Magnetism and electromagnetic induction

In short

Magnetism deals with the fields produced by moving charges and currents, the forces they exert, and how matter responds to them; electromagnetic induction deals with the emf generated whenever magnetic flux through a circuit changes. Together they explain motors, generators, transformers, inductors, induction heating, metal detectors, MRI and magnetic storage. In B.Sc Electricity and Magnetism papers this section carries the heaviest weight after electrostatics, and it is standard in GATE (PH), JAM, CSIR-NET, JEST, TIFR and in technical rounds for BHEL, BEL, DRDO, ISRO, NPCIL, CSIR labs, transformer and motor manufacturers, and EDA/semiconductor-test roles. The mathematics is vector calculus — line and surface integrals, curl and divergence — applied to B, H and M.

Explained

Magnetic field of steady currents

• Source of B: moving charge. There are no magnetic monopoles, so ∇·B = 0 everywhere and magnetic field lines close on themselves. Flux through any closed surface is zero.
• Biot-Savart law: each current element I dl contributes dB = (μ₀/4π) I dl × r̂ / r². μ₀ = 4π × 10⁻⁷ T·m/A (SI, exact in the pre-2019 definition; now fixed by the measured value of the fine-structure constant, μ₀ ≈ 1.25663706 × 10⁻⁶ T·m/A).
• Standard results: infinite straight wire B = μ₀I/2πr (circular lines, right-hand rule); circular loop on axis B = μ₀IR²/2(R²+z²)^{3/2}; long solenoid inside B = μ₀nI (n = turns per metre), ends ≈ half that; toroid B = μ₀NI/2πr inside, zero outside.
• Ampère's circuital law: ∮B·dl = μ₀I_enc, differential form ∇ × B = μ₀J. Valid for steady currents; Maxwell's displacement-current correction belongs to the electromagnetic-waves topic.
• Vector potential: B = ∇ × A, with A = (μ₀/4π)∫ J dV/r in Coulomb gauge (∇·A = 0). Useful because A is a single vector equation and because flux Φ = ∮A·dl.
• Magnetic scalar potential exists only in current-free regions, since ∇ × B ≠ 0 where current flows.

Forces, torques and magnetic dipoles

• Lorentz force: F = q(E + v × B). The magnetic part does no work (always ⊥ v), so it changes direction, not kinetic energy.
• Circular motion: r = mv/qB, cyclotron frequency ω = qB/m (independent of speed — the basis of the cyclotron), period T = 2πm/qB. With a velocity component along B the path is a helix; pitch = v∥T.
• Force on a current: dF = I dl × B; on a straight wire F = BIL sinθ. Two parallel wires carrying currents in the same direction attract, with F/L = μ₀I₁I₂/2πd — the old definition of the ampere.
• Magnetic dipole moment: m = IA (A·m²); for a planar coil of N turns, m = NIA. Torque τ = m × B, energy U = −m·B, force in a non-uniform field F = ∇(m·B).
• Dipole field: B = (μ₀/4π)[3(m·r̂)r̂ − m]/r³; on-axis B_ax = μ₀·2m/4πr³, equatorial B_eq = μ₀m/4πr³, so B_ax = 2B_eq. Tangent galvanometer and deflection-magnetometer experiments (Tan A and Tan B positions) use exactly these.
• Hall effect: a transverse field on a current-carrying slab gives V_H = IB/nqt; the sign of V_H gives the carrier sign, used in Hall probes and gaussmeters.
• Gyromagnetic ratio and Bohr magneton: μ_B = eħ/2m_e = 9.274 × 10⁻²⁴ A·m²; orbital m = −(e/2m_e)L, spin m = −g_s(e/2m_e)S with g_s ≈ 2 — the microscopic origin of magnetism in matter.

