SECTION 5.1 – 5.2
The Bar Magnet
Introduction to Magnetism
Magnetism is universal in nature — from distant galaxies to tiny atoms. The word magnet is derived from Magnesia, a Greek island where magnetic ore was found around 600 BC.
- Earth behaves as a magnet — geographic south to north direction.
- A freely suspended bar magnet points north-south.
- Like poles repel; unlike poles attract.
- Magnetic monopoles do NOT exist — cutting a magnet gives two smaller magnets.
Magnetic Field Lines — Properties
- Field lines form continuous closed loops (unlike electric field lines).
- Tangent to field line gives direction of B⃗ at that point.
- Denser lines → stronger magnetic field.
- Field lines never intersect each other.
Fig 5.1 — Iron filings pattern around bar magnet (NCERT Figure 5.1)
Fig 5.2 — Field lines: bar magnet vs solenoid vs electric dipole (NCERT Figure 5.2)
Bar Magnet as Equivalent Solenoid
A bar magnet behaves like a solenoid — circulating currents (Ampere's hypothesis). The axial magnetic field at large distance r:
Axial Field of Bar Magnet (r >> l)
B = (μ₀ / 4π) × (2m / r³) — along axis
B = −(μ₀ / 4π) × (m / r³) — along equator
Torque: τ = m × B = mB sin θ
Potential Energy: U = −m·B = −mB cos θ
B = −(μ₀ / 4π) × (m / r³) — along equator
Torque: τ = m × B = mB sin θ
Potential Energy: U = −m·B = −mB cos θ
Dipole Analogy (Electric vs Magnetic)
- Replace: E → B, p → m, 1/4πε₀ → μ₀/4π
- Stable equilibrium: m parallel to B (θ = 0°), U = −mB (minimum)
- Unstable equilibrium: m anti-parallel to B (θ = 180°), U = +mB (maximum)
Fig 5.3(b) — Magnetic needle in uniform field showing torque (NCERT Figure 5.3)
🎯 Practice MCQs — Bar Magnet
Q1. A short bar magnet has magnetic moment m = 0.48 J T⁻¹. The magnetic field on its axial line at r = 10 cm from its centre is:
B_axial = (μ₀/4π)(2m/r³) = 10⁻⁷ × (2 × 0.48)/(0.1)³ = 10⁻⁷ × 0.96/10⁻³ = 0.96 × 10⁻⁴ T
Q2. A bar magnet of magnetic moment 1.5 J T⁻¹ is placed in a uniform field of 0.22 T. The work done to rotate it from stable to unstable equilibrium is:
W = U_final − U_initial = (+mB) − (−mB) = 2mB = 2 × 1.5 × 0.22 = 0.66 J
Q3. The ratio of the axial field to equatorial field of a short bar magnet at the same distance r >> l is:
B_axial = (μ₀/4π)(2m/r³); B_equatorial = (μ₀/4π)(m/r³). Ratio = 2 : 1
SECTION 5.3
Magnetism and Gauss's Law
Gauss's Law for Magnetism
Unlike electrostatics where net flux through a closed surface equals enclosed charge / ε₀, in magnetism:
- The net magnetic flux through any closed surface is always zero.
- This reflects the non-existence of magnetic monopoles.
- There are no sources or sinks of B⃗.
Gauss's Law for Magnetism
φ_B = ∮ B⃗ · ΔS⃗ = 0
Compare with Gauss's law (electrostatics):
∮ E⃗ · ΔS⃗ = q/ε₀ (net charge enclosed)
Compare with Gauss's law (electrostatics):
∮ E⃗ · ΔS⃗ = q/ε₀ (net charge enclosed)
Why is Magnetic Flux Always Zero?
- Magnetic field lines form closed loops — they exit and re-enter the same closed surface.
- No isolated N or S pole (monopole) exists, so no net outward or inward flux is possible.
- Even for a current-carrying solenoid or bar magnet, every field line that exits a surface re-enters it.
Fig 5.5 — Closed surface with area element ΔS and field B — Gauss's law (NCERT Figure 5.5)
Fig 5.6 — Correct and incorrect magnetic field line diagrams (NCERT Figure 5.6)
🎯 Practice MCQs — Gauss's Law of Magnetism
Q4. The net magnetic flux through any closed Gaussian surface is:
Gauss's law for magnetism: φ_B = ∮ B·dS = 0 always, because magnetic monopoles do not exist.
