Magnetism and Matter

MAGNETISM AND MATTER – COMPREHENSIVE CBSE CLASS 12 NOTES

5.1 INTRODUCTION AND BASIC CONCEPTS

**Magnetism** is a fundamental phenomenon present throughout nature, from distant galaxies to atomic scales. The word "magnet" derives from Magnesia, an ancient Greek island where magnetic ore deposits were discovered around 600 BC.

Key Historical and Conceptual Points

**Oersted's discovery (early 19th century)**: Moving electric charges and currents produce magnetic fields, linking electricity and magnetism.

**Universal properties of magnets**:

Earth behaves as a magnet with field pointing approximately south to north geographically

A freely suspended bar magnet aligns north-south

**North pole**: End pointing toward geographic north

**South pole**: End pointing toward geographic south

Unlike-poles attract; like-poles repel

**Magnetic monopoles do NOT exist**: Unlike electric charges, isolated N or S poles cannot be separated. Breaking a bar magnet produces two complete magnets, each with N and S poles

Magnetic materials can be synthesized from iron and alloys

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5.2 THE BAR MAGNET

5.2.1 Magnetic Field Lines

When iron filings are sprinkled around a bar magnet on glass, they arrange in a distinctive pattern revealing the **magnetic field structure**.

**Properties of Magnetic Field Lines**:

1. **Form continuous closed loops** — Unlike electric field lines (which begin on positive charges and end on negative charges), magnetic field lines form complete closed paths through space and within the magnet

2. **Tangent direction represents field direction** — At any point, the tangent to a field line shows the direction of magnetic field **B** at that point

3. **Density indicates field strength** — More field lines crossing per unit area = stronger magnetic field. In Fig. 5.2(a), region (ii) has stronger B than region (i)

4. **Field lines never intersect** — If they did, the field direction would be ambiguous at the crossing point

**Visual method for field mapping**: Place a small magnetic compass needle at various positions and observe its orientation. The needle aligns with the local magnetic field direction.

5.2.2 Bar Magnet as an Equivalent Solenoid

**Ampere's hypothesis**: All magnetic phenomena arise from circulating microscopic currents within atoms and materials.

**Key analogy**: A bar magnet can be modeled as a large collection of current loops (like a solenoid) because:

The magnetic field patterns of a bar magnet and finite solenoid are identical

Cutting a bar magnet creates two smaller magnets (like cutting a solenoid creates two shorter solenoids)

Field lines remain continuous, emerging from one pole and re-entering at the other

**Axial magnetic field of a finite solenoid** (derived from Biot-Savart law):

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

where:

**μ₀** = 4π × 10⁻⁷ T·m/A (permeability of free space)

**m** = magnetic moment of the solenoid

**r** = distance along axis (for r >> length)

This equation exactly matches the far-field axial magnetic field of a bar magnet **experimentally**, confirming the solenoid equivalence.

5.2.3 Dipole in a Uniform Magnetic Field

When a magnetized needle (magnetic dipole) is placed in an external uniform magnetic field **B**, it experiences:

**Torque on the needle**:

$$\vec{\tau} = \vec{m} \times \vec{B}$$

Magnitude: $$\tau = mB\sin\theta$$

where:

**m** = magnetic moment of needle

**θ** = angle between **m** and **B**

Torque is **restoring** (tends to align **m** with **B**)

**Magnetic Potential Energy**:

$$U_m = -\vec{m} \cdot \vec{B} = -mB\cos\theta$$

Derivation (by integration):

$$U_m = -\int \tau \, d\theta = -\int mB\sin\theta \, d\theta = -mB\cos\theta + C$$

Taking U = 0 at θ = 90° (perpendicular position): **U_m = –mB cos θ**

**Energy analysis**:

**Minimum potential energy** (most stable): θ = 0° (dipole parallel to field), U = –mB

**Maximum potential energy** (unstable): θ = 180° (antiparallel), U = +mB

**Important note**: No net force acts on the dipole in a *uniform* field (only torque). The net force is zero because forces on the two poles are equal and opposite.

