Chapter 5 — Magnetism And Matter

Fig 5.1 — Educational Diagram: Magnetic dipole and magnetic moment

Magnetic dipole: A magnetic dipole is a system that produces a magnetic field similar to that of a tiny bar magnet having a north and a south pole. Common examples are a small current loop and a bar magnet. At large distances the field of such a system falls off as 1/r^3 and has the same angular pattern as an electric dipole field.

Magnetic dipole moment (m): The magnetic dipole moment is a vector quantity that measures the strength and orientation of a magnetic dipole. For a current loop of area A carrying current I, the magnetic dipole moment is m = I A n, where n is a unit vector normal to the loop given by the right-hand rule (curl fingers in direction of current; thumb gives n). For a coil with N turns, m = N I A. The SI unit is ampere-square metre (A m^2), which is equivalent to joule per tesla (J/T).

Direction: The direction of m is from the south to the north pole of an equivalent bar magnet (or given by the right-hand rule for a current loop).

Interaction with external magnetic field:

• Torque: A magnetic dipole in a uniform magnetic field B experiences a torque τ = m × B. Magnitude τ = m B sinθ, which tends to align m with B (θ is angle between m and B).
• Potential energy: U = -m · B = -m B cosθ. Minimum energy when m is aligned with B.
• Force in a non-uniform field: A dipole in a non-uniform magnetic field feels a net force that tends to pull it toward stronger field regions. For constant m, F = ∇(m · B) (often written F = (m · ∇) B in component form).

Magnetic field of a dipole (far field): At distance r (r much larger than dipole size), the magnetic field of a dipole m is B(r) = (μ0 / 4π) * [ (3 (m · r̂) r̂ - m) / r^3 ], where r̂ is the unit vector from the dipole to the field point. On the axis of the dipole (along m) the magnitude is B_axial = (μ0 / 4π) * (2 m / r^3). On the equatorial plane (perpendicular to m) B_equatorial = (μ0 / 4π) * ( - m / r^3 ).

Microscopic origin: At atomic scale magnetic moments arise from orbital motion of electrons and electron spin. The orbital magnetic moment is proportional to orbital angular momentum L: μ_orbital = (e / 2 m_e) L (sign conventions: electron charge is negative). The natural scale is the Bohr magneton μ_B = e ħ / (2 m_e) ≈ 9.27 × 10^-24 A m^2. Electron spin gives an intrinsic magnetic moment (spin magnetic moment) with a g-factor ≈ 2.

Remarks: A bar magnet behaves like a dipole only at distances large compared to its size. Magnetic dipole moment is used to quantify strength of magnets, current loops, coils, atoms and nuclei (nuclear magnetic moments play a role in NMR and MRI).

Fig 5.2 — Educational Diagram: Torque and potential energy of a magnetic dipole

Definition of magnetic dipole moment
A magnetic dipole (for example a small current loop or a bar magnet) is characterized by its magnetic dipole moment m. For a planar current loop, m = I·A (vector), where I is current and A is the area vector whose direction is given by the right‑hand rule.

Torque on a magnetic dipole in a uniform magnetic field
When a magnetic dipole with moment m is placed in a uniform magnetic field B, it experiences a torque τ that tends to align m with B. Vectorially,

τ = m × B

The magnitude is

|τ| = m B sinθ

where θ is the angle between m and B. The direction of the torque is given by the right‑hand rule and is perpendicular to the plane containing m and B. For a rectangular/current loop derivation: opposite sides of the loop experience forces that form a couple giving net torque τ = I·A·B·sinθ = mB sinθ.

Potential energy of a magnetic dipole
Work must be done to rotate the dipole against the torque. The magnetic potential energy U(θ) of a dipole in field B is defined so that the torque equals the negative derivative of U with respect to angle:

τ = -dU/dθ

Integrating gives (choosing zero-point appropriately):

U = - m · B = - m B cosθ

This expression shows that U is minimum (U = -mB) when m is parallel to B (θ = 0, stable equilibrium) and maximum (U = +mB) when m is antiparallel (θ = π, unstable equilibrium). The energy difference to flip the dipole from parallel to antiparallel is ΔU = 2 m B.

Physical meaning and notes
In a uniform field the dipole experiences no net translational force (only torque); in a non‑uniform field it can experience a net force proportional to the gradient of the field, which is why bar magnets are pulled into regions of stronger field. The magnetic potential energy expression is analogous to the electric dipole in an external electric field.

Small‑angle motion
For small oscillations about θ = 0, τ ≈ -mB θ and the dipole (if free to oscillate) undergoes simple harmonic motion with effective restoring constant k = mB (if moment of inertia is Irot then angular frequency ω = sqrt(mB/Irot)).

Directions and sign conventions
The sign in U = -m·B means energy is lowest when dipole and field are aligned. When using the vector formula, ensure m points along the chosen direction of the loop's area vector.

Summary sentence
A magnetic dipole in a magnetic field experiences a torque τ = m × B that tends to align it with the field and has potential energy U = -m·B which is minimum when the dipole is parallel to the field.

