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State Board · Class 12 · Physics

Lesson 11: Magnetic Materials

7 concepts · 56 practice questions with answers · Free to read

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Torque and potential energy of a magnetic dipole

From daily life

A compass needle turns to point north because of the torque on it.

Simple definition

A magnetic dipole is a pair of N and S poles or a current loop.

A magnetic dipole in a fieldSNmBτ = m B sinθ ; U = −m B cosθ

In the figure : A dipole in a field; τ = mB sinθ.

Rule 1: Torque — τ = m × B, τ = m B sin θ; m = magnetic moment (A m²); θ = angle between m and B.

  • Easy : m = 0.5 A m², B = 0.2 T, θ = 30° ⇒ τ = 0.050 N m.
  • Medium : θ = 90° ⇒ τ is maximum = mB.
  • Exam (CET) : When is τ zero? (1) θ = 0° (2) 90° (3) 45° (4) never → answer 1)

Rule 2: Potential energy — U = −m · B = −m B cos θ. θ = 0 : U = −mB (stable); θ = 180° : +mB (unstable).

  • Easy : θ from 0 to 60° : work = mB(1 − cos60°) = 0.05 J.
  • Medium : 0 to 180° : work = 2mB = 0.2 J.
  • Exam (CET) : Stable equilibrium at θ = ? (1) 0° (2) 90° (3) 180° (4) 45° → answer 1)

Rule 3: Magnetic moment — Loop : m = N I A; bar magnet : m = m_p × 2l (pole strength × length).

  • Easy : N = 50, I = 0.2 A, A = 20 cm² ⇒ m = 0.02 A m².
  • Medium : m_p = 5 A m, 2l = 0.1 m ⇒ m = 0.5 A m².
  • Exam (CET) : Unit of m? (1) A m² (2) A m (3) T (4) N → answer 1)

Derivation : U = −m B cos θ

  • Step 1 : Work to turn by dθ : dW = τ dθ = m B sin θ dθ.
  • Step 2 : θ₁ to θ₂ : W = m B (cos θ₁ − cos θ₂).
  • Step 3 : W = U₂ − U₁.
  • Step 4 : Taking U = 0 at θ₁ = 90° gives U = −m B cos θ.
Remember

τ = mB sinθ; U = −mB cosθ; m = NIA.

Common mistake / CET trap

θ is between m and B; the unit of m is A m².

Shortcut

Memory clue : "just like the electric dipole — p ↔ m, E ↔ B".

Spot the difference

QuantityFormulaUnit / dimensions
τmB sinθN m
U−mB cosθJ
mNIAA m²; [L² A]

Practice questions

Easy
  1. 1Formula for τ?
  2. 2Unit of m?
  3. 3Stable equilibrium?
Medium
  1. 4m = 2, B = 0.1, θ = 90° : τ?
  2. 5m = 1, B = 0.5, θ = 60° : U?
  3. 6N = 100, I = 0.1, A = 0.01 : m?
Challenging
  1. 7mB = 0.4 J : work from 0 to 180°?
  2. 8Released at θ = 30° : which way does it turn?
Show answers (try them yourself first)
  1. 1mB sinθ
  2. 2A m²
  3. 3θ = 0°
  4. 4τ = 0.2 N m
  5. 5U = −0.25 J
  6. 6m = 0.1 A m²
  7. 70.8 J
  8. 8Toward B (θ = 0)

Oscillations of a bar magnet and the vibration magnetometer

From daily life

Like a swing, a compass needle oscillates a little about north.

Simple definition

A vibration magnetometer finds B_H from the oscillation period.

Vibration magnetometerSNT = 2π √(I ÷ mB)B_H = 4π² I ÷ m T²

In the figure : T = 2π√(I/mB).

Rule 1: Oscillations of a bar magnet — Small oscillations are SHM : T = 2π √(I ÷ m B), I = moment of inertia; larger B ⇒ shorter T.

  • Easy : I = 5 × 10⁻⁷ kg m², m = 0.5 A m², B = 3.5 × 10⁻⁵ T ⇒ T = 1.06 s.
  • Medium : B four times ⇒ T halves.
  • Exam (CET) : T ∝ ? (1) 1/√B (2) 1/B (3) B (4) √B → answer 1)

Rule 2: Vibration magnetometer — Horizontal field : B_H = 4π² I ÷ m T²; for two magnets m₁/m₂ = (T₂/T₁)² (same I, B).

  • Easy : With that T ⇒ B_H = 3.5 × 10⁻⁵ T.
  • Medium : T₁ = 2 s, T₂ = 4 s ⇒ m₁ : m₂ = 4 : 1.
  • Exam (CET) : B_H is measured with? (1) vibration magnetometer (2) ammeter (3) galvanometer (4) voltmeter → answer 1)

Rule 3: Earth's field — Earth acts like a large bar magnet; B_H = B cos δ, B_V = B sin δ; tan δ = B_V/B_H (δ = dip angle). Over India B ≈ 4 × 10⁻⁵ T.

  • Easy : B = 4 × 10⁻⁵, δ = 30° ⇒ B_H = 3.46 × 10⁻⁵ T.
  • Medium : At the poles δ = 90°, B_H = 0.
  • Exam (CET) : At the equator δ = ? (1) 0° (2) 90° (3) 45° (4) 30° → answer 1)

Derivation : T = 2π √(I/mB)

  • Step 1 : τ = −m B sin θ ≈ −m B θ (small θ).
  • Step 2 : I α = −m B θ ⇒ α = −(mB/I) θ.
  • Step 3 : SHM : ω² = m B ÷ I.
  • Step 4 : T = 2π √(I ÷ m B).
Remember

T = 2π√(I/mB); B_H = 4π²I/mT²; tanδ = B_V/B_H.

Common mistake / CET trap

The relevant field is B_H, not the total B.

Shortcut

Memory clue : "large B — quick oscillation".

Spot the difference

ComponentFormulaEquator / pole
B_HB cosδmax / 0
B_VB sinδ0 / max
δtan⁻¹(B_V/B_H)0° / 90°

Practice questions

Easy
  1. 1Formula for T?
  2. 2What is B_H?
  3. 3What is δ?
Medium
  1. 4I = 10⁻⁶, m = 1, B = 4 × 10⁻⁵ : T?
  2. 5T = 3 s, B_H = 3 × 10⁻⁵ : B/T²?
  3. 6B = 5 × 10⁻⁵, δ = 60° : B_V?
Challenging
  1. 7T₁ : T₂ = 1 : 3 : m₁ : m₂?
  2. 8m doubled : T?
Show answers (try them yourself first)
  1. 12π√(I/mB)
  2. 2Horizontal component of Earth's field
  3. 3Angle of dip
  4. 4T = 0.993 s
  5. 53.33 × 10⁻⁶
  6. 6B_V = 4.33 × 10⁻⁵ T
  7. 79 : 1
  8. 8T ÷ √2

Origin of magnetism : orbital moment and the Bohr magneton

From daily life

A current makes a magnet; an orbiting electron is a tiny current — so materials are magnetic.

Simple definition

The Bohr magneton is the natural unit of magnetic moment.

Orbital moment of an electronvμLμ = (e ÷ 2m) Lμ_B = 9.27 × 10⁻²⁴ A m²

In the figure : μ and L are opposite.

Rule 1: Orbital magnetic moment — An electron in an orbit is a current loop : μ = (e ÷ 2 m_e) L; L = m v r; μ and L are opposite in direction.

  • Easy : L = h/2π = 1.055 × 10⁻³⁴ ⇒ μ = 9.26 × 10⁻²⁴ A m².
  • Medium : μ/L = e/2m = 8.8 × 10¹⁰ C/kg (gyromagnetic ratio).
  • Exam (CET) : μ/L = ? (1) e/2m (2) e/m (3) 2e/m (4) m/e → answer 1)

Rule 2: Bohr magneton — For n = 1 : μ_B = e h ÷ 4π m_e = 9.27 × 10⁻²⁴ A m²; orbital moment = n μ_B.

  • Easy : μ_B = 9.266 × 10⁻²⁴ A m².
  • Medium : n = 2 : μ = 2 μ_B.
  • Exam (CET) : μ_B = ? (1) 9.27 × 10⁻²⁴ (2) 9.1 × 10⁻³¹ (3) 1.6 × 10⁻¹⁹ (4) 6.63 × 10⁻³⁴ → answer 1)

Rule 3: Spin magnetic moment — An electron also has a spin moment ≈ 1 μ_B. The atom's total moment is orbital + spin; paired electrons cancel.

  • Easy : Paired electrons : total 0.
  • Medium : Unpaired electrons ⇒ paramagnetic.
  • Exam (CET) : With unpaired electrons? (1) a magnetic moment (2) zero (3) diamagnetism (4) insulation → answer 1)

Derivation : μ = (e/2m) L

  • Step 1 : Period T = 2π r ÷ v; current I = e ÷ T = e v ÷ 2π r.
  • Step 2 : μ = I A = (e v ÷ 2π r) π r² = e v r ÷ 2.
  • Step 3 : L = m v r ⇒ v r = L/m.
  • Step 4 : μ = (e ÷ 2m) L.
Remember

μ = (e/2m)L; μ_B = eh/4πm = 9.27 × 10⁻²⁴ A m².

Common mistake / CET trap

It is e/2m, not e/m; μ and L are opposite.

Shortcut

Memory clue : "half of e/m".

Spot the difference

MomentOriginValue
OrbitalL = nh/2πn μ_B
Spins = ½≈ μ_B
μ_Beh/4πm9.27 × 10⁻²⁴

Practice questions

Easy
  1. 1What is the Bohr magneton?
  2. 2μ/L ?
  3. 3Unpaired electrons give?
Medium
  1. 4n = 3 : orbital μ?
  2. 5L = 2 × 10⁻³⁴ : μ?
  3. 6Compute e/2m.
Challenging
  1. 7Derive μ_B.
  2. 8Effect of paired electrons?
Show answers (try them yourself first)
  1. 1Orbital moment for n = 1
  2. 2e/2m
  3. 3A net magnetic moment
  4. 42.78 × 10⁻²³ A m²
  5. 5μ = 1.76 × 10⁻²³ A m²
  6. 68.78 × 10¹⁰ C/kg
  7. 7μ = (e/2m)(h/2π) = eh/4πm
  8. 8Moments cancel

Magnetisation, magnetic intensity and susceptibility

From daily life

An iron nail near a magnet becomes a magnet itself — magnetisation.

Simple definition

Magnetisation M; susceptibility χ = M/H.

Magnetisation M, intensity H, susceptibility χM = m_net ÷ VB = μ₀ (H + M)M = χ Hμ_r = 1 + χB = μ₀ μ_r H

In the figure : B = μ₀(H + M); χ and μ_r.

Rule 1: Magnetisation — M = m_net ÷ V (A/m); magnetic moment per unit volume.

  • Easy : m_net = 2 A m², V = 10⁻⁴ m³ ⇒ M = 2 × 10⁴ A/m.
  • Medium : N atoms of moment m ⇒ M = N m ÷ V.
  • Exam (CET) : Unit of M? (1) A/m (2) A m² (3) T (4) Wb → answer 1)

Rule 2: Intensity and susceptibility — B = μ₀ (H + M), H = n I; χ = M ÷ H; μ_r = 1 + χ; B = μ₀ μ_r H.

  • Easy : H = 1000 A/m, χ = 2 × 10⁻⁵ ⇒ M = 0.020 A/m; μ_r = 1.00002.
  • Medium : B = μ₀(H + M) = 1.2567 × 10⁻³ T.
  • Exam (CET) : Unit of χ? (1) none (2) A/m (3) T (4) H → answer 1)

Rule 3: Material in a solenoid — n = 500/m, I = 2 A ⇒ H = 1000 A/m; a core with μ_r = 800 gives B = μ₀ μ_r H = 1.0 T.

  • Easy : B = 4π × 10⁻⁷ × 800 × 1000 = 1.01 T.
  • Medium : H depends only on n I, not on the core.
  • Exam (CET) : H depends on? (1) n I (2) the core (3) μ_r (4) M → answer 1)

Derivation : B = μ₀ μ_r H

  • Step 1 : B = μ₀ (H + M) and M = χ H.
  • Step 2 : B = μ₀ H (1 + χ).
  • Step 3 : μ_r = 1 + χ.
  • Step 4 : B = μ₀ μ_r H = μ H.
Remember

M = m/V; B = μ₀(H + M); χ = M/H; μ_r = 1 + χ.

Common mistake / CET trap

H is external (nI), B total; their units differ.

Shortcut

Memory clue : "B = μ₀ μ_r H".

Spot the difference

QuantityFormulaUnit / dimension
Mm/VA/m
HnIA/m
χM/Hnone
μ_r1 + χnone

Practice questions

Easy
  1. 1What is M?
  2. 2Formula for χ?
  3. 3Formula for μ_r?
Medium
  1. 4n = 800, I = 1.5 : H?
  2. 5χ = 10⁻³, H = 500 : M?
  3. 6μ_r = 100, H = 400 : B?
Challenging
  1. 7μ_r = 1.002 : χ?
  2. 8What does μ_r ≫ 1 in iron mean?
Show answers (try them yourself first)
  1. 1Moment per unit volume
  2. 2M/H
  3. 31 + χ
  4. 4H = 1200 A/m
  5. 5M = 0.5 A/m
  6. 6B = 0.0503 T
  7. 7χ = 0.002
  8. 8The field is greatly enhanced

Diamagnetism, paramagnetism and Curie's law

From daily life

Water is pushed very slightly away from a magnet while liquid oxygen is slightly drawn in — two faint kinds of magnetism.

Simple definition

Diamagnetic : opposes the field; paramagnetic : lines up with the field.

Diamagnetic and paramagnetic materialsDiamagneticParamagneticχ < 0 ; μ_r < 1χ > 0 ; μ_r > 1χ = C ÷ T (paramagnetic)

In the figure : Lines are expelled by a diamagnet and crowd into a paramagnet.

Rule 1: Diamagnetism — Present in all materials; χ negative and tiny (≈ −10⁻⁵); induced moment opposes the field; pushed out of a strong field; independent of temperature. E.g. bismuth, copper, water.

  • Easy : χ = −1.6 × 10⁻⁵ ⇒ μ_r = 0.999984 (< 1).
  • Medium : H = 2000 A/m, χ = −2 × 10⁻⁵ ⇒ M = −0.04 A/m (opposite to H).
  • Exam (CET) : μ_r of a diamagnet? (1) < 1 (2) > 1 (3) ≫ 1 (4) 0 → answer 1) Trick : dia = against.

Rule 2: Paramagnetism and Curie's law — Permanent atomic moments, slightly aligned by the field; χ small positive (10⁻⁵ – 10⁻³); drawn into the field. Curie's law : χ = C ÷ T (C = Curie constant, T in kelvin).

  • Easy : C = 7.2 × 10⁻³ K, T = 300 K ⇒ χ = 2.4 × 10⁻⁵.
  • Medium : T 300 → 400 K ⇒ χ = 1.8 × 10⁻⁵; χ ∝ 1/T.
  • Exam (CET) : If T doubles, paramagnetic χ becomes? (1) half (2) double (3) four times (4) same → answer 1)

Rule 3: Comparison and identification — Hang the sample near a weak magnet : moves away = diamagnetic; comes closer = paramagnetic; sticks strongly = ferromagnetic.

  • Easy : Aluminium is weakly attracted ⇒ paramagnetic.
  • Medium : Water moves out of a field ⇒ diamagnetic.
  • Exam (CET) : Copper is? (1) diamagnetic (2) paramagnetic (3) ferromagnetic (4) all → answer 1)

Derivation : Curie's law (qualitative derivation)

  • Step 1 : Field B aligns the atomic moments; thermal motion randomises them.
  • Step 2 : Alignment ∝ B ÷ T (stronger field, lower T ⇒ more alignment).
  • Step 3 : M ∝ B ÷ T ⇒ M = C' H ÷ T.
  • Step 4 : χ = M ÷ H ⇒ χ = C ÷ T.
Remember

Dia : χ < 0; para : χ = C/T.

Common mistake / CET trap

Diamagnetism is temperature-independent; paramagnetism depends on T. χ and μ_r are unitless.

Shortcut

Memory clue : "Dia = against; para = along".

Spot the difference

PropertyDiamagneticParamagnetic
χnegative, tinypositive, small
μ_r< 1> 1
In fieldrepelledattracted
Effect of Tnoneχ ∝ 1/T
E.g.Bi, Cu, H₂OAl, Pt, O₂

Practice questions

Easy
  1. 1χ of a diamagnet?
  2. 2State Curie's law.
  3. 3Two paramagnetic examples?
Medium
  1. 4C = 0.9 K, T = 300 K : χ?
  2. 5χ = 3 × 10⁻⁵ at 200 K : χ at 300 K?
  3. 6χ = −3 × 10⁻⁶ : μ_r?
Challenging
  1. 7χ = 4.5 × 10⁻⁵ at 300 K : χ at 450 K?
  2. 8Why is diamagnetism present in all materials?
Show answers (try them yourself first)
  1. 1negative and tiny
  2. 2χ = C/T
  3. 3Al, Pt
  4. 43 × 10⁻³
  5. 52 × 10⁻⁵
  6. 60.999997
  7. 73 × 10⁻⁵
  8. 8Every atom develops an opposing induced moment

Ferromagnetism, domains and Curie temperature

From daily life

Nails stick to a magnet, but a strongly heated magnet loses its strength.

Simple definition

Ferromagnetic : huge χ due to domains; paramagnetic above T_c.

Domains and Curie temperatureDomainsT_cχ ↑T →χ = C ÷ (T − T_c) (T > T_c)

In the figure : Domains, and χ versus T (change at T_c).

Rule 1: Ferromagnetism and domains — χ very large (10² – 10⁵); strongly attracted. Atomic moments are aligned inside domains (~10⁻⁶ – 10⁻⁹ m³); with no external field the domains point randomly ⇒ net M ≈ 0. E.g. Fe, Co, Ni.

  • Easy : Fe : n = 8.5 × 10²⁸ /m³, m = 2.2 μ_B ⇒ saturation M_s = n m = 1.73 × 10⁶ A/m.
  • Medium : In an external field favourable domains grow and others rotate.
  • Exam (CET) : A domain in a ferromagnet is? (1) a small region of aligned moments (2) a hole (3) an atom (4) a fibre → answer 1)

Rule 2: Curie temperature — Above T_c the domains vanish and the material turns paramagnetic. Curie–Weiss law : χ = C ÷ (T − T_c), T > T_c. Fe : 1043 K, Ni : 627 K, Co : 1388 K.

  • Easy : Fe, T = 1143 K, C = 2 K ⇒ χ = 2 ÷ (1143 − 1043) = 0.02.
  • Medium : T = 1243 K ⇒ χ = 0.01; χ falls as T rises.
  • Exam (CET) : Above T_c a ferromagnet becomes? (1) paramagnetic (2) diamagnetic (3) non-magnetic (4) stays ferromagnetic → answer 1)

Rule 3: Magnetisation curve — As H increases, M rises quickly, then saturates at M_s once all domains are aligned.

  • Easy : Near saturation B ≈ μ₀(H + M_s).
  • Medium : M_s = 1.73 × 10⁶ A/m, H small ⇒ B ≈ μ₀M_s = 2.18 T.
  • Exam (CET) : After saturation M is? (1) constant (2) rising (3) falling (4) zero → answer 1)

Derivation : Curie–Weiss law (qualitative)

  • Step 1 : Above T_c the material is paramagnetic but a residual exchange interaction remains.
  • Step 2 : Replace T in Curie's law by (T − T_c).
  • Step 3 : As T → T_c, χ → ∞.
  • Step 4 : χ = C ÷ (T − T_c).
Remember

χ = C/(T − T_c); M saturates at M_s; Fe T_c = 1043 K.

Common mistake / CET trap

At T_c the material is not destroyed, it only loses ferromagnetism; for T > T_c use C/(T−T_c), not C/T.

Shortcut

Memory clue : "Cross T_c, turn paramagnetic".

Spot the difference

PropertyParamagneticFerromagnetic
χsmallvery large
Domainsnonepresent
T dependenceC/TC/(T−T_c)
E.g.AlFe, Co, Ni

Practice questions

Easy
  1. 1What is a domain?
  2. 2Three ferromagnetic examples?
  3. 3What is Curie temperature?
Medium
  1. 4T_c = 627 K, T = 727 K, C = 1.5 K : χ?
  2. 5n = 9 × 10²⁸, m = 2 μ_B : M_s?
  3. 6T = T_c + 50, C = 2.5 : χ?
Challenging
  1. 7χ = 0.05 : T − T_c? (C = 2 K)
  2. 8Why does heating weaken a magnet?
Show answers (try them yourself first)
  1. 1A small region of aligned moments
  2. 2Fe, Co, Ni
  3. 3Temperature above which domains vanish
  4. 40.015
  5. 5M_s = 1.67 × 10⁶ A/m
  6. 60.05
  7. 740 K
  8. 8Heat disorders the domains

Hysteresis, permanent magnets, electromagnets and shielding

From daily life

Rub a nail with a magnet and it stays magnetic — the material 'remembers'.

Simple definition

Hysteresis : the lag of B behind H, tracing a closed loop.

Hysteresis loop (B – H)HBB_rH_csaturationarea = loss / cycle

In the figure : The B – H loop; B_r, H_c and area.

Live figure

Hysteresis, permanent magnets, electromagnets and shielding

In the app you can move this figure with your finger.

Open in the app

Rule 1: Hysteresis loop — B lags behind H. Retentivity B_r : B left at H = 0. Coercivity H_c : reverse H needed to bring B to zero. Loop area = energy lost per cycle per unit volume (J/m³).

  • Easy : H_c = 100 A/m, n = 1000/m ⇒ I = H_c ÷ n = 0.1 A.
  • Medium : Area 400 J/m³, 50 Hz, V = 2 × 10⁻³ m³ ⇒ P = 40 W.
  • Exam (CET) : The loop area represents? (1) energy loss (2) B_r (3) H_c (4) χ → answer 1)

Rule 2: Permanent magnet and electromagnet — Permanent magnet : large B_r and large H_c, wide loop (steel, alnico). Electromagnet / transformer core : high μ_r, small H_c, narrow loop (soft iron) ⇒ low loss.

  • Easy : Steel (wide loop) for a speaker magnet.
  • Medium : Soft iron (narrow loop) for an AC transformer.
  • Exam (CET) : Best for a transformer core? (1) soft iron (2) steel (3) alnico (4) copper → answer 1) Trick : AC = narrow loop.

Rule 3: Magnetic shielding and demagnetisation — Inside a soft-iron shell the field lines pass through the iron, leaving a weak field inside. Demagnetise by reverse H, a decreasing AC, or by heating above T_c.

  • Easy : A watch inside a soft-iron case sees a much weaker magnetic field.
  • Medium : Hammering or heating disorders the domains and weakens the magnet.
  • Exam (CET) : A shielding case is made of? (1) soft iron (2) copper (3) glass (4) plastic → answer 1)

Derivation : Loss = area of the hysteresis loop

  • Step 1 : In one cycle ∮ H dB = work done per unit volume.
  • Step 2 : This equals the area A of the loop.
  • Step 3 : For volume V and frequency f there are f cycles per second.
  • Step 4 : P = A f V (watt).
Remember

P = A f V; permanent magnet : wide loop; soft iron : narrow loop.

Common mistake / CET trap

Do not mix B_r (B at H = 0) with H_c (H at B = 0); area = loss.

Shortcut

Memory clue : "Wide = lasting; narrow = easy".

Spot the difference

PropertyPermanent magnetElectromagnet core
B_rlargelarge
H_clargesmall
Loopwidenarrow
Materialsteel, alnicosoft iron

Practice questions

Easy
  1. 1What is retentivity?
  2. 2What is coercivity?
  3. 3What does the loop area give?
Medium
  1. 4Area 250 J/m³, f = 50 Hz, V = 10⁻³ m³ : P?
  2. 5H_c = 60 A/m, n = 600 : I?
  3. 6Area 100 J/m³, V = 0.01 m³ : energy per cycle?
Challenging
  1. 7Why steel for a permanent magnet?
  2. 8What happens to a magnet when heated?
Show answers (try them yourself first)
  1. 1B left at H = 0
  2. 2Reverse H that brings B to zero
  3. 3Energy loss per cycle
  4. 4P = 12.5 W
  5. 50.1 A
  6. 61 J
  7. 7Large B_r and H_c retain the magnetism
  8. 8Weakens; vanishes at T_c

Quick revision of the whole lesson

ConceptKey formula / factUnit / dimensions
Dipoleτ = mB sinθ; U = −mB cosθN m; J
OscillationT = 2π√(I/mB); B_H = 4π²I/mT²s; T
Bohr magnetonμ_B = eħ/2m = 9.27×10⁻²⁴A m²
MagnetisationM = m/V; B = μ₀(H+M)A/m; T
Susceptibilityχ = M/H; μ_r = 1 + χunitless
Paramagneticχ = C/Tunitless
Ferromagneticχ = C/(T − T_c)unitless
HysteresisP = A f VW

🎯 Most important points for the exam

  • τ = mB sinθ; stable equilibrium at θ = 0.
  • T = 2π√(I/mB).
  • μ_B = 9.27 × 10⁻²⁴ A m².
  • B = μ₀(H + M); μ_r = 1 + χ.
  • Dia χ < 0; para χ = C/T.
  • Ferromagnet : domains; paramagnetic above T_c.
  • Permanent magnet : wide loop; core : narrow loop.
Study mantra

Identify the type → sign and size of χ → formula → units.