Notes on Electromagnetic Induction and Alternating Current (Unit 4)
4.1 Introduction
Electric currents produce magnetic fields (Oersted).
A current-carrying loop behaves like a magnet (Ampere).
Question: can magnetic fields produce electric currents? Yes — Faraday and Henry discovered electromagnetic induction independently (1831, UK/USA).
Faraday credited with discovery of electromagnetic induction; premise: changing magnetic flux linked with a closed circuit induces an emf and hence current.
Faraday’s experiments underpin the phenomena of induction, mutual induction, self-induction, and their applications (generators, transformers).
Notable quote highlighting experimental spirit: Faraday’s remark about the use of a newborn baby as an analogy for discovery.
4.1 Magnetic Flux (Φ_B)
Magnetic flux through area A in a magnetic field B is the number of field lines passing through normally:
\PhiB = \intA \mathbf{B} \cdot d\mathbf{A}If B is uniform and perpendicular to A, the flux becomes
\Phi_B = B A \cos\theta
where θ is the angle between B and the outward normal to the area.For a uniform field perpendicular to area (θ = 0°), Φ_B = B A.
SI unit: weber (Wb) = T·m².
Example 4.1: A circular antenna of area A = 3 m² has magnetic field magnitude B = 4.1×10⁻⁵ T with angle θ = 43°. Flux linked:
\Phi_B = B A \cos\theta = (4.1\times 10^{-5})\times 3\times \cos 43^{\circ} \approx 89.96\ \text{mWb}
4.1.2 Magnetic Flux (FB)
Flux through area A in a magnetic field: same definition as above.
θ is the angle between the magnetic field and the area’s outward normal.
Note the terminology in this chapter uses Φ_B for flux through a surface.
4.1.3 Faraday’s Experiments on Electromagnetic Induction
First experiment (Fig. 4.2): coil C, galvanometer G. Movement of a magnet relative to coil causes a transient current in the coil (galvanometer deflection). If the magnet approaches, flux through coil increases → induced emf and current in a given direction.
If magnet is moved away, flux decreases → induced emf in opposite direction.
Change in current in the primary circuit of a coupled system induces a current in a secondary circuit via changing magnetic flux linking the secondary.
Observations summarized: a changing magnetic flux linked with a closed circuit induces emf; the emf magnitude depends on the rate of change of flux.
Example 4.2 and 4.3 illustrate flux through a loop and rotation/motion of coil producing different flux linkages and thus emf.
Faraday’s qualitative idea: a changing flux through a coil links an emf; motion of magnet or change in current in nearby coils changes flux through secondary.
4.1.4 Faraday’s Law of Electromagnetic Induction
When the magnetic flux linked with a closed circuit changes, an emf is induced in the circuit.
The induced emf is proportional to the time rate of change of magnetic flux:
\mathcal{E} = -\frac{d\Phi_B}{dt}If the coil has N turns and each turn sees the same flux ΦB, the total emf in the coil is \mathcal{E} = -N\frac{d\PhiB}{dt}
Flux linkage is defined as the product of the number of turns and the flux through each turn:
\mathcal{N}\PhiB = N\PhiBThus Faraday’s law with flux linkage: \mathcal{E} = -\frac{d(N\PhiB)}{dt} = -N\frac{d\PhiB}{dt}
4.1.5 Lenz’s Law
Lenz’s law: the direction of the induced current is such that it opposes the cause that produced it (i.e., opposes the change in flux).
The law is consistent with conservation of energy.
Combined with Faraday’s law, the emf can be written with a negative sign to indicate opposition to the change in flux:
\mathcal{E} = -\frac{d(N\Phi_B)}{dt}.Illustrations (Fig. 4.6–4.7) show opposing directions of induced current when flux increases or decreases or when a magnet moves toward/away from a coil.
Energy conservation rationale: moving a magnet against the induced emf requires external work; if the emf aided the motion, it would violate energy conservation (no perpetual motion).