Fig 5.3 — Educational Diagram: Magnetic field of a dipole and a bar magnet

Overview
A magnetic dipole is the simplest magnetic source: it has a magnetic dipole moment m and produces a magnetic field that falls off rapidly with distance. A small bar magnet can be modelled as a dipole at distances large compared with its size. The dipole model explains the shape of magnetic field lines, the torque on magnets and current loops, and why the Earth's field is approximately dipolar.

Magnetic dipole moment
Two common definitions:
- Current loop: m = I·A (vector perpendicular to the loop given by right-hand rule).
- Pole model (bar magnet treated as two poles ±q_m separated by distance 2l): m = q_m·(2l) directed from south to north pole.

Vector field of an ideal magnetic dipole (far-field)
For a dipole with moment m located at the origin, the magnetic induction at position vector r (|r| = r) in free space is
B(r) = (μ₀/4π r³) [3 (m·r̂) r̂ − m] , where r̂ = r/ r and μ₀ is the permeability of free space. This expression is valid when the observation distance r is much larger than the physical size of the source (the dipole approximation).

Components in spherical coordinates
If θ is the angle between m and r̂, the dipole field components are

• Radial: B_r = (μ₀/4π) · (2 m cos θ) / r³
• Polar (θ-direction): B_θ = (μ₀/4π) · (m sin θ) / r³ (points toward decreasing θ)

Special directions
- On the axis of the dipole (θ = 0): B = (μ₀/4π) · (2 m / r³) directed along +m.
- In the equatorial plane (θ = 90°): B = (μ₀/4π) · (m / r³) directed opposite to m (i.e. into the plane).

Bar magnet as two poles (pole model) and dipole approximation
A bar magnet can be approximated by two magnetic poles +q_m and −q_m separated by distance 2l. The exact field is the superposition of the fields of the two poles; at distances r >> 2l this reduces to the dipole expression with m = q_m·2l. Thus, far from a bar magnet the field falls as 1/r³ and has the angular form given above.

Torque and potential energy
A magnetic dipole m placed in an external magnetic field B feels a torque τ = m × B which tends to align m with B. The potential energy is U = −m·B (minimum when m is parallel to B).

Field lines and qualitative picture
Field lines emerge from the north pole and enter the south pole outside the magnet; inside the magnet they run from south to north, forming closed loops. Near-field (close to the magnet) the two-pole structure is visible; far-field the pattern is the familiar dipole shape (closed loops symmetric about the dipole axis).

Key ideas for Class 12
- Use the dipole formula for any small magnet or current loop when r is much larger than the magnet size.
- Remember the angular dependence and the 1/r³ distance dependence.
- Use torque and energy formulas to analyse compass behavior and current-loop dynamics.

Fig 5.4 — Educational Diagram: Magnetic field B and magnetic intensity H

Overview
In magnetism, two closely related vector fields are used to describe the magnetic effect: the magnetic flux density B (also called magnetic induction) and the magnetic field intensity H (also called magnetic field strength). B describes the total magnetic field in space (including the contribution of material), while H is used to describe the part of the magnetic field produced by 'free' currents and is convenient when materials (magnetized media) are present.

Definitions

• Magnetic flux density (B): B is the magnetic field measured in tesla (T). It appears in the Lorentz force law F = q(v × B) and determines the magnetic force on moving charges.
• Magnetic intensity (H): H is measured in ampere per metre (A/m). It is the field that appears naturally when applying Ampère's law in media and is related to free currents.

Relation between B, H and magnetization M
When a material is placed in a magnetic field it becomes magnetized; magnetization M (magnetic moment per unit volume, units A/m) represents the contribution of bound currents inside the material. The fields are related by:

B = µ0 (H + M)

For linear, isotropic materials M is proportional to H: M = χ_m H, where χ_m is the magnetic susceptibility (dimensionless). Combining gives:

B = µ0 (1 + χ_m) H = µ H

Here µ = µ0 µ_r, with µ_r = 1 + χ_m the relative permeability. µ0 is the permeability of free space (µ0 = 4π × 10^-7 H/m).

Ampère's law in terms of H
The integral form that is most useful in presence of matter is:

∯ H · dl = I_{free,enclosed}

This states that the circulation of H around a closed path equals the free current through the loop (bound currents from magnetization are accounted for via M in the B–H relation).

Energy density
For linear media the magnetic energy density (energy per unit volume) is:

u = 1/2 B · H

Boundary conditions (useful at interfaces)

• The normal component of B is continuous across a boundary (no magnetic monopoles): (B₂ − B₁) · n = 0.
• The tangential component of H is discontinuous where a surface free current K_free exists: n × (H₂ − H₁) = K_free.

Material types (qualitative)

• Diamagnetic: χ_m < 0 (very small magnitude) — slightly repelled by magnetic fields.
• Paramagnetic: χ_m > 0 (small) — weakly attracted, B increases a little more than in vacuum.
• Ferromagnetic: χ_m large and nonlinear — strong magnetization, exhibits hysteresis (memory of magnetization).

Practical use
H is convenient to use when designing electromagnets, coils, transformers and when accounting for currents you control (free currents). B is the field that determines forces on charges and the flux through coils (Faraday's law).

