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State Board · Class 11 · Physics

Lesson 12: Magnetism

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

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Magnetic field, field lines and flux

From daily life

A compass needle always turns north — because of the Earth's magnetic field. Link to earlier classes : magnets, field lines and the compass of classes 9-10 now extend to dipoles, fields and torque formulas.

Simple definition

A magnetic field (B) is the region around a magnet where it exerts a force on other magnets or currents.

Field lines and fluxNSABθΦ = B A cos θ1 Wb = 1 T m²

In the figure : Field lines on the left, flux on the right.

Live figure

Magnetic field, field lines and flux

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

Open in the app

Rule 1: Field lines — Lines leave N, enter S; inside the magnet S → N; closed loops; never cross; dense = stronger B.

  • Easy : The tangent to a line gives the direction of B.
  • Medium : Two lines crossing would give two directions at a point — impossible.
  • Exam (CET) : Field lines are? (1) closed (2) open (3) crossing (4) straight only → answer 1)

Rule 2: Magnetic flux — Φ = B A cos θ; unit weber (Wb) = T m²; θ = angle between B and the area normal.

  • Easy : B = 0.2 T, A = 0.05 m², θ = 0 ⇒ Φ = 0.01 Wb.
  • Medium : θ = 60° ⇒ Φ = 0.005 Wb.
  • Exam (CET) : Φ is maximum at? (1) 0° (2) 90° (3) 45° (4) 60° → answer 1) Trick : area perpendicular to B ⇒ maximum.

Rule 3: Units and dimensions — Unit of B is tesla (T) = Wb/m²; 1 T = 10⁴ gauss. [B] = M T⁻² A⁻¹; [Φ] = M L² T⁻² A⁻¹.

  • Easy : 0.5 T = 5000 gauss.
  • Medium : 2 × 10⁻⁴ Wb ÷ 0.01 m² = 0.02 T.
  • Exam (CET) : 1 T = ? (1) 10⁴ G (2) 10² G (3) 10⁻⁴ G (4) 1 G → answer 1)

Derivation : Φ = B A cos θ

  • Step 1 : Area normal n̂; angle θ with B.
  • Step 2 : Component of B normal to the surface is B cos θ.
  • Step 3 : Φ = (normal component) × A.
  • Step 4 : Φ = B A cos θ.
Remember

Φ = BA cos θ; 1 Wb = 1 T m²; lines are closed.

Common mistake / CET trap

Flux depends on angle; θ is measured from the normal.

Shortcut

Memory clue : "Normal = maximum (cos 0)".

Spot the difference

QuantityUnitDimensions
BTM T⁻² A⁻¹
ΦWbM L² T⁻² A⁻¹

Practice questions

Easy
  1. 1What is a magnetic field?
  2. 2Unit of Φ?
  3. 3Do field lines cross?
Medium
  1. 4B = 0.4 T, A = 0.02 m², θ = 0 : Φ?
  2. 5Same, θ = 90° : Φ?
  3. 63 × 10⁻³ Wb, A = 0.06 m² : B?
Challenging
  1. 7By what factor does Φ change at θ = 60°?
  2. 8What do dense lines mean?
Show answers (try them yourself first)
  1. 1The space around a magnet where its force is felt
  2. 2Wb
  3. 3No
  4. 48 × 10⁻³ Wb
  5. 50
  6. 60.05 T
  7. 7½
  8. 8Stronger B

Bar magnet as a magnetic dipole

From daily life

However you cut a magnet, it still has two ends.

Simple definition

A magnetic dipole is two equal opposite poles close together; m = pole strength × separation.

Bar magnet dipoleNSm2l−q_m+q_mm = q_m × 2l

In the figure : m points from S to N.

Rule 1: Dipole moment — m = q_m × 2l; direction S → N; unit A m² (or J/T); [m] = L² A.