Magnetic materials

• Magnetisation M = dipole moment per unit volume (A/m). Bound currents: J_b = ∇ × M (volume), K_b = M × n̂ (surface).
• Auxiliary field H = B/μ₀ − M (A/m), with ∮H·dl = I_free. For linear media M = χ_mH, B = μ₀(1+χ_m)H = μ₀μ_rH = μH.
• Diamagnetic: χ_m small and negative (≈ −10⁻⁵), from induced orbital currents opposing B (Langevin diamagnetism); bismuth, copper, water, superconductors (χ_m = −1, perfect diamagnet, Meissner effect). Field-independent and nearly temperature-independent.
• Paramagnetic: χ_m small and positive (10⁻⁵–10⁻³), from permanent moments partially aligned against thermal disorder. Curie law χ = C/T; Langevin function M = NμL(x), x = μB/k_BT; aluminium, oxygen, salts of transition metals.
• Ferromagnetic: χ_m very large (10²–10⁵), non-linear, with domains, hysteresis and remanence; Fe, Co, Ni, and alloys. Above the Curie temperature T_C (Fe 1043 K, Co 1388 K, Ni 627 K) it becomes paramagnetic obeying the Curie–Weiss law χ = C/(T − T_C). Weiss molecular-field theory explains spontaneous alignment; exchange interaction is its quantum origin.
• Antiferromagnetic (MnO, below Néel temperature, adjacent moments antiparallel and cancelling) and ferrimagnetic (ferrites, antiparallel but unequal sublattices, net moment with high resistivity — hence ferrite cores).
• Hysteresis loop: B (or M) versus H shows saturation B_s, remanence B_r (B at H = 0) and coercivity H_c (H needed to bring B to zero). Loop area = energy lost per unit volume per cycle (J/m³).
• Soft versus hard: soft magnetic materials (silicon steel, ferrites, amorphous/nanocrystalline alloys) have narrow loops and low loss — used in transformer and motor cores; hard materials (alnico, ferrite magnets, NdFeB, SmCo) have wide loops and large H_c — used as permanent magnets. Transformer lamination grades and test methods are covered by the Indian Standards on cold-rolled grain-oriented (CRGO) electrical steel; BEE star-labelling under the Energy Conservation Act sets minimum distribution-transformer loss levels in India.

Electromagnetic induction

• Flux: Φ = ∫B·dA (weber, Wb = T·m²).
• Faraday's law: emf = −dΦ/dt; for N turns, emf = −N dΦ/dt. Differential form ∇ × E = −∂B/∂t, so an induced E is non-conservative — ∮E·dl ≠ 0 and "potential" is not defined for it.
• Lenz's law: the induced current opposes the change in flux — a statement of energy conservation; the minus sign is not arbitrary.
• Two mechanisms: (i) motional emf, conductor moving in a static field, explained by the v × B force (emf = Blv for a rod of length l moving ⊥ to B); (ii) transformer emf, stationary circuit in a time-varying field, explained by the induced E. Faraday's law covers both; relativity shows they are one phenomenon in different frames.
• Rotating coil (AC generator): Φ = NBA cos ωt → emf = NBAω sin ωt, peak emf = NBAω. Indian supply is 50 Hz; phase-to-phase LV 415 V, phase-to-neutral 240 V nominal (IS 12360 / the Electricity Rules permit ±6% voltage variation at the consumer end for LV supply).
• Self-inductance: Φ_link = LI, emf = −L dI/dt; L in henry. Solenoid L = μ₀N²A/l; coaxial cable per unit length L′ = (μ₀/2π)ln(b/a); toroid L = μ₀N²h ln(b/a)/2π.
• Mutual inductance: Φ₂ = M₂₁I₁, emf₂ = −M dI₁/dt, and reciprocity M₁₂ = M₂₁ (a standard exam proof via the double line integral / Neumann formula). Coupling coefficient k = M/√(L₁L₂), 0 ≤ k ≤ 1.
• Energy: U = ½LI², or U = ½L₁I₁² + ½L₂I₂² ± MI₁I₂ for coupled coils; energy density u = B²/2μ₀ = ½B·H.
• Eddy currents: circulating currents induced in bulk conductors; loss ∝ B_m²f²t² (t = lamination thickness) and ∝ 1/ρ. Reduced by thin laminations and high-resistivity cores; exploited in induction heating, induction furnaces, eddy-current brakes, metal detectors, induction cooktops and non-destructive testing.
• Hysteresis loss: P_h = k_h f B_mⁿ per unit volume (Steinmetz, n ≈ 1.6–2). Core loss = hysteresis + eddy current; copper loss = I²R.
• Transformer: V₂/V₁ = N₂/N₁, I₂/I₁ ≈ N₁/N₂ for an ideal unit; works only on AC because it needs changing flux. Efficiency of large power transformers exceeds 99%.
• LR circuit: growth I = (V/R)(1 − e^{−t/τ}), decay I = I₀e^{−t/τ}, τ = L/R. An inductor opposes change in current, so current cannot jump — breaking an inductive circuit causes a large −L dI/dt spike (arcing; hence freewheel diodes).
• LC and LCR: natural frequency ω₀ = 1/√(LC); resonance, Q = ω₀L/R. Reactance X_L = ωL, impedance Z = √(R² + (X_L − X_C)²).
• Magnetic circuits (analogy): mmf F = NI (ampere-turns) ↔ emf; flux Φ ↔ current; reluctance S = l/μA ↔ resistance; Φ = NI/S. Reluctances add in series. Air gaps dominate because μ_air is thousands of times smaller than μ_core; fringing and saturation break the analogy.
• Betatron/induction accelerators, Faraday disc (homopolar generator), and the AC generator/motor pair are the examined applications. The motor–generator duality: back-emf in a motor is Faraday's law acting on its own rotation.