Fig 5.5 — Educational Diagram: Magnetization

Magnetization (M) is a vector quantity defined as the magnetic dipole moment per unit volume of a material. If a sample of volume V has net magnetic moment m, its magnetization is M = m/V. In SI units M is measured in ampere per metre (A/m).

Microscopic origin: Atoms and ions possess magnetic moments from electron orbital motion and spin. In absence of an external field these moments are randomly oriented (in most materials) so the net M ≈ 0. An applied magnetic field tends to align these moments, producing a net magnetization. Thermal agitation opposes alignment; the competition gives different magnetic behaviours.

Relation with B and H: In materials the magnetic induction B, magnetic field H and magnetization M are related by
B = μ₀(H + M), where μ₀ ≈ 4π × 10⁻⁷ N A⁻² is the permeability of free space. For linear, isotropic materials one often writes M = χH, where χ is the (volume) magnetic susceptibility, and then B = μ₀(1 + χ)H = μH with relative permeability μ_r = 1 + χ.

Types of magnetic response:

• Diamagnetism: Induced magnetic moments oppose the applied field (weak, temperature independent). χ is small and negative.
• Paramagnetism: Unpaired atomic moments tend to align with the field (weak, decreases with temperature). χ is small and positive. For many paramagnets, χ follows Curie's law: χ = C/T.
• Ferromagnetism: Strong alignment due to exchange interactions even without an external field. Materials show spontaneous magnetization, saturation (M_s), remanence (M_r), coercivity (H_c), and hysteresis. Ferromagnets lose spontaneous magnetization above the Curie temperature T_C.

Nonlinear & hysteretic behaviour: Ferromagnets are not linear: M rises rapidly with small H and then saturates (M_s) as most moments align. When H is cycled, M (and B) follow a hysteresis loop — an important feature for permanent magnets and magnetic memory.

Practical importance: Magnetization is central to design and function of transformers, inductors, electric motors, generators, magnetic storage, magnetic sensors and permanent magnets. Understanding M(H) and hysteresis determines core losses, remanence, and coercivity required in applications.

Fig 5.6 — Educational Diagram: Relation among B, H and M

Definitions

B (magnetic flux density or magnetic induction): total magnetic field inside a material, measured in tesla (T). It represents the force on moving charges and magnetic dipoles.

H (magnetic field intensity): the applied magnetic field produced by free currents (such as currents in coils), measured in amperes per metre (A/m).

M (magnetization): magnetic dipole moment per unit volume of the material, measured in A/m. It describes the magnetic response of the material (alignment of microscopic magnetic moments).

Fundamental relation (SI)

The macroscopic fields are related by

B = μ0 (H + M)

where μ0 is the permeability of free space (μ0 = 4π × 10⁻⁷ H/m).

Linear (magnetic) materials

For many materials under moderate fields, magnetization is proportional to H:

M = χm H

where χm is the magnetic susceptibility (dimensionless). Substitute into the fundamental relation to get

B = μ0 (1 + χm) H = μ0 μr H = μ H

where μr = 1 + χm is the relative permeability and μ = μ0 μr is the permeability of the material.

Nonlinear and ferromagnetic behaviour

Ferromagnetic materials (iron, cobalt, nickel) show strong, nonlinear dependence: M(H) grows rapidly and saturates (Ms, saturation magnetization). There is hysteresis: magnetization (and B) depends on history of applied H, producing a hysteresis loop with remanence and coercivity.

Microscopic view

M arises from alignment of atomic magnetic moments. If N atoms per unit volume each have average moment μatom, then M ≈ N μatom (up to saturation).

Units

• B: tesla (T)
• H: ampere per metre (A/m)
• M: ampere per metre (A/m)

Summary

B is the total magnetic effect inside material; H is the external cause (free currents); M is the material's response (bound currents). For linear media B is proportional to H through μ; for ferromagnets M and B are nonlinear and show hysteresis.

Fig 5.7 — Educational Diagram: Magnetic susceptibility and permeability

Overview
Magnetic susceptibility and permeability are two basic parameters that describe how a material responds to an applied magnetic field. They connect the applied magnetic field (H), the material's induced magnetization (M), and the resulting magnetic induction (B).

Definitions and fundamental relations

• Magnetization (M) — magnetic dipole moment per unit volume: M = (total magnetic moment)/V.
• Magnetic susceptibility (χ) — a measure of how much a material becomes magnetized in response to H: χ = M/H. (χ is dimensionless in SI.)
• Magnetic induction (B) and H are related by: B = μ₀(H + M).
• Permeability (μ) — how easily magnetic flux is established in the material: μ = B/H = μ₀(1 + χ). Often we use relative permeability μ_r = μ/μ₀ = 1 + χ.

Notes on signs and magnitudes
- Diamagnetic materials: χ < 0 (small negative), μ_r < 1. Examples: bismuth, graphite, water. - Paramagnetic materials: χ > 0 (small positive), μ_r slightly > 1. Examples: Al, O₂ (gas), Pt. - Ferromagnetic materials: χ large positive (can be very large), μ_r ≫ 1; show nonlinear and history-dependent behavior (hysteresis). Examples: Fe, Co, Ni and many alloys.