5.2.4 Electrostatic-Magnetic Analogy

By comparing dipole equations in electrostatics and magnetism, we can map one to the other using substitutions:

| Substitution | Symbol |

|---|---|

| E → B | Electric field to magnetic field |

| p → m | Electric dipole moment to magnetic moment |

| 1/(4πε₀) → μ₀/(4π) | Constant replacement |

**Equatorial field of bar magnet** (perpendicular bisector, r >> magnet length):

$$B_E = -\frac{\mu_0 m}{4\pi r^3}$$

(Negative sign indicates field points opposite to moment direction)

**Axial field of bar magnet** (along axis, r >> magnet length):

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

(Field points in same direction as moment)

**Comparison Table**:

| Property | Electrostatics | Magnetism |

|---|---|---|

| Dipole moment | p | m |

| Constant | 1/(4πε₀) | μ₀/(4π) |

| Equatorial field | –p/(4πε₀r³) | –μ₀m/(4πr³) |

| Axial field | 2p/(4πε₀r³) | 2μ₀m/(4πr³) |

| Torque in field | **p** × **E** | **m** × **B** |

| Potential energy | –**p**·**E** | –**m**·**B** |

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WORKED EXAMPLE 5.1: BAR MAGNET PROPERTIES

**Problem**: (a) What happens when a bar magnet is cut transversely or lengthwise? (b) Why does a magnetized needle in uniform field experience torque but no net force, while an iron nail experiences both? (c) Must every magnetic field configuration have north and south poles? (d) How can you identify which of two identical-looking iron bars is magnetized?

**Solution**:

(a) **Both cases produce two complete magnets**, each with N and S poles (unlike breaking an electric dipole). Cutting transversely or lengthwise always results in poles at the broken surfaces.

(b) **Uniform field**: On a dipole in uniform field, forces on N and S poles cancel (F_net = 0), but torque remains nonzero. **Non-uniform field**: Iron nail develops *induced* magnetic moment aligned with external field. The induced S-pole (say) is closer to magnet's N-pole than the induced N-pole, so net attractive force acts on nail along with torque.

(c) **Not necessarily**. A toroid produces magnetic field but has no net magnetic moment at large distances — no overall N or S pole detectable. Similarly, an infinite straight current-carrying wire has field but no net dipole character.

(d) **Method 1 (Repulsion test)**: Bring different ends of bars near each other. If repulsion occurs, both are magnetized. If always attraction, one is unmagnetized.

**Method 2 (Field strength test)**: Magnetic field is strongest at poles and weakest at center. Take one bar (say A), lower its end first on the end of bar B, then on the center of B. If A experiences no force at B's center but experiences force at B's ends, then B is magnetized (field concentrated at poles). If force is uniform, A is magnetized.

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WORKED EXAMPLE 5.2: STABLE AND UNSTABLE EQUILIBRIUM

**Problem**: Figure 5.4 shows needle P at origin and needle Q in various configurations. Which configurations are (a) not in equilibrium, (b) stable, (c) unstable, and (d) lowest energy?

**Given**: Magnetic field due to dipole P:

On perpendicular bisector: $B_P = -\frac{\mu_0 m_P}{4\pi r^3}$ (perpendicular to axis)

On axis: $B_P = \frac{\mu_0 \cdot 2m_P}{4\pi r^3}$ (along axis)

**Solution**:

Equilibrium occurs when **mQ aligns with the local field B_P**. Potential energy: $U = -\vec{m_Q} \cdot \vec{B_P}$

**Stable equilibrium**: m_Q parallel to B_P (lowest energy) — Q_3, Q_6

**Unstable equilibrium**: m_Q antiparallel to B_P (highest energy) — Q_4, Q_5

**Not in equilibrium**: Q_1, Q_2 (m_Q neither parallel nor antiparallel to B_P)

**Lowest potential energy**: Q_6 (fully aligned with field)

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5.3 MAGNETISM AND GAUSS'S LAW

Statement of Gauss's Law for Magnetism

**"The net magnetic flux through any closed surface is zero."**

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

or in summation form:

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

Why is Magnetic Gauss's Law Different from Electric Gauss's Law?

**In electrostatics** (Gauss's law):

$$\oint \vec{E} \cdot d\vec{S} = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$

Non-zero flux emerges from enclosed charges.

**In magnetism** (Gauss's law):

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

Always zero because **magnetic monopoles do not exist**.