  • Easy : q_m = 5 A m, 2l = 0.1 m ⇒ m = 0.5 A m².
  • Medium : m = 0.5, 2l = 0.2 m ⇒ q_m = 2.5 A m.
  • Exam (CET) : Direction of m? (1) S → N (2) N → S (3) outward (4) zero → answer 1)

Rule 2: Current loop — The m of a loop = N I A; a current loop is a magnetic dipole.

  • Easy : I = 2 A, A = 0.01 m², N = 1 ⇒ m = 0.02 A m².
  • Medium : N = 50 ⇒ m = 1 A m².
  • Exam (CET) : m of a loop? (1) NIA (2) I/A (3) NI/A (4) IA² → answer 1)

Rule 3: Cutting a magnet — However a magnet is cut, each piece keeps N and S (no monopole). Cut across the length : m halves, q_m same.

  • Easy : m = 0.8 A m²; cut across the middle ⇒ m = 0.4.
  • Medium : Cut across : q_m same, 2l halves.
  • Exam (CET) : A piece with one pole only? (1) never (2) always (3) sometimes (4) in small pieces → answer 1)

Derivation : Loop as a dipole

  • Step 1 : Far from a loop, its field looks like that of a dipole.
  • Step 2 : A dipole has m = q_m × 2l.
  • Step 3 : For a loop m = I A.
  • Step 4 : m = N I A.
Remember

m = q_m 2l = NIA; unit A m².

Common mistake / CET trap

Cutting never gives a monopole.

Shortcut

Memory clue : "S to N = m".

Spot the difference

FeatureBar magnetElectric dipole
PolesN, S (no monopole)+q, −q (can be separated)
Momentm = q_m 2lp = q 2l
DirectionS → N−q → +q

Practice questions

Easy
  1. 1Unit of m?
  2. 2Direction of m?
  3. 3Do monopoles exist?
Medium
  1. 4q_m = 4 A m, 2l = 0.05 m : m?
  2. 5I = 3 A, A = 0.02 m², N = 10 : m?
  3. 6A magnet of m = 1.2 cut across the middle : m?
Challenging
  1. 7Why is a loop a dipole?
  2. 8Dimensions of m?
Show answers (try them yourself first)
  1. 1A m²
  2. 2S → N
  3. 3No
  4. 40.2 A m²
  5. 50.6 A m²
  6. 60.6
  7. 7Same far field
  8. 8L² A

Axial and equatorial fields of a short magnet

From daily life

Move away from a magnet and its effect falls quickly.

Simple definition

Axial = along the magnet's line; equatorial = on the perpendicular bisector.

Axial and equatorialNSB_axialB_eqB = (μ₀/4π)·2m/r³B = (μ₀/4π)·m/r³

In the figure : Green axial, purple equatorial.

Rule 1: Axial line — B = (μ₀/4π) · 2m / r³; along m.

  • Easy : m = 0.5, r = 0.2 m ⇒ B = 1.25 × 10⁻⁵ T.
  • Medium : r doubled ⇒ B becomes ⅛.
  • Exam (CET) : B ∝ ? (1) 1/r³ (2) 1/r² (3) 1/r (4) r → answer 1)

Rule 2: Equatorial line — B = (μ₀/4π) · m / r³; opposite to m. B_axial = 2 B_eq.

  • Easy : Same m, r : B = 6.250 × 10⁻⁶ T.
  • Medium : Axial : equatorial = 2 : 1.
  • Exam (CET) : B_axial / B_eq = ? (1) 2 (2) 1 (3) ½ (4) 4 → answer 1)

Rule 3: Short magnet — Valid only when r ≫ l; μ₀/4π = 10⁻⁷ T m/A.