Q5. If magnetic monopoles existed, the net magnetic flux through a closed surface enclosing a monopole of magnetic charge q_m would be:
By analogy with Gauss's law of electrostatics (∮E·dS = q/ε₀), if monopoles existed: ∮B·dS = μ₀ q_m.
Q6. A closed surface encloses a bar magnet completely. The total magnetic flux through this surface is:
By Gauss's law for magnetism, the net magnetic flux through ANY closed surface is always zero, regardless of what is enclosed.
SECTION 5.4
Magnetisation and Magnetic Intensity
Magnetisation (M⃗)
Magnetisation is defined as the net magnetic moment per unit volume of the material:
Key Formulae
Magnetisation: M⃗ = m_net / V (units: A m⁻¹)
Magnetic field in material:
B⃗ = μ₀(H⃗ + M⃗)
Magnetic intensity (H⃗):
H⃗ = B⃗/μ₀ − M⃗
Magnetic susceptibility:
M⃗ = χ H⃗
Relative permeability:
μᵣ = 1 + χ and B = μ₀μᵣH = μH
Magnetic field in material:
B⃗ = μ₀(H⃗ + M⃗)
Magnetic intensity (H⃗):
H⃗ = B⃗/μ₀ − M⃗
Magnetic susceptibility:
M⃗ = χ H⃗
Relative permeability:
μᵣ = 1 + χ and B = μ₀μᵣH = μH
Important Terms Summary
- χ (Magnetic susceptibility): Dimensionless. Measures how a material responds to external field.
- μᵣ (Relative permeability): Dimensionless. μᵣ = 1 + χ
- μ (Magnetic permeability): μ = μ₀μᵣ — units same as μ₀ (T·m·A⁻¹)
- H⃗: Due to external sources (solenoid current). Units: A m⁻¹
- M⃗: Due to material's own nature. Units: A m⁻¹
Solved Example — Solenoid with Magnetic Core
A solenoid (μᵣ = 400, n = 1000 turns/m, I = 2A):
- H = nI = 1000 × 2 = 2000 A/m
- B = μᵣμ₀H = 400 × 4π×10⁻⁷ × 2000 = 1.0 T
- M = (μᵣ − 1)H = 399 × 2000 ≈ 8 × 10⁵ A/m
- Magnetising current I_M = 794 A
🎯 Practice MCQs — Magnetisation & Intensity
Q7. A solenoid has 500 turns/m and carries a current of 4A. The magnetic intensity H inside the solenoid is:
H = nI = 500 × 4 = 2000 A/m. H is independent of the core material.
Q8. A material has magnetic susceptibility χ = 399. Its relative magnetic permeability μᵣ is:
μᵣ = 1 + χ = 1 + 399 = 400
Q9. A solenoid with a ferromagnetic core (μᵣ = 500) has n = 800 turns/m and I = 3A. The magnetic field B inside is approximately:
H = nI = 800 × 3 = 2400 A/m
B = μᵣμ₀H = 500 × 4π×10⁻⁷ × 2400 ≈ 500 × 3.016×10⁻³ ≈ 1.508 T
B = μᵣμ₀H = 500 × 4π×10⁻⁷ × 2400 ≈ 500 × 3.016×10⁻³ ≈ 1.508 T
SECTION 5.5.1
Diamagnetism
Diamagnetic Materials
- Tendency to move from stronger to weaker part of external field → repelled by magnet.
- Susceptibility χ is small and negative (−1 ≤ χ < 0).
- μᵣ < 1; μ < μ₀
- Field lines are repelled/expelled from inside the material.
Cause of Diamagnetism
Orbiting electrons have orbital magnetic moments. In a diamagnetic substance, the net magnetic moment of an atom is zero without external field. When B is applied:
- Electrons with moment in same direction as B — slow down.
- Electrons with moment opposite to B — speed up.
- Net magnetic moment develops opposite to applied field → repulsion.
- This is governed by Lenz's law.
Examples and Special Cases
- Diamagnetic materials: Bismuth, Copper, Lead, Silicon, Water, NaCl, Nitrogen (STP)
- Superconductors: Perfect diamagnets — χ = −1, μᵣ = 0 (field completely expelled).