Linear vs nonlinear behavior
For many materials (diamagnetic and paramagnetic), M is proportional to H (M = χH) and χ is independent of H (linear response). For ferromagnets, M vs H is nonlinear: at low H M rises steeply, then approaches saturation (M_s), and shows hysteresis (remanence and coercivity).

Temperature dependence
Paramagnetic susceptibility follows Curie's law: χ = C/T (C = Curie constant). For ferromagnets above the Curie temperature T_C, susceptibility often follows Curie–Weiss law: χ = C/(T − θ) (where θ ≈ T_C).

Units and constants
In SI: B in tesla (T), H in ampere per meter (A/m), μ₀ = 4π × 10⁻⁷ H/m. χ is dimensionless; μ has units H/m (same as μ₀).

Practical significance
Permeability of core materials determines inductance and magnetic flux concentration in transformers, inductors and motors. Susceptibility indicates whether a material will be attracted to or repelled from a magnetic field and how strong that effect will be. Ferromagnetic materials are used for permanent magnets and soft magnetic cores (depending on hysteresis properties). Diamagnetic materials are used in magnetic levitation demonstrations and for tiny magnetic shielding effects, while high-μ alloys (mu-metal) are used for magnetic shielding.

Fig 5.8 — Educational Diagram: Classification of magnetic materials

Introduction: Magnetic behaviour of a material arises from atomic magnetic moments (orbital + spin). When placed in an external magnetic field H, materials develop magnetization M and show different responses. Classification is based on sign and magnitude of magnetic susceptibility χ (= M/H) and temperature dependence.

1. Diamagnetic materials
- Origin: Lenz's-law induced change in orbital motion produces a small magnetic moment opposite to applied field.
- Properties: χ < 0 (negative, very small magnitude ∼ −10⁻⁶ to −10⁻⁵). Response is weak and independent of temperature.
- Behaviour: Slightly repelled by magnetic fields. No permanent magnetic moments.
- Examples: Bismuth, copper, gold, silver, water, most organic compounds. Superconductors are perfect diamagnets (Meissner effect) with χ = −1.

2. Paramagnetic materials
- Origin: Atoms/ions have permanent magnetic moments but are randomly oriented without field; thermal agitation tends to randomize orientations.
- Properties: χ > 0 (small, ∼ 10⁻³ to 10⁻⁵), decreases with increasing temperature.
- Temperature dependence: Obeys Curie law: χ = C/T for ideal paramagnets (C = Curie constant). More generally Curie–Weiss law χ = C/(T − θ) with θ ≈ 0 for ideal paramagnets.
- Behaviour: Weakly attracted by magnetic field; no hysteresis; magnetization proportional to H at ordinary fields.
- Examples: Aluminium, platinum, sodium, oxygen (O₂).

3. Ferromagnetic materials
- Origin: Exchange interactions cause parallel alignment of neighbouring spins in regions called domains; spontaneous magnetization below the Curie temperature Tc.
- Properties: Very large positive susceptibility (nonlinear, can be huge). Show spontaneous magnetization, hysteresis, and saturation magnetization Ms.
- Temperature dependence: Below Tc there is spontaneous M; above Tc material becomes paramagnetic and susceptibility follows Curie–Weiss with θ ≈ Tc.
- Hysteresis: When H is cycled, B (or M) shows a loop characterized by saturation Ms, remanent magnetization Mr, and coercive field Hc.
- Applications: Permanent magnets, transformers, electric motors.
- Examples: Iron (Fe), cobalt (Co), nickel (Ni), steel, many ferrites.

4. Antiferromagnetic materials
- Origin: Exchange interactions favor antiparallel alignment of equal magnetic moments on neighbouring atoms, yielding zero net magnetization below the Néel temperature T_N.
- Properties: Net M ≈ 0 below T_N; above T_N they behave paramagnetically. Susceptibility shows a peak or cusp at T_N; Weiss constant θ is negative in Curie–Weiss fit.
- Examples: MnO, Cr, some oxides and alloys.

5. Ferrimagnetic materials
- Origin: Similar to antiferromagnetism, but the opposing sublattice moments are unequal, so a net spontaneous magnetization remains.
- Properties: Show spontaneous magnetization and hysteresis like ferromagnets; common in ceramic ferrites with high resistivity (useful at high frequencies).
- Examples: Magnetite (Fe₃O₄), ferrites used in transformers and inductors.

Useful relations and physical ideas (brief):
- Magnetization: M = χH.
- Magnetic induction in vacuum: B = μ₀(H + M) = μ₀μ_r H, where μ_r = 1 + χ.
- Curie law (paramagnets): χ = C/T and Curie–Weiss law: χ = C/(T − θ) (θ ≈ Tc for ferromagnets; θ < 0 for antiferromagnets).
- Hysteresis parameters: saturation (Ms), remanence (Mr), coercivity (Hc).

Microscopic picture: Diamagnetism is a universal weak effect. Paramagnetism arises from independent permanent moments randomized by thermal motion. Ferromagnetism/ferrimagnetism/antiferromagnetism result from exchange interactions that correlate spins (domain formation in ferromagnets explains hysteresis).

Summary table (conceptual): Diamagnetic (χ < 0, T independent, repelled); Paramagnetic (χ > 0 small, χ ∝ 1/T); Ferromagnetic (large χ, spontaneous M, hysteresis, Tc); Antiferromagnetic (opposite equal moments, T_N); Ferrimagnetic (opposite unequal moments, net M).