Physical Interpretation

Magnetic field lines form **closed loops** (unlike electric field lines which emanate from positive charges and terminate on negative charges)

For any closed surface, as many field lines enter as exit — net flux = 0

The simplest magnetic element is a **dipole** (or current loop), not a monopole

All magnetic phenomena arise from arrangements of dipoles and current loops

**Visual proof**: Consider closed surfaces (i) and (ii) around a bar magnet:

Surface (i): Encloses both N and S poles equally → equal lines enter and exit → Φ_B = 0

Surface (ii): Encloses portion with more N-pole character → still equal entry and exit because magnetic field lines recirculate inside magnet → Φ_B = 0

Example: Field Line Diagrams

**Wrong diagrams** (violate Gauss's law):

Field lines emanating from a point (net outward flux nonzero)

Field lines not forming closed loops around current-carrying region

Straight parallel lines between pole pieces without fringing at ends

**Right diagrams**:

Closed loops through space and inside magnet (like bar magnet or solenoid)

Closed loops within toroid (each loop encloses current)

Field lines inside magnet run from S to N (completing external N→S loops)

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5.4 MAGNETISATION AND MAGNETIC INTENSITY

Definition of Magnetization

**Magnetization (M)** is the **net magnetic moment per unit volume** of a material:

$$M = \frac{m_{\text{net}}}{V}$$

**SI Units**: A/m (amperes per meter)

**Physical meaning**: Measure of the degree to which a material becomes magnetized (aligned magnetic moments per unit volume).

How Magnetization Arises

**Orbital motion of electrons**: Electrons circulating in atoms produce magnetic dipole moments

**Electron spin**: Intrinsic spin angular momentum contributes magnetic moment

**Vector addition**: In bulk material, these microscopic moments vectorially add

$$\vec{M} = \sum_i \frac{\vec{m}_i}{V}$$

where **m_i** = magnetic moment of i-th atomic dipole, V = sample volume

Magnetic Field Inside and Outside a Material

In presence of magnetizable material, magnetic field **B** has two sources:

$$\vec{B} = \vec{B}_{\text{ext}} + \vec{B}_{\text{induced}}$$

where:

**B_ext** = field due to external sources (currents, magnets)

**B_induced** = field due to magnetization of material itself

**Magnetic field intensity (H)**: Defined to separate external and internal effects:

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

or equivalently:

$$\vec{H} = \frac{\vec{B}}{\mu_0} - \vec{M}$$

**SI Units of H**: A/m (amperes per meter)

**Key distinction**:

**H**: Depends only on free currents (sources we control)

**B**: Total field including effects of material magnetization

**M**: Property of the material itself

Linear Magnetic Materials

For many materials (paramagnetic, diamagnetic, ferromagnetic at unsaturated fields):

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

where **χ** (chi) = **magnetic susceptibility** (dimensionless)

**Combining equations**:

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

where **μ_r = 1 + χ** = **relative permeability** (dimensionless)

**Alternative form**:

$$\vec{B} = \mu \vec{H}$$

where **μ = μ₀μ_r** = permeability of material (SI units: T·m/A or H/m)

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5.5 CLASSIFICATION OF MAGNETIC MATERIALS

Materials are classified into three categories based on susceptibility and response to external fields.

5.5.1 DIAMAGNETIC MATERIALS

**Definition**: Materials with **χ < 0** (negative susceptibility), weakly repelled by magnetic field

**Atomic mechanism**:

Electrons orbit nuclei with zero net angular momentum in ground state

External magnetic field induces orbital motion (Lenz's law)

Induced magnetic moment **opposes** external field direction

No permanent atomic dipole moments

**Susceptibility**: χ ≈ –10⁻⁵ to –10⁻⁶ (very small, negative)

**Relative permeability**: μ_r < 1 (slightly less than 1)

**Behavior in non-uniform field**: **Weak repulsion** — diamagnetic materials are pushed out of regions of strong field

**Examples**:

Bismuth, copper, water, organic compounds

**Clinical example**: Diamagnetic properties of water allow levitation of frogs and humans in strong magnetic fields (dramatic but temporary effect)