4.1.6 Fleming’s Right-Hand Rule (Generator Rule)
When a conductor moves in a magnetic field, the direction of motion, field, and induced current are related by Fleming’s right-hand rule:
Thumb: direction of motion of the conductor.
Forefinger (index): direction of the magnetic field (B).
Middle finger: direction of the induced current (I).
The three mutually perpendicular directions give the induced current direction for a given motion through a magnetic field.
4.1.7 Motional emf from Lorentz Force
A moving conductor in a magnetic field experiences a Lorentz force on charges: for electrons, F = −e(v × B).
Charge separation due to v × B establishes an electric field E until equilibrium is reached; the induced emf across a rod of length l moving with velocity v perpendicular to B is
\mathcal{E} = B\,l\,vIf the rod ends are connected in a circuit of resistance R, current is
i = \frac{\mathcal{E}}{R} = \frac{B l v}{R}Direction of current from the induced emf is given by the right-hand rule (or Fleming’s rule).
Motional emf is produced only when there is relative motion between the conductor and the magnetic field.
4.2 EDDY CURRENTS
When a conductor (not necessarily a closed loop) experiences a change in magnetic flux, currents induced in the conductor flow in closed loops called eddy currents (Foucault currents).
Example: a conducting disc or plate in a changing magnetic field, or a pendulum disc swinging in a magnetic field while electromagnetic damping occurs.
Demonstrations include: a pendulum in a strong magnet demonstrating damping due to eddy currents in the disc; laminating transformer cores to reduce eddy currents; eddy currents in rotors and brake discs.
Uses of eddy currents include: induction stove (hence high-frequency field induces currents in the pan), eddy-current brakes in trains, eddy-current testing for nondestructive testing, electromagnetic damping.
Drawbacks: energy loss as heat; efficiency is reduced if eddy currents are not minimized.
Methods to reduce losses: laminated cores (thin insulated laminations) for transformers; laminated motor iron cores; use of slots to interrupt circular current paths.
4.3 SELF-INDUCTION
Inductor: a device (coil, solenoid, toroid) that stores energy in a magnetic field when current flows through it.
Self-induction means a changing current in a coil induces an emf in the same coil; the induced emf opposes the change in current (Lenz’s law).
Flux linkage for a coil with N turns: N\PhiB is proportional to the current i in the coil: N\PhiB = L i
Self-inductance L is defined as the proportionality constant between the flux linkage and the current:
N\PhiB = L i\quad\Rightarrow\quad L = \frac{N\PhiB}{i}For a long solenoid, with turn density n = N/l and cross-sectional area A, the magnetic field inside is
B = \mu_0 n iThe total flux linkage is
N\PhiB = N (B A) = N (\mu0 n i) A = \mu0\frac{N^2 A}{l} i hence inductance L = \frac{\mu0 N^2 A}{l}Energy stored in an inductor is
U = \tfrac{1}{2} L i^2Energy density in magnetic field inside an inductor/solenoid:
u = \frac{U}{Al} = \frac{B^2}{2\mu_0}
4.3.2 Self-Inductance of a Long Solenoid
General result for a long solenoid: if it has n turns per unit length, cross-sectional area A, and length l, then
L = \frac{\mu0 N^2 A}{l} = \mu0 n^2 A lIf a material with relative permeability μr fills the core, then L = \frac{\mu0\mur N^2 A}{l} = \mu0\mu_r\frac{N^2 A}{l}
The inductance increases with more turns, larger cross-section, and higher permeability; it decreases with longer length.
Energy stored and the role of inductance in opposing current changes are discussed with the analogy to inertia in mechanical systems.
4.3.3 Mutual Induction
When current in one coil (coil 1) changes with time, emf is induced in a neighboring coil (coil 2).