Fig 5.9 — Educational Diagram: Diamagnetism and paramagnetism (microscopic origin)

Overview
Diamagnetism and paramagnetism are two fundamental magnetic responses of materials to an applied magnetic field. Their microscopic origins lie in the behavior of electrons in atoms: diamagnetism arises from changes induced in the orbital motion of paired electrons (no permanent moment), while paramagnetism arises from permanent magnetic moments of atoms or ions with unpaired electrons that tend to align with an applied field.

Microscopic origin — Diamagnetism
In an atom with all electrons paired, there is no net permanent magnetic moment. When an external magnetic field B is applied, the orbital motion of electrons is altered (Lenz's law). Classically this is described by the Larmor precession: the orbital angular momentum acquires an extra contribution proportional to B, which produces an induced magnetic moment opposing the applied field. The induced moment per atom is linear in B and opposite in direction, so the material shows a weak negative magnetization (M opposite to H).

Key physical points:

• All materials have some diamagnetic response because closed electron shells oppose changes in flux.
• Diamagnetism is temperature independent (thermal agitation does not affect induced orbital currents significantly).
• The effect is generally very weak, but can be strong in materials like pyrolytic graphite or bismuth.

Microscopic origin — Paramagnetism
Paramagnetic atoms or ions have one or more unpaired electrons giving a permanent magnetic moment μ (from spin and/or orbital angular momentum). In absence of a field these moments are randomly oriented by thermal motion, so net M = 0. An applied magnetic field produces a partial alignment of these moments, producing a net magnetization in the direction of the field. Thermal agitation competes with alignment, so the magnetization depends strongly on temperature.

Key physical points:

• Paramagnetism requires permanent moments (unpaired electrons).
• Magnetization increases with field and decreases with temperature.
• At low fields (or high T) the magnetization is linear in field; at high fields (or low T) alignment saturates when most moments point along the field.

Relation between M, B and H
In SI units B = μ0(H + M). For weak responses one often approximates B ≈ μ0 H to relate M and H and define the susceptibility χ = M/H. Diamagnetic materials have χ < 0; paramagnetic materials have χ > 0.

Quantum remarks
A correct microscopic treatment of paramagnetism uses quantized magnetic moments (spin and orbital angular momentum) and statistical mechanics. This leads to Curie's law (χ ∝ 1/T) for localized moments. Metals with conduction electrons show Pauli paramagnetism (temperature independent, from Fermi-level density of states) in addition to any localized contributions.

Practical summary
Diamagnetism: induced, negative susceptibility, temperature independent, very weak. Paramagnetism: due to permanent moments, positive susceptibility, follows Curie law (χ ∝ 1/T) for localized moments, stronger than diamagnetism but still usually small compared with ferromagnetism.

Fig 5.10 — Educational Diagram: Ferromagnetism and Weiss molecular field

Ferromagnetism — basic idea
Ferromagnetism is a form of magnetic ordering in which atomic magnetic moments (spins) in a material tend to align parallel to each other even in the absence of an external magnetic field, producing a large spontaneous magnetization. Common ferromagnetic materials include iron, nickel and cobalt.

Domains
To minimize magnetostatic energy, a ferromagnet breaks into regions called magnetic domains. Inside each domain spins are aligned, but neighbouring domains may be oriented differently. Application of an external field causes domain walls to move and domains aligned with the field to grow, producing macroscopic magnetization.

Origin of alignment — exchange interaction
The fundamental cause of parallel alignment is a quantum-mechanical exchange interaction between neighbouring electron spins (not classical dipole–dipole). Exchange energy favors parallel alignment for certain materials, overcoming thermal agitation at low temperature.

Weiss molecular field (phenomenological)
Pierre Weiss proposed that each magnetic moment feels, in addition to any externally applied field H, an internal 'molecular' field H_m proportional to the magnetization M of the material: H_m = λ M, where λ (Weiss constant) is a positive material-dependent parameter. The effective field acting on a moment is therefore H_eff = H + λ M. This mean-field idea explains spontaneous magnetization and the existence of a Curie temperature below which ferromagnetic order appears.

Mean-field/Curie–Weiss derivation (outline)
For paramagnetic behaviour in a small effective field, M ≈ (C/T) H_eff (Curie law form). Using H_eff = H + λ M gives
M = (C/T) (H + λ M) ⇒ M (1 - C λ / T) = (C/T) H. Hence the susceptibility χ = M/H is

χ = C / (T - T_c) where T_c = C λ is the Curie temperature.
Below T_c (T < T_c) the denominator changes sign and the system can have a nonzero spontaneous M even when H = 0. Thus Weiss molecular field predicts spontaneous magnetization below T_c.

Spontaneous magnetization near T_c
In mean-field theory, the spontaneous magnetization M_s decreases continuously to zero as T approaches T_c from below, with approximate behaviour M_s(T) ≈ M_0 (1 - T/T_c)^{1/2} (mean-field exponent 1/2).