  • Easy : l = 1 cm, r = 50 cm ⇒ r ≫ l, the formula works.
  • Medium : m = 2, r = 1 m ⇒ B_ax = 4 × 10⁻⁷ T.
  • Exam (CET) : μ₀/4π = ? (1) 10⁻⁷ (2) 10⁻⁵ (3) 4π×10⁻⁷ (4) 10⁷ → answer 1)

Derivation : Axial field

  • Step 1 : Point at r from the centre; r − l from N, r + l from S.
  • Step 2 : B = (μ₀/4π) q_m [1/(r − l)² − 1/(r + l)²].
  • Step 3 : = (μ₀/4π) q_m · 4 r l / (r² − l²)².
  • Step 4 : r ≫ l, m = q_m 2l ⇒ B = (μ₀/4π) 2m / r³.
Remember

B_ax = (μ₀/4π) 2m/r³; B_eq = (μ₀/4π) m/r³.

Common mistake / CET trap

B_eq is opposite to m; axial is double.

Shortcut

Memory clue : "Axis = 2, equator = 1".

Spot the difference

LineBDirection
Axial(μ₀/4π) 2m/r³Along m
Equatorial(μ₀/4π) m/r³Opposite to m

Practice questions

Easy
  1. 1Formula for axial B?
  2. 2Direction of equatorial B?
  3. 3How does B vary with r?
Medium
  1. 4m = 1, r = 0.1 m : B_axial?
  2. 5Same : B_eq?
  3. 6r tripled ⇒ B?
Challenging
  1. 7B_ax = 8 × 10⁻⁶ T at r; at 2r?
  2. 8Why r ≫ l in the formula?
Show answers (try them yourself first)
  1. 1(μ₀/4π) 2m/r³
  2. 2Opposite to m
  3. 31/r³
  4. 42 × 10⁻⁴ T
  5. 51 × 10⁻⁴ T
  6. 61/27
  7. 710⁻⁶ T
  8. 8Because r² − l² ≈ r²

Field at any point, torque and neutral points

From daily life

A compass needle turns and settles in a field — because of torque.

Simple definition

Torque τ turns m in a field; a neutral point is where the net B is zero.

Torque and neutral pointsmBθτ = m B sin θU = − m B cos θB_d = B_H ⇒ Nnet B = 0

In the figure : θ between m and B.

Rule 1: B at any point — B = (μ₀/4π)(m/r³) √(1 + 3cos²θ); θ = angle from the axis.

  • Easy : m = 0.5, r = 0.2, θ = 60° ⇒ B = 8.27 × 10⁻⁶ T.
  • Medium : θ = 0 ⇒ 2m/r³ ; θ = 90° ⇒ m/r³.
  • Exam (CET) : B at θ = 90°? (1) m/r³ (2) 2m/r³ (3) 0 (4) 3m/r³ (times μ₀/4π) → answer 1)

Rule 2: Torque and potential energy — τ = m B sin θ (m × B); U = − m B cos θ. θ = 0 stable equilibrium; θ = 180° unstable.

  • Easy : m = 0.5, B = 0.2 T, θ = 30° ⇒ τ = 0.05 N m.
  • Medium : θ = 0 ⇒ U = − 0.1 J; θ = 180° ⇒ + 0.1 J; difference 0.2 J.
  • Exam (CET) : τ maximum at? (1) 90° (2) 0° (3) 180° (4) 45° → answer 1) Trick : sin 90° = 1.

Rule 3: Neutral points — Where the magnet's B equals the Earth's B_H. N pointing north ⇒ on the equatorial line (μ₀/4π)m/d³ = B_H; N pointing south ⇒ on the axial line (μ₀/4π)2m/d³ = B_H.

  • Easy : m = 0.35, B_H = 3.5 × 10⁻⁵ ⇒ d = 0.10 m.
  • Medium : N pointing south ⇒ same m gives d = 0.126 m.
  • Exam (CET) : N pointing north ⇒ neutral points lie on? (1) equatorial (2) axial (3) anywhere (4) none → answer 1)

Derivation : τ = m B sin θ

  • Step 1 : +q_m B on N, −q_m B on S; a couple.
  • Step 2 : Arm = 2l sin θ.
  • Step 3 : τ = q_m B × 2l sin θ.
  • Step 4 : τ = m B sin θ.
Remember

τ = mB sin θ; U = −mB cos θ; neutral : B_d = B_H.