**Temperature dependence**: χ is **independent of temperature** (no paramagnetic contribution)

**Magnetization**: M is very weak even in strong fields, always antiparallel to H

5.5.2 PARAMAGNETIC MATERIALS

**Definition**: Materials with **χ > 0** (positive susceptibility), weakly attracted to magnetic field

**Atomic mechanism**:

Atoms have **permanent magnetic dipole moments** from unpaired electrons

In absence of external field, dipoles point randomly → M_net = 0

External field **partially aligns** dipoles (thermal motion opposes alignment)

Net magnetization parallel to applied field

**Susceptibility**: χ ≈ 10⁻⁵ to 10⁻³ (positive, small)

**Relative permeability**: μ_r > 1 (slightly greater than 1)

**Behavior in non-uniform field**: **Weak attraction** — paramagnetic materials are pulled into regions of strong field

**Examples**:

Aluminum, platinum, tungsten, oxygen (O₂), iron salts like FeCl₃

**Temperature dependence (Curie's Law)**:

$$\chi = \frac{C}{T}$$

where **C** = Curie constant, **T** = absolute temperature (Kelvin)

**Physical interpretation**: Higher temperature increases thermal motion, reduces alignment → χ decreases

At **T → 0**: χ → ∞ (perfect alignment, limited by quantum mechanics)

At **high T**: χ → 0 (randomization dominates)

**Magnetization**: M is proportional to H and increases with field strength, but remains small

5.5.3 FERROMAGNETIC MATERIALS

**Definition**: Materials with **χ >> 1** (very large positive susceptibility), strongly attracted to magnetic field

**Atomic mechanism**:

Atoms have permanent magnetic dipoles

**Exchange interaction** (quantum mechanical effect) causes **spontaneous alignment** of adjacent atomic dipoles

Large regions of aligned moments called **magnetic domains** (size ~10⁻⁶ m)

Each domain acts as a microscopic magnet with moment pointing in one direction

In unmagnetized state, domains point randomly → overall M ≈ 0

External field **aligns domains** → enormous net magnetization

**Susceptibility**: χ ≈ 10³ to 10⁶ (very large, positive)

**Relative permeability**: μ_r >> 1 (very large)

**Behavior in non-uniform field**: **Strong attraction** — ferromagnetic materials are powerfully pulled into strong field regions

**Examples**:

Iron (Fe), cobalt (Co), nickel (Ni), their alloys

**Permanent magnets**: Can retain magnetization after external field removed due to domain alignment

**Electromagnets**: Ferromagnetic cores in coils amplify field by factor of hundreds

**Hysteresis Effect**: The relationship between B and H is **non-linear and path-dependent**

**Hysteresis loop description**:

1. Start unmagnetized (origin)

2. Apply increasing H → B increases rapidly (domains align)

3. At saturation → B plateaus (all domains aligned)

4. Decrease H → B does NOT retrace original curve (domain alignment partially retained)

5. At H = 0 → **Remanence (B_r)**: Material retains magnetic field

6. Reverse H → B decreases further

7. At **H = H_c (coercivity)**: B becomes zero (domains randomized but field remains)

8. Continue reversing → complete loop forms

**Key features**:

**Remanence (B_r)**: Magnetic field remaining when H = 0 (practical permanent magnet property)

**Coercivity (H_c)**: Reverse field needed to reduce B to zero (measure of magnetization "stickiness")

**Area of loop**: Represents energy dissipated as heat per unit volume per cycle

**Temperature dependence (Curie temperature T_C)**:

$$\text{For } T > T_C: \text{ Ferromagnetic → Paramagnetic}$$

At Curie temperature, spontaneous domain alignment breaks down

Exchange interaction insufficient to overcome thermal randomization

Material converts to paramagnetic behavior

**Curie temperatures**:

Fe: T_C = 770°C

Co: T_C = 1115°C

Ni: T_C = 358°C

**Practical implications**:

A steel magnet heated above Curie temperature loses magnetization permanently

Cooling below T_C re-establishes ferromagnetism

Summary Table: Magnetic Material Properties

| Property | Diamagnetic | Paramagnetic | Ferromagnetic |

|---|---|---|---|

| **χ** | Negative (~10⁻⁶) | Positive (~10⁻⁴) | Very positive (~10³) |

| **μ_r** | < 1 | > 1 | >> 1 |

| **Atomic dipoles** | None | Permanent (aligned randomly) | Permanent (aligned by exchange) |