Hysteresis and practical properties
Ferromagnets show hysteresis: when H is cycled, M does not retrace the same path. Key parameters from the hysteresis loop are saturation magnetization M_s, remanent magnetization M_r (remaining M when H → 0 after saturation), and coercive field H_c (reverse field needed to reduce M to zero). Materials with small H_c are soft magnetic (transformer cores), large H_c are hard magnets (permanent magnets).

Limitations and physical meaning
Weiss molecular field is a phenomenological, mean-field description. The constant λ effectively summarizes exchange interactions and the local environment. A full microscopic explanation comes from quantum exchange (Heisenberg model), but Weiss theory gives correct qualitative predictions (Curie–Weiss law, Tc, spontaneous M).

Important practical consequences
Ferromagnetic materials are used for permanent magnets, transformer cores, read/write heads, electric motors and generators. Their temperature dependence (Curie temperature) determines operating limits: above T_c the material becomes paramagnetic and loses magnetization.

Fig 5.11 — Educational Diagram: Domain theory and hysteresis

Domain theory (Weiss domains)
Ferromagnetic materials consist of many small regions called magnetic domains. Inside each domain atomic magnetic moments (spins) are aligned parallel due to strong exchange interaction, giving a spontaneous magnetisation. Different domains are oriented in different directions so that the net magnetisation of an unmagnetised piece is nearly zero. Domain formation reduces magnetostatic (external) energy though it costs some domain-wall energy.

Key features of domains and walls:

• Domain: region of uniform magnetisation.
• Domain wall: thin transition region between domains where magnetisation rotates (Bloch wall in bulk materials).
• Domain size and patterns result from competition between exchange energy, magnetocrystalline anisotropy energy and magnetostatic energy.
• Applying an external magnetic field moves domain walls and grows domains favourably aligned with the field; rotation of magnetisation inside domains also contributes.
• Barkhausen effect: discrete jumps in magnetisation when domain walls jump past defects — observed as noise.

Hysteresis
When a ferromagnetic specimen is subjected to a cyclic magnetic field H, the flux density B (or magnetisation M) does not retrace the same path on reversal of H. The resultant closed curve B vs H (or M vs H) is the hysteresis loop. Hysteresis is due to irreversible processes such as domain wall pinning and unpinning, and rotation of domain magnetisation.

Important points on the hysteresis loop:

• Saturation (Bs or Ms): maximum B (or M) when practically all domains align with H.
• Retentivity or remanence (Br): residual B when H is reduced to zero after saturation.
• Coercivity (Hc): negative H required to reduce B (or M) to zero after saturation.
• Loop area: proportional to energy loss per cycle (hysteresis loss) per unit volume; dissipated as heat.
• Initial magnetisation curve: path from demagnetised state to saturation for the first increase of H.
• Minor loops: loops obtained when H is cycled between intermediate values — useful for describing behaviour under small-signal alternating fields.

Why hysteresis matters
Hysteresis causes energy loss in AC magnetic devices (transformers, inductors). Materials are therefore chosen as soft magnetic (narrow loop, low Hc) for cores where low loss is needed, and hard magnetic (wide loop, high Hc and high Br) for permanent magnets where high remanence and coercivity are desired.

Thermal effects: Above the Curie temperature (Tc) a ferromagnet becomes paramagnetic; domains disappear and hysteresis vanishes.

Fig 5.12 — Educational Diagram: Magnetic field of common current configurations

This topic treats how steady electric currents produce magnetic fields for several common geometries using the Biot–Savart law and Ampère’s circuital law. Key configurations are: long straight wire, finite/ semi‑infinite wire, circular loop (single and multi‑turn), solenoid, and toroid. Direction of the field is given by the right‑hand rule (curl fingers in current direction; thumb gives field direction along axis).

• Biot–Savart law (fundamental): For a small current element I dℓ, the contribution to magnetic field at point P is
  dB = (μ0 / 4π) (I dℓ × r̂) / r^2
  (vector form). Integrate over the current path to get total B.

• Infinite straight wire: Magnetic field at perpendicular distance r from an infinitely long straight wire carrying current I is
  B = μ0 I / (2π r)
  Field lines are concentric circles around the wire. Direction by right‑hand rule.

• Finite straight or semi‑infinite wire: For a straight segment, if the two end points subtend angles θ1 and θ2 at the perpendicular from the point (angles measured between the line to each end and the perpendicular), then
  B = (μ0 I / 4π r) (sin θ1 + sin θ2)
  Special cases: semi‑infinite wire ⇒ B = μ0 I / (4π r); finite symmetric segment uses θ1 = θ2.

• Circular current loop (single turn): Magnetic field at the center of a loop of radius R carrying current I is
  B_center = μ0 I / (2 R)
  For N tightly wound turns: B_center = μ0 N I / (2 R). On the axis at distance x from center:
  B_axis(x) = (μ0 I R^2) / (2 (R^2 + x^2)^(3/2))
  (multiply by N for N turns). The field is maximum at x = 0 and falls off ≈ 1/x^3 at large x.

• Long solenoid (ideal): For a long solenoid of N turns and length l carrying current I, with turn density n = N/l, the field inside (near center) is nearly uniform:
  B_inside ≈ μ0 n I = μ0 (N/l) I
  Outside the ideal long solenoid B ≈ 0. Real solenoids have fringing fields near ends.