If the flux through coil 2 due to current in coil 1 is Φ21 per turn, the total flux through coil 2 is N2 Φ21, and the induced emf in coil 2 is
\mathcal{E}2 = -N2 \frac{d\Phi{21}}{dt} = -M \frac{di1}{dt}The proportionality constant M is the mutual inductance, defined as the flux linkage in coil 2 per unit current change in coil 1:
M = \frac{N2 \Phi{21}}{i_1}Example: If i1 changes from 2 A to 6 A in 0.04 s in a coplanar pair of coils A and B, the induced emf in B is ε_B = -M (di1/dt) with M = 0.06 H (example from the text).
For two long, coaxial solenoids of N1 and N2 turns, cross-sectional areas A1, A2, length l, flux linkages satisfy:
\mathcal{E}2 = -M \frac{di1}{dt},\quad M = \frac{\mu0 \; n1 n2 Al}{?}
In standard form for coaxial solenoids with large length and uniform B, the mutual inductance is
M = \frac{\mu0 N1 N2 A}{l} = \mu0 \mur \frac{N1 N_2 A}{l}General mutual-inductance relation for two long coaxial solenoids:
M = \dfrac{\mu0 N1 N2 A}{l} (with possible μr if a core is present)Important property: M12 = M21 (mutual inductance is symmetric).
The unit of mutual inductance is the henry (H).
Equations that relate mutual inductance to emf in the second coil and the rate of change of current in the first: ε2 = -M di1/dt; ε1 = -M di2/dt.
4.3.4 Mutual Inductance for Two Long Coaxial Solenoids (General Case)
If two long solenoids have areas A1, A2, turn densities n1, n2, length l, currents i1, i2, the mutual inductance can be expressed as
M = \frac{\mu0 n1 n2 A1 A2 l}{?} (Note: the text provides a detailed derivation showing dependence on turns, areas and length; the standard compact form for tightly coupled, coaxial solenoids is M = \dfrac{\mu0 N1 N2 A}{l}
for uniform cross-section and full coupling; with μr, replace μ0 by μ0 μr.)In general, mutual inductance depends on size, shape, number of turns, relative orientation, and medium permeability.
4.4 METHODS OF PRODUCING INDUCED EMF
Induced emf, ε, results from changing magnetic flux ΦB through a circuit; the four primary ways are: 1) By changing the magnetic field B (varying the field strength or using an electromagnet): ε ∝ dΦB/dt.
2) By changing area A of the coil relative to the field (sliding a rod to change loop area): ε ∝ d/dt (B A).
3) By changing the relative orientation θ between coil and magnetic field (rotating coil): ΦB = B A cosθ, hence ε ∝ d/dt (cosθ). 4) By altering the number of turns N (in a coil, via flux linkage NΦB).Production by changing magnetic field (4.4.2): first experiment (moving magnet toward/away) and second experiment (changing current in adjacent coil, changing flux through secondary).
Production by changing area (4.4.3): a conducting rod of length l moving in a frame changes the enclosed area, hence flux; induced emf ε = d/dt (Φ_B) with ε = B l v in the “motional” case (area change at constant B).
Production by changing relative orientation (4.4.4): a coil rotating in a uniform magnetic field; emf varies as the coil turns, producing sinusoidal emf. The instantaneous emf for a rectangular loop of N turns rotates with angular velocity ω:
\mathcal{E}(t) = -N \frac{d}{dt}[B A \cos(\omega t)] = -N B A (-\omega \sin(\omega t)) = N B A \omega \sin(\omega t).
4.4.1 Faraday’s Law in Production of EMF
The magnitude of induced emf when the flux changes: \mathcal{E} = -\dfrac{d\PhiB}{dt}. For N turns: \mathcal{E} = -N \dfrac{d\PhiB}{dt}.
In cases where flux ΦB depends on B, A, θ: ΦB = B A cosθ, so
\mathcal{E} = -N \dfrac{d}{dt}[B A \cos\theta].In practical devices, Faraday’