| **M direction** | Opposes H | Parallel to H | Parallel to H (large) |

| **Response** | Weak repulsion | Weak attraction | Strong attraction |

| **Temperature effect** | Independent | χ ∝ 1/T (Curie law) | χ decreases; vanishes at T_C |

| **External field needed** | To induce M | To partially align | To saturate domains |

| **Permanent magnet?** | No | No | Yes |

| **Examples** | Bi, Cu, H₂O | Al, O₂, Fe³⁺ | Fe, Co, Ni, steel |

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WORKED EXAMPLE 5.3: IDENTIFYING WRONG FIELD DIAGRAMS

**Problem**: Analyze magnetic field line diagrams (a)–(g) from Fig. 5.6. Identify wrong diagrams and explain violations. Point out which represent electrostatic fields correctly.

**Solution**:

**(a) WRONG** — Field lines emanate radially from point. **Violation**: Gauss's law for magnetism; net outward flux nonzero. **Actual**: This represents an electric field of positive point charge (electrostatic field diagram). Actual magnetic field of straight wire forms concentric circles around wire (Chapter 4, Ampere's law).

**(b) WRONG** — Intersecting field lines. **Violation 1**: Field direction ambiguous at intersection. **Violation 2**: Closed loops form around empty space (no current enclosed). Actual magnetic field requires current inside closed loop.

**(c) CORRECT** — Field lines confined within toroid forming closed loops. **Why right**: Each closed loop encloses toroid winding carrying current. Gauss's law satisfied: equal flux enters and exits any closed surface inside toroid.

**(d) WRONG** — Solenoid field lines are straight and confined within solenoid, completely absent outside. **Violation**: Violates Ampere's law for solenoid. **Actual**: Field lines must curve out at both ends and form closed loops outside (field exists outside solenoid, though weaker). Fringing of field lines occurs at ends.

**(e) CORRECT** — Bar magnet field lines. **Key feature**: Field lines inside magnet run from **S-pole to N-pole** (opposite to external direction). This completes closed loops: external lines run N→S, internal lines run S→N. **Why right**: Gauss's law satisfied; net flux through any closed surface = 0.

**(f) WRONG** — Field lines emanate completely from upper shaded plate. **Violation**: Net outward flux nonzero; violates magnetic Gauss's law. **Actual magnetic field**: Cannot show net flux out of closed surface. **Type**: This is **electrostatic field** of parallel plate capacitor (positive upper plate, negative lower plate).

**Contrast (e) vs (f)**: Bar magnet (e) has field lines exiting N-pole AND field lines entering from inside — net zero flux. Capacitor (f) has field lines exiting positive plate only — net nonzero flux. This fundamental difference distinguishes magnetic from electrostatic fields.

**(g) WRONG** — Field lines between pole pieces are perfectly straight and parallel at edges. **Violation**: Violates Ampere's law. **Actual**: Some fringing (curving outward) must occur at ends of pole pieces. Completely confined straight-line fields between extended electrodes are impossible in real systems; edge effects create curved field lines.

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WORKED EXAMPLE 5.4: CONCEPTUAL QUESTIONS ON MAGNETIC FIELDS

**Problem**: (a) Do magnetic field lines represent force lines on moving charged particles? (b) How would Gauss's law change if magnetic monopoles existed? (c) Does a bar magnet exert torque on itself? (d) Can neutral systems (net charge = 0) have magnetic moments?

**Solution**:

**(a) NO** — **Common misconception**: Field lines show force direction. **Correct**: Magnetic force on moving charge is:

$$\vec{F} = q\vec{v} \times \vec{B}$$

Force is **always perpendicular to B** (and to velocity). Field lines are parallel (or tangent) to B. Therefore, magnetic field lines do NOT represent force direction. **Electrostatic field lines DO represent electric force direction** (F = qE parallel to E), so this difference is crucial.

**(b) Modified Gauss's Law with Monopoles**:

Current Gauss's law (no monopoles):

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

**If monopoles existed** with monopole charge q_m:

$$\oint \vec{B} \cdot d\vec{S} = \mu_0 q_{m,\text{enclosed}}$$

**Analogy to electrostatics**: Just as $\oint \vec{E} \cdot d\vec{S} = q_{\text{enclosed}}/\varepsilon_0$, the magnetic version would have monopole charge multiplied by μ₀. The constant μ₀ appears (not 1/μ₀) by dimensional analysis and physical analogy.