- The phenomenon of perfect diamagnetism in superconductors is called Meissner effect.
- Diamagnetism is universal (present in all substances) but usually masked by other effects.
Fig 5.7(a) — Diamagnetic bar in external field — field lines repelled (NCERT Figure 5.7a)
🎯 Practice MCQs — Diamagnetism
Q10. For a perfectly diamagnetic superconductor, the value of magnetic susceptibility χ is:
For a superconductor (perfect diamagnet): χ = −1, μᵣ = 0, μ = 0. The external field is completely expelled (Meissner effect).
Q11. A diamagnetic material placed in a non-uniform magnetic field will:
Diamagnetic materials develop a magnetic moment opposite to the applied field, so they are repelled and move towards the region of low field.
Q12. The relative magnetic permeability μᵣ of a diamagnetic substance is:
For diamagnetic: −1 ≤ χ < 0, so μᵣ = 1 + χ is between 0 and 1. (μ < μ₀)
SECTION 5.5.2 – 5.5.3
Paramagnetism & Ferromagnetism
Paramagnetism
- Weakly magnetised in external field; move from weak to strong field region.
- Susceptibility χ is small and positive (0 < χ < ε).
- Each atom has permanent magnetic dipole moment.
- Random thermal motion → no net magnetisation without field.
- With strong external field and low temperature → dipoles align with B.
- Examples: Aluminium, Sodium, Calcium, Oxygen (STP), Copper chloride.
- χ and μᵣ depend on material and temperature.
Fig 5.7(b) — Paramagnetic bar — field lines concentrated inside (NCERT Figure 5.7b)
Ferromagnetism
- Strongly magnetised; χ >> 1; μᵣ >> 1 (often >1000).
- Atoms in domains — each domain ~1 mm, ~10¹¹ atoms, with common alignment.
- Without external field: domains randomly oriented → no bulk magnetisation.
- With external field: domains align with B and grow in size.
- Hard ferromagnets: Retain magnetisation after field is removed (permanent magnets) — Alnico, Lodestone.
- Soft ferromagnets: Lose magnetisation when field is removed — Soft iron.
- Examples: Iron, Cobalt, Nickel, Gadolinium.
- Above Curie temperature: ferromagnet becomes paramagnetic.
Fig 5.8(a)(b) — Randomly oriented domains vs aligned domains in ferromagnet (NCERT Figure 5.8)
Classification Table — Magnetic Materials
Diamagnetic: −1 ≤ χ < 0 | 0 ≤ μᵣ < 1 | μ < μ₀
Paramagnetic: 0 < χ < ε | 1 < μᵣ < 1+ε | μ > μ₀
Ferromagnetic: χ >> 1 | μᵣ >> 1 | μ >> μ₀
Paramagnetic: 0 < χ < ε | 1 < μᵣ < 1+ε | μ > μ₀
Ferromagnetic: χ >> 1 | μᵣ >> 1 | μ >> μ₀
🎯 Practice MCQs — Para & Ferromagnetism
Q13. A ferromagnetic material has relative permeability μᵣ = 1000. Its magnetic susceptibility χ is:
χ = μᵣ − 1 = 1000 − 1 = 999
Q14. At temperatures above the Curie temperature, a ferromagnetic material becomes:
Above Curie temperature, the domain structure disintegrates and the material becomes paramagnetic.
Q15. The typical size of a magnetic domain in a ferromagnetic material is:
As per NCERT: each domain is typically 1 mm in size and contains about 10¹¹ atoms.
📋 Chapter Summary
🧲 Bar Magnet
Acts like solenoid. Axial field = μ₀2m/4πr³. Torque = mB sinθ. U = −mB cosθ.
🔮 Gauss's Law
Net magnetic flux through any closed surface = 0. No magnetic monopoles exist.
⚡ Magnetisation
M = m_net/V. B = μ₀(H+M). χ = M/H. μᵣ = 1+χ.
🔵 Diamagnetic
χ negative (–1 to 0). Repelled. Superconductors: χ = –1 (Meissner effect).
🟡 Paramagnetic
χ small positive. Weakly attracted. Permanent dipole moments. Temperature dependent.
🔴 Ferromagnetic
χ >> 1. Domains. Hard (permanent) vs soft (temporary). Becomes paramagnetic above Curie temp.
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