• Toroid: A solenoid bent into a closed ring (toroid) of mean radius r with N turns carrying current I produces inside the core (for r between inner and outer radii):
  B(r) = μ0 N I / (2π r)
  Outside an ideal toroid B ≈ 0. Field varies approximately as 1/r across the cross‑section if thickness is not negligible.

• Ampère’s circuital law (integral form): ∮ B · dl = μ0 I_enc (in vacuum). Useful for symmetrical configurations (infinite wire, ideal solenoid, toroid) to obtain B quickly without integration.

Practical points: always identify symmetry, choose an appropriate Amperian loop when possible, or apply Biot–Savart for less symmetric cases. Use right‑hand rule for direction and superposition for multiple currents.

Fig 5.13 — Educational Diagram: Measurement techniques and magnetometers

Overview
Measurement techniques for magnetism determine the magnetic moment (M) of magnets and the magnetic field (B) of sources (including Earth's field). Two standard laboratory magnetometers used in Class 12 are the tangent (deflection) magnetometer and the vibration (oscillation) magnetometer. Both exploit the torque on a magnetic dipole in a magnetic field and the superposition of fields.

Tangent (deflection) magnetometer — principle and method
A magnetic needle (dipole moment M_n) placed in the horizontal component B_h of Earth's magnetic field aligns with it. If an additional horizontal field B_x (from a bar magnet or a current loop) is applied perpendicular to B_h, the needle deflects by an angle θ until the torque due to B_h (M_n B_h sinθ) balances the torque due to B_x (M_n B_x cosθ). For equilibrium, tanθ = B_x / B_h (tangent law). If the applied field is produced by a bar magnet (modelled as a dipole) at distance x on its axis, B_x = (μ0/4π)·(2M)/x^3, so the bar magnet’s magnetic moment is M = (2π x^3 B_h / μ0) · tanθ. This method requires measurement of θ and distance x (and knowledge of B_h). If B_x is from a circular coil carrying current I, use B_x = μ0 N I /(2R) (field at centre) and tanθ = B_x/B_h to find I or B_h.

Vibration (oscillation) magnetometer — principle and method
A small magnet of magnetic moment M placed in a magnetic field B_h undergoes small-angle torsional oscillations. For small θ, restoring torque ≈ −M B_h θ, so the equation of motion gives angular frequency ω^2 = M B_h / I (I = moment of inertia of the magnet about the oscillation axis). The time period is T = 2π √(I / (M B_h)). Hence the magnetic moment is M = 4π^2 I / (T^2 B_h). If the magnet is approximated by a uniform rod of length L and mass m (axis through the centre, perpendicular to length), use I = (1/12) m L^2. This method only needs measurement of T and knowledge of B_h and I.

Other practical magnetometers
Modern instruments include fluxgate magnetometers (sensitive vector measurement), Hall-effect sensors (local field measurement), proton precession and optically pumped magnetometers (absolute scalar B measurement), and SQUIDs (extremely sensitive superconducting devices). Laboratory techniques discussed above are the classical mechanical methods used to teach dipole behavior and measure magnetic moments.

Key experimental considerations
Careful alignment to isolate the horizontal component B_h, minimizing friction and eddy-current damping, accurate distance or coil-geometry measurements, and small-angle conditions for oscillation method improve accuracy. For field vs distance studies, ensure distance is much larger than magnet size so dipole approximation (B ∝ 1/r^3) is valid.

Fig 5.14 — Educational Diagram: Applications and technological aspects

Magnetism and magnetic materials are central to many modern devices. The chapter’s applications and technological aspects explain how the magnetic properties of materials (diamagnetic, paramagnetic, ferromagnetic) and macroscopic quantities (magnetisation M, susceptibility χ, permeability μ, and the B–H relation) are used in real devices and how materials and design choices reduce losses and improve performance.

Key ideas:

• Material types and choice: Soft magnetic materials (high μr, low coercivity Hc, narrow hysteresis) such as silicon steel and permalloy are used where magnetisation must change easily (transformer cores, electromagnets). Hard magnetic materials (high coercivity and remanence) such as AlNiCo, ferrites and rare-earth alloys (NdFeB) are used for permanent magnets (speakers, motors).
• Hysteresis and core loss: Ferromagnets show hysteresis. The area of the B–H loop equals magnetic energy lost per cycle (hysteresis loss). In AC applications this loss + eddy current loss determine core heating and efficiency.
• Eddy currents and lamination: Changing magnetic flux induces circulating currents (eddy currents) in conducting cores, causing I^2R heating. Cores are laminated or made of low-conductivity magnetic materials (ferrites) to reduce eddy losses. At high frequencies ferrites are preferred.
• Curie temperature: Above the Curie temperature (Tc) ferromagnetic materials lose spontaneous magnetisation and become paramagnetic — an important design constraint for permanent magnets and magnetic devices operating at elevated temperatures.
• Magnetic circuits and flux control: Magnetic circuits (cores, air gaps) concentrate flux; air gaps are used in electromagnets, inductors, and motors to control reluctance and store magnetic energy.
• Advanced technologies: Superconducting magnets (Meissner effect) produce extremely high fields with negligible resistive loss (MRI, particle accelerators). Rare-earth permanent magnets (NdFeB, SmCo) enable compact high-torque motors for electric vehicles and appliances.