**(c) Self-Torque — NO**:

**On itself**: A bar magnet does NOT exert torque on itself due to its own field. Each element of magnetic dipole experiences forces from its own field, but by Newton's third law and symmetry, net torque = 0.

**Between elements**: One element of a current-carrying wire DOES exert force on another element of same wire. **Exception**: For a straight wire, the forces are collinear (pass through same axis) → net torque on wire is zero, but forces still exist.

**Physical reason**: Ampere's force between current elements depends on geometry. For circular loops or solenoids, net self-force may be zero, but elements certainly interact.

**(d) Neutral Systems with Magnetic Moment — YES**:

**Example**: Paramagnetic atom with unpaired electron:

**Net charge**: 0 (equal protons and electrons)

**Magnetic moment**: Nonzero (from electron spin and orbital angular momentum)

**Another example**: Oxygen molecule (O₂):

**Charge**: 0

**Magnetic moment**: Nonzero (two unpaired electrons in π* orbitals)

**General principle**: Magnetic moment arises from **circulating currents** (moving charges), not from net charge. A neutral atom/molecule can have:

Zero net charge Q = 0

Nonzero net magnetic dipole moment **m** ≠ 0

This is because magnetism depends on **charge distribution and motion**, not total charge.

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KEY FORMULAS SUMMARY (Chapter 5)

| Formula | Description | SI Units |

|---|---|---|

| **τ = mB sin θ** | Torque on magnetic dipole in external field | N·m |

| **U_m = –mB cos θ** | Magnetic potential energy of dipole | J |

| **B_A = (μ₀·2m)/(4πr³)** | Axial field of bar magnet (r >> size) | T |

| **B_E = –(μ₀m)/(4πr³)** | Equatorial field of bar magnet (r >> size) | T |

| **∮ B·dS = 0** | Gauss's law for magnetism | — |

| **M = m_net/V** | Magnetization (moment per unit volume) | A/m |

| **B = μ₀(H + M)** | Relation between B, H, M | T |

| **M = χH** | Linear magnetization (susceptibility) | — |

| **B = μ₀μ_r H** | Permeability relation | T |

| **χ = C/T** | Curie's law (paramagnetic) | — |

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IMPORTANT POINTS FOR BOARD EXAM

1. **Magnetic field lines form closed loops** — fundamental difference from electric field lines

2. **No magnetic monopoles exist** — Gauss's law for magnetism always gives zero flux

3. **Bar magnet ≈ solenoid** — equivalent due to circulating atomic currents (Ampere's hypothesis)

4. **Magnetic dipole in uniform field**: Experiences **torque but no net force**

5. **Magnetic potential energy**: Minimum at alignment (θ = 0°), maximum at antiparallel (θ = 180°)

6. **Three material classes**:

**Diamagnetic** (χ < 0): Weak repulsion, independent of T, induced moments

**Paramagnetic** (χ > 0, small): Weak attraction, χ ∝ 1/T, permanent atoms

**Ferromagnetic** (χ >> 1): Strong attraction, domains, hysteresis, T_C transition

7. **Temperature effects**: Paramagnetic χ decreases with T; ferromagnetic breaks down at Curie temperature

8. **Magnetization M**: Net dipole moment per unit volume, always produced by circulating currents (not monopoles)

9. **Magnetic intensity H**: Related to free currents only; B = μ₀(H + M)

10. **Hysteresis**: Nonlinear B-H relation in ferromagnets; remanence and coercivity are key parameters

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TYPICAL CBSE BOARD QUESTIONS

**Type A (Definition/Conceptual)**:

Define magnetization, magnetic intensity, susceptibility

Distinguish diamagnetic, paramagnetic, ferromagnetic materials

Explain why magnetic monopoles don't exist

State and explain Gauss's law for magnetism

**Type B (Numerical)**:

Calculate torque/energy for dipole in external field

Find axial/equatorial field of bar magnet at given distance

Determine energy required to rotate dipole from one angle to another

Compare field strengths at different points

**Type C (Analysis)**:

Identify incorrect magnetic field diagrams and explain violations

Analyze stability of dipole configurations

Discuss temperature dependence of magnetic properties

Explain domain alignment in ferromagnetics