Common mistake / CET trap

N north ⇒ equatorial points; N south ⇒ axial points.

Shortcut

Memory clue : "North → equatorial; south → axial".

Spot the difference

StateθτU
Stable00−mB
Max τ90°mB0
Unstable180°0+mB

Practice questions

Easy
  1. 1Formula for τ?
  2. 2When is equilibrium stable?
  3. 3What is a neutral point?
Medium
  1. 4m = 2, B = 0.1, θ = 90° : τ?
  2. 5Same, θ = 0 : U?
  3. 6Work to turn θ = 0 to 180°?
Challenging
  1. 7τ = 0.05 N m, m = 0.5, B = 0.2 : θ?
  2. 8N north : where are neutral points?
Show answers (try them yourself first)
  1. 1m B sin θ
  2. 2θ = 0
  3. 3A point where the net field is zero
  4. 40.2 N m
  5. 5−0.2 J
  6. 60.4 J
  7. 730° (or 150°)
  8. 8On the equatorial line

Gauss's law for magnetism

From daily life

A single pole of a magnet never comes alone.

Simple definition

Gauss for magnetism : the net magnetic flux through a closed surface is 0.

Closed surface and GaussNSΦ_in = Φ_outΣ B·ΔA = 0∴ no monopole

In the figure : Flux in = flux out.

Rule 1: Gauss for magnetism — ∮ B · dA = 0 — the net magnetic flux through any closed surface is zero.

  • Easy : A sphere around a bar magnet : flux in = flux out.
  • Medium : Φ_out = 3 × 10⁻³ Wb ⇒ Φ_in = 3 × 10⁻³ Wb.
  • Exam (CET) : Net Φ through a closed surface? (1) 0 (2) B A (3) q/ε₀ (4) μ₀I → answer 1)

Rule 2: Electric vs magnetic Gauss — Electric : ∮E·dA = q/ε₀ (isolated charges exist). Magnetic : 0 (no isolated pole).

  • Easy : Closed surface around an electric dipole : q_net = 0 ⇒ Φ_E = 0.
  • Medium : For a magnet it is always 0.
  • Exam (CET) : Magnetic monopole? (1) never found (2) found (3) proven (4) of iron → answer 1)

Rule 3: Why closed lines — Lines leave N, enter S and return inside — no start or end.

  • Easy : Outside N → S; inside S → N.
  • Medium : Electric lines run from +q to −q (not closed).
  • Exam (CET) : Magnetic lines are? (1) closed (2) from + pole (3) to infinity (4) straight → answer 1)

Derivation : No monopole

  • Step 1 : Take any closed surface.
  • Step 2 : As many lines enter as leave.
  • Step 3 : ∴ Σ B·ΔA = 0.
  • Step 4 : ∮ B · dA = 0.
Remember

∮ B·dA = 0.

Common mistake / CET trap

Electric gives q/ε₀; magnetic gives 0 — do not mix.

Shortcut

Memory clue : "A magnet always comes in pairs".

Spot the difference

Gauss's lawElectricMagnetic
Net fluxq/ε₀0
SourceIsolated chargeDipole
LinesOpenClosed

Practice questions

Easy
  1. 1Gauss's law for magnetism?
  2. 2Has a monopole been found?
  3. 3How are the lines?
Medium
  1. 4Φ in = 5 × 10⁻⁴ Wb : Φ out?
  2. 5Net Φ of a closed surface?
  3. 6Φ_E of an electric dipole?
Challenging
  1. 7Why do new poles appear on cutting?
  2. 8Electric vs magnetic Gauss difference?
Show answers (try them yourself first)
  1. 1∮B·dA = 0
  2. 2No
  3. 3Closed
  4. 45 × 10⁻⁴ Wb
  5. 50
  6. 60
  7. 7Always dipoles; no monopole
  8. 8q/ε₀ versus 0

Earth's magnetism and its elements

From daily life

In India a compass stays nearly flat; near the poles it is useless.