Measurement, units and standards

• SI units: B in tesla (1 T = 10⁴ gauss), H and M in A/m, Φ in weber, L and M in henry, μ₀ in T·m/A, χ_m dimensionless (SI volume susceptibility; CGS values differ by 4π — a classic trap).
• Typical magnitudes: Earth's field 25–65 μT (≈ 40–47 μT over most of India, with dip angle rising from near 0° at the magnetic equator through central India to ~45–50° in the north); a fridge magnet ≈ 5 mT; transformer core 1.2–1.7 T; MRI 1.5–3 T clinically; superconducting lab magnets 10–20 T.
• Instruments: search coil with fluxmeter/ballistic galvanometer, Hall probe gaussmeter, vibrating-sample magnetometer, SQUID (highest sensitivity), B–H curve tracer, Epstein frame for electrical-steel loss measurement.
• Indian practice: NPL India (CSIR-NPL, New Delhi) maintains national electrical standards; the Indian Institute of Geomagnetism (Navi Mumbai) runs geomagnetic observatories; transformer testing follows the IS 2026 series (power transformers) and loss grading under BEE labelling. Where an IS number may have been revised, verify the current edition before quoting it.

Key points

• No magnetic monopoles — ∇·B = 0; magnetic field lines are closed loops and total flux through any closed surface is zero.
• Biot–Savart law — gives B from a current element; the magnetic analogue of Coulomb's law but with a cross product.
• Ampère's law — ∮B·dl = μ₀I_enc; fast route to B for straight wires, solenoids, toroids and planar sheets with symmetry.
• Lorentz force — F = q(E + v × B); the magnetic part does zero work, so speed is unchanged.
• Cyclotron frequency — ω = qB/m, independent of speed and radius (until relativistic speeds).
• Dipole moment — m = NIA; τ = m × B, U = −m·B; axial field is twice the equatorial field at equal distance.
• H versus B — H is sourced by free current only (∮H·dl = I_free); B includes the material's contribution through M.
• Susceptibility — χ_m < 0 diamagnetic, small positive paramagnetic, large and non-linear ferromagnetic; χ = C/T (Curie), C/(T−T_C) (Curie–Weiss).
• Curie temperature — above it ferromagnetism vanishes; Fe 1043 K, Ni 627 K, Co 1388 K.
• Hysteresis — loop area is energy loss per cycle per unit volume; coercivity and remanence distinguish hard from soft magnets.
• Faraday's law — emf = −N dΦ/dt; flux can change through B, area or orientation.
• Lenz's law — the induced effect opposes the cause; it is energy conservation, not a separate rule.
• Induced E is non-conservative — ∇ × E = −∂B/∂t, so no scalar potential for it.
• Self and mutual inductance — L from a coil's own flux linkage, M between coils with M₁₂ = M₂₁ and k = M/√(L₁L₂) ≤ 1.
• Magnetic energy — U = ½LI², density u = B²/2μ₀; this is where an inductor stores energy.
• Eddy-current loss ∝ B_m²f²t² — the reason cores are laminated or made of ferrite.
• LR time constant — τ = L/R; current reaches 63.2% of final value in one τ.
• Magnetic circuit — Φ = NI/S with S = l/μA; air gaps dominate reluctance.

Formulas and examples

Example 1 — Force between parallel conductors (busbar spacing). Two long parallel busbars 0.20 m apart each carry 2,000 A in the same direction. Find the force per metre.

• F/L = μ₀I₁I₂/2πd = (4π × 10⁻⁷ × 2000 × 2000)/(2π × 0.20)
• Numerator: 4π × 10⁻⁷ × 4 × 10⁶ = 16π × 10⁻¹ = 5.0265 N·m/A²·... ; dividing by 2π × 0.20 = 1.2566
• F/L = 5.0265/1.2566 ≈ 4.0 N/m, attractive. Under a 20× short-circuit current the force scales by 400 → about 1,600 N/m, which is why switchgear busbar supports are designed for fault forces.

Example 2 — Solenoid inductance and stored energy. A solenoid has N = 500 turns, length l = 0.25 m, cross-section A = 4.0 cm² = 4.0 × 10⁻⁴ m², air core, carrying 3 A.