Practical design measures and consequences:

• Use soft magnetic core materials with high permeability and low hysteresis for transformers and AC inductors to minimize hysteresis losses.
• Laminated cores (thin insulated sheets) or powdered-core/ferrite cores reduce eddy currents in AC and high-frequency devices.
• Choose permanent magnet materials with appropriate coercivity and temperature stability for motors, loudspeakers, sensors, and data-storage heads.
• Account for saturation (Bs): when B approaches saturation, permeability falls and devices become non‑linear; design to operate below saturation in transformer cores and inductors.

Techniques & devices that use magnetic phenomena:

• Transformers, power inductors, and chokes (magnetic cores to concentrate flux and transfer energy).
• Electric motors and generators (torque from current-carrying coils in magnetic fields; permanent magnets or field windings provide magnetic fields).
• Electromagnets, relays, and solenoids (controlled magnetic force for switching and actuation).
• Magnetic recording and sensors: magnetic hard disks, MR/ GMR/TMR read heads, Hall-effect sensors.
• Speakers and microphones (coil + magnet convert electrical signals ↔ mechanical motion).
• MRI scanners and particle-accelerator magnets (large, stable magnetic fields from superconducting coils).
• Magnetic levitation and magnetic separation (industrial sorting, maglev transport).

This overview connects the physical laws (M = χH, B = μ0(H+M), hysteresis) to engineering choices (material, core shape, laminations, cooling) that make magnetic devices efficient, compact, and reliable.

Fig 5.15 — Educational Diagram: Units, dimensions and key formulae

Overview
This topic collects the SI units, physical dimensions and the main formulae used in magnetism of matter (Class 12). It covers magnetic quantities such as magnetic moment, magnetization, magnetic field (B), magnetic field intensity (H), susceptibility (χ) and permeability (μ) and explains how they relate inside materials.

Definitions & relations (concise)

• Magnetic moment (m or μ): For a planar current loop m = I·A (vector normal to loop by right-hand rule). Unit: ampere·metre^2 (A·m^2). Dimension: [I][L]^2.
• Magnetization (M): Magnetic moment per unit volume, M = (total magnetic moment)/V. Unit: A·m^−1. Dimension: [I][L]^−1.
• Magnetic field intensity (H): Source-related field produced by currents and poles. Unit: A·m^−1. Dimension: [I][L]^−1.
• Magnetic flux density (B): Also called magnetic induction; B measures force on moving charges. Relation in SI: B = μ0(H + M) = μH. Unit: tesla (T) = kg·s^−2·A^−1. Dimension: [M][T]^−2[I]^−1.
• Magnetic susceptibility (χ): M = χH. Dimensionless (pure number). χ > 0 for paramagnets, χ < 0 for diamagnets, large positive for ferromagnets (nonlinear).
• Permeability (μ): μ = μ0 μr = μ0(1 + χ). Unit: henry per metre (H·m^−1) or N·A^−2. Dimension: [M][L][T]^−2[I]^−2. μ0 (vacuum permeability) = 4π × 10^−7 H·m^−1.

Important physical relations

• Inside linear, isotropic materials: B = μH, where μ = μ0(1 + χ) and μr = 1 + χ.
• Magnetization: M = χH (volume susceptibility).
• Magnetic dipole moment of loop: m = I·A (A = loop area).
• Torque on a magnetic dipole in uniform B: τ = m × B. Magnitude τ = mB sinθ (θ between m and B).
• Potential energy of dipole: U = −m·B (lowest when m aligned with B).
• Force on a dipole in non-uniform field: F = ∇(m·B) (for a small dipole often written as F = (m·∇)B).
• Magnetic field on axis of a dipole (at distance r » loop size): B_on-axis ≈ (μ0/4π)·(2m/r^3) (shows 1/r^3 dependence).
• Magnetic field at centre of circular loop (radius R): B = (μ0 I R^2)/(2(R^2 + x^2)^{3/2}) on axis at distance x; for x = 0, B_center = μ0 I/(2R) .
• Field inside long solenoid (ideal): B = μ0 n I (in vacuum) where n = turns per unit length. With material B = μ n I.
• Curie law (paramagnets): χ = C/T (C = Curie constant), so χ ∝ 1/T.

Units & dimensions summary (SI)

• B (magnetic flux density): unit = tesla (T) = N·A^−1·m^−1 = kg·s^−2·A^−1; dimension = [M][T]^−2[I]^−1.
• H, M: unit = A·m^−1; dimension = [I][L]^−1.
• Magnetic moment m: unit = A·m^2; dimension = [I][L]^2.
• Permeability μ: unit = H·m^−1 or N·A^−2; μ0 = 4π×10^−7 H·m^−1.
• Susceptibility χ, relative permeability μr: dimensionless.

Notes on behaviour of materials
Diamagnetic: small negative χ (weakly repelled by B). Paramagnetic: small positive χ (weak attraction, χ ~ 1/T). Ferromagnetic: large positive χ, nonlinear and history-dependent (hysteresis). These differences determine how B and M respond to H.