($B=0, H ≠ 0$).
• Point D: Saturation in the opposite direction.

Soft Magnetic Materials (e.g., Soft Iron) have high retentivity but low coercivity and small hysteresis loss, making them suitable for transformer cores and electromagnets. Hard Magnetic Materials (e.g., Steel, Alnico) have high retentivity and high coercivity, making them ideal for permanent magnets.