Simple definition

Earth's field is described by three elements : declination (α), dip (δ), horizontal component (B_H).

Earth's elementsB_HB_VB_EδB_H = B_E cos δB_V = B_E sin δtan δ = B_V / B_H

In the figure : Horizontal and vertical parts of B_E.

Rule 1: Three elements — Declination α = angle between geographic and magnetic meridians; dip δ = angle of B_E with the horizontal; B_H = horizontal component.

  • Easy : δ = 0 at the equator; δ = 90° at the poles.
  • Medium : B_H = 0 at the poles; B_V = 0 at the equator.
  • Exam (CET) : δ at the magnetic equator? (1) 0 (2) 90° (3) 45° (4) 60° → answer 1)

Rule 2: Relations — B_H = B_E cos δ; B_V = B_E sin δ; tan δ = B_V / B_H; B_E = √(B_H² + B_V²).

  • Easy : B_E = 5 × 10⁻⁵, δ = 60° ⇒ B_H = 2.5 × 10⁻⁵, B_V = 4.33 × 10⁻⁵ T.
  • Medium : B_H = B_V = 3.5 × 10⁻⁵ ⇒ δ = 45°, B_E = 4.95 × 10⁻⁵ T.
  • Exam (CET) : δ = 45° ⇒ B_V / B_H = ? (1) 1 (2) 0 (3) 2 (4) ∞ → answer 1) Trick : tan 45° = 1.

Rule 3: Dip circle — A dip circle measures δ; the needle is vertical at the poles and horizontal at the equator.

  • Easy : δ is small in India (near the equator).
  • Medium : Larger δ ⇒ larger B_V.
  • Exam (CET) : Needle at the poles? (1) vertical (2) horizontal (3) at 45° (4) spins → answer 1)

Derivation : tan δ = B_V / B_H

  • Step 1 : Resolve B_E : horizontal B_H, vertical B_V.
  • Step 2 : B_H = B_E cos δ, B_V = B_E sin δ.
  • Step 3 : Divide : B_V / B_H = tan δ.
  • Step 4 : Square and add : B_E = √(B_H² + B_V²).
Remember

B_H = B_E cos δ; B_V = B_E sin δ; tan δ = B_V/B_H.

Common mistake / CET trap

δ is 0 at the equator, 90° at the poles — do not swap.

Shortcut

Memory clue : "cos = horizontal, sin = vertical".

Spot the difference

PlaceδB_HB_V
Magnetic equator0Max0
Mid latitude0-90°MediumMedium
Magnetic pole90°0Max

Practice questions

Easy
  1. 1What is declination?
  2. 2What is dip?
  3. 3Formula for B_H?
Medium
  1. 4B_E = 6 × 10⁻⁵, δ = 30° : B_H?
  2. 5B_H = 3 × 10⁻⁵, B_V = 3 × 10⁻⁵ : δ?
  3. 6B_H = 3, B_V = 4 (units) : B_E?
Challenging
  1. 7δ = 60° : B_V / B_H?
  2. 8Why is a compass useless at the poles?
Show answers (try them yourself first)
  1. 1The angle between the geographic and magnetic meridians
  2. 2Angle of B_E with the horizontal
  3. 3B_E cos δ
  4. 45.2 × 10⁻⁵ T
  5. 545°
  6. 65
  7. 71.73
  8. 8B_H = 0

Magnetic maps and the Earth as a dipole

From daily life

A ship's captain corrects the compass using a map of declination.

Simple definition

The Earth is a large dipole; magnetic maps show lines of equal α, δ and B_H.

Earth as a dipoleN_gS←Ngeographic axismagnetic axis (~11°)tan δ = 2 tan λB_E = B₀ √(1 + 3 sin²λ)

In the figure : Magnetic axis tilted about 11°.

Rule 1: Earth is a dipole — Treat the Earth as a huge magnet : its S pole is near the geographic north, its N pole near the south; the axis is tilted about 11° from the geographic axis.