• n = 500/0.25 = 2,000 turns/m → B = μ₀nI = 4π × 10⁻⁷ × 2000 × 3 = 7.54 × 10⁻³ T.
• L = μ₀N²A/l = (4π × 10⁻⁷ × 250000 × 4.0 × 10⁻⁴)/0.25 = (1.2566 × 10⁻⁶ × 100)/0.25 ... compute: 250000 × 4.0 × 10⁻⁴ = 100; 1.2566 × 10⁻⁶ × 100 = 1.2566 × 10⁻⁴; ÷0.25 → L = 5.03 × 10⁻⁴ H (0.503 mH).
• U = ½LI² = 0.5 × 5.03 × 10⁻⁴ × 9 = 2.26 × 10⁻³ J.
• Check by energy density: u = B²/2μ₀ = (7.54 × 10⁻³)²/(2 × 1.2566 × 10⁻⁶) = 5.685 × 10⁻⁵/2.513 × 10⁻⁶ = 22.6 J/m³; volume = 4.0 × 10⁻⁴ × 0.25 = 1.0 × 10⁻⁴ m³ → U = 2.26 × 10⁻³ J. ✔ Same answer.
• If an iron core of μ_r = 1,000 is inserted, L becomes ≈ 0.503 H — a thousand-fold rise, which is why real inductors and transformers use cores.

Example 3 — Motional emf on a rail. A 0.80 m conducting rod slides at 5 m/s on frictionless rails in a vertical field B = 0.50 T; the circuit resistance is 2.0 Ω.

• ε = Blv = 0.50 × 0.80 × 5 = 2.0 V.
• I = ε/R = 2.0/2.0 = 1.0 A.
• Retarding force F = BIl = 0.50 × 1.0 × 0.80 = 0.40 N; mechanical power = Fv = 0.40 × 5 = 2.0 W = I²R = 1² × 2.0 = 2.0 W. ✔ All the mechanical work appears as heat — Lenz's law as energy conservation.

Example 4 — LR circuit transient. L = 0.40 H, R = 20 Ω, V = 12 V, switch closed at t = 0.

• τ = L/R = 0.40/20 = 0.020 s; final current I_f = 12/20 = 0.60 A.
• At t = 0.020 s: I = 0.60(1 − e⁻¹) = 0.60 × 0.6321 = 0.379 A.
• At t = 0.060 s (3τ): I = 0.60(1 − e⁻³) = 0.60 × 0.9502 = 0.570 A (95%).
• Energy finally stored = ½ × 0.40 × 0.60² = 0.072 J. Initial rate of rise dI/dt = V/L = 30 A/s.

Example 5 — Hysteresis loss in a transformer core. A core of volume 0.015 m³ has a hysteresis loop of area 320 J/m³ per cycle, run at 50 Hz.

• Loss per cycle = 320 × 0.015 = 4.8 J → P_h = 4.8 × 50 = 240 W.
• Over 24 h: 240 × 24 = 5,760 Wh = 5.76 kWh/day. At an industrial tariff of ₹8/kWh that is about ₹46/day, roughly ₹16,800/year — the commercial reason CRGO steel and BEE star-rated transformers are specified.
• If the frequency were 60 Hz with the same loop, P_h = 4.8 × 60 = 288 W (hysteresis loss ∝ f, eddy loss ∝ f²).

Where it is used

• Power equipment manufacturing: transformers, motors and generators at BHEL, Crompton, Kirloskar, Siemens India, ABB India — core-loss design, CRGO lamination selection, insulation and short-circuit force calculation.
• Electric mobility and drives: PMSM and induction motors for Tata, Mahindra, Ather, Ola and TVS EVs — NdFeB magnet grades, back-emf constants, eddy-current loss in rotors.
• Research and defence labs: DRDO, ISRO, BARC, IPR Gandhinagar (tokamak and superconducting magnet coils), IUAC Delhi (accelerator magnets), CSIR-NPL magnetic metrology; materials roles in VSM/SQUID magnetometry.
• Medical and instrumentation: MRI magnets and gradient coils (installed and serviced across Indian hospitals; GE, Siemens, Wipro GE), Hall-effect current sensors, fluxgate magnetometers, ECG shielding.
• Industrial heating and NDT: induction furnaces in steel re-rolling mills, induction hardening, magnetic-particle and eddy-current testing of welds, rails and boiler tubes (an NDT Level-II skill in demand at L&T, BHEL, Indian Railways).
• Geophysics and exploration: Indian Institute of Geomagnetism observatories, GSI and ONGC magnetic surveys, magnetometer-based mineral prospecting.
• Teaching and testing roles: B.Sc/M.Sc lecturing, PSU technical exams, and electronics design roles where inductor selection, EMI filtering and transformer sizing are everyday work.

Interview questions

• Q: Why does the magnetic part of the Lorentz force do no work? A: Because F = qv × B is always perpendicular to v, so F·v = 0 and the kinetic energy is unchanged — only the direction of motion changes.
• Q: What is the physical meaning of the minus sign in Faraday's law? A: It is Lenz's law: the induced emf drives a current whose flux opposes the change that created it, so energy must be supplied to sustain the change — otherwise you would have a perpetual-motion machine.
• Q: Difference between B and H? A: B is the total magnetic flux density including the material's magnetisation; H is defined so that its circulation depends only on free current. In vacuum B = μ₀H; in matter B = μ₀(H + M).
• Q: Why can a transformer not work on DC? A: A transformer needs dΦ/dt; steady DC gives constant flux and zero induced emf, and the primary draws only I = V/R, which usually burns the winding.
• Q: Why are transformer and motor cores laminated? A: Eddy-current loss scales as the square of the lamination thickness, so thin insulated laminations break up the induced current paths and cut that loss drastically.
• Q: Why is the cyclotron frequency independent of particle speed, and when does that fail? A: r and v both scale with momentum so ω = qB/m cancels the speed; it fails at relativistic speeds where m → γm, which is why synchrocyclotrons and synchrotrons vary frequency or field.
• Q: Is the induced electric field conservative? A: No — ∇ × E = −∂B/∂t is non-zero, so ∮E·dl ≠ 0 and no single-valued scalar potential exists for it; a voltmeter reading then depends on lead routing.
• Q: What does the area of a hysteresis loop represent, and what material do you want for a transformer? A: Energy dissipated per unit volume per cycle; a transformer wants a soft material with a narrow loop and low coercivity, such as CRGO silicon steel or ferrite at high frequency.
• Q: Why does a magnet dropped through a copper pipe fall slowly, when copper is not magnetic? A: The changing flux induces eddy currents in the pipe wall; by Lenz's law their field opposes the magnet's motion, producing a retarding force — the principle of eddy-current braking.
• Q: Prove or explain why M₁₂ = M₂₁. A: The Neumann double line integral for mutual inductance is symmetric in the two circuit paths, so the flux linked in coil 2 per unit current in coil 1 equals the reverse; it also follows from the energy ½L₁I₁² + ½L₂I₂² + MI₁I₂ being a well-defined state function.
• Q: What happens if you suddenly open a switch in a highly inductive circuit? A: dI/dt is very large, so −L dI/dt produces a high voltage spike that arcs across the contacts; freewheeling diodes, snubbers or varistors are used to give the current a path.
• Q: How would you measure an unknown magnetic field in the lab? A: A Hall probe gaussmeter for steady fields, a search coil with a fluxmeter or ballistic galvanometer for pulsed or AC fields, and a VSM or SQUID when measuring a sample's magnetisation rather than the field itself.
• Q: Why do air gaps dominate a magnetic circuit's reluctance? A: S = l/μA, and μ_air is of the order of a thousand times smaller than μ_core, so even a millimetre of gap can exceed the reluctance of a long iron path.

Common mistakes

• Treating H as "the field in vacuum" and B as "the field in matter" — both exist everywhere; the real distinction is free current versus total current as the source.
• Mixing SI and CGS susceptibility (they differ by 4π) or quoting B in gauss while using μ₀ in SI units.
• Using Ampère's law where there is no symmetry, or forgetting to count the enclosed current with its sign and the number of turns.
• Dropping the minus sign or the N in ε = −N dΦ/dt, and ignoring that Φ can change through B, A or the angle between them.
• Assuming induced emf needs relative motion — a stationary coil in a time-varying field also has emf; and assuming an induced electric field has a potential.
• Writing L = μ₀N²A/l for a short solenoid or an iron core without the end correction or μ_r, and confusing turns N with turns per metre n.
• Confusing hysteresis loss (∝ f) with eddy-current loss (∝ f²), or claiming laminating reduces hysteresis loss.
• Saying a magnetic force can change a charged particle's speed, or forgetting that the magnetic force on a stationary charge is zero.

Related topics

Electrostatics · Maxwell's equations and electromagnetic waves · AC circuits and resonance · Electrical machines and transformers · Solid state physics: magnetic properties of materials · Special relativity and field transformations