  • Easy : A compass N is drawn to the Earth's magnetic S pole.
  • Medium : m ≈ 8 × 10²² A m².
  • Exam (CET) : Near the geographic north the Earth's pole is? (1) S (2) N (3) both (4) none → answer 1)

Rule 2: Magnetic latitude — B_E = B₀ √(1 + 3 sin²λ); B₀ = (μ₀/4π) m/R³; tan δ = 2 tan λ.

  • Easy : B₀ = 10⁻⁷ × 8 × 10²² ÷ (6.4 × 10⁶)³ = 3.05 × 10⁻⁵ T; at the pole 2B₀ = 6.10 × 10⁻⁵ T.
  • Medium : λ = 45° ⇒ tan δ = 2 ⇒ δ = 63.4°.
  • Exam (CET) : λ = 0 ⇒ δ? (1) 0 (2) 90° (3) 45° (4) 63° → answer 1)

Rule 3: Magnetic maps — Isogonic lines (equal α), agonic line (α = 0), isoclinic lines (equal δ), aclinic line = magnetic equator.

  • Easy : Sailors correct for declination using such maps.
  • Medium : δ = 0 on the aclinic line.
  • Exam (CET) : The line of δ = 0? (1) aclinic (2) isogonic (3) agonic (4) isoclinic → answer 1)

Derivation : tan δ = 2 tan λ

  • Step 1 : Dipole : radial and tangential parts B_r = 2B₀ sin λ, B_θ = B₀ cos λ.
  • Step 2 : Vertical = B_r, horizontal = B_θ.
  • Step 3 : tan δ = B_r / B_θ.
  • Step 4 : tan δ = 2 tan λ.
Remember

B_E = B₀√(1 + 3sin²λ); tan δ = 2 tan λ; B₀ ≈ 3 × 10⁻⁵ T.

Common mistake / CET trap

λ is magnetic latitude, not geographic.

Shortcut

Memory clue : "tan δ = twice tan λ".

Spot the difference

LineMeaning
IsogonicEqual α
Agonicα = 0
IsoclinicEqual δ
Aclinicδ = 0 (magnetic equator)

Practice questions

Easy
  1. 1Earth's pole near geographic north?
  2. 2What is an isoclinic line?
  3. 3Formula for B₀?
Medium
  1. 4λ = 30° : δ?
  2. 5λ = 0 : B_E?
  3. 6λ = 90° : B_E?
Challenging
  1. 7Prove tan δ = 2 tan λ.
  2. 8m = 8 × 10²² : B₀?
Show answers (try them yourself first)
  1. 1Magnetic S
  2. 2A line joining places of equal dip
  3. 3(μ₀/4π) m/R³
  4. 449.1°
  5. 5B₀
  6. 62B₀
  7. 7Use B_r / B_θ
  8. 83.1 × 10⁻⁵ T

Quick revision of the whole lesson

ConceptKey formula / factUnit / dimensions
FluxΦ = BA cos θWb; M L² T⁻² A⁻¹
FieldBT; M T⁻² A⁻¹
Momentm = q_m 2l = NIAA m²; L² A
Axial(μ₀/4π) 2m/r³T
Equatorial(μ₀/4π) m/r³T
Torqueτ = mB sin θN m
EnergyU = −mB cos θJ
Gauss∮B·dA = 0Wb
EarthB_H = B_E cos δ; tan δ = 2 tan λT

🎯 Most important points for the exam

  • Φ = BA cos θ; lines are closed; 1 T = 10⁴ G.
  • m = q_m 2l = NIA, S → N.
  • Axial = 2 × equatorial; ∝ 1/r³.
  • τ = mB sin θ; U = −mB cos θ.
  • Neutral point : B_d = B_H.
  • ∮B·dA = 0 — no monopole.
  • B_H = B_E cos δ; tan δ = 2 tan λ.
Study mantra

B → Φ → m → axial/equatorial → τ, U → Gauss 0 → Earth (δ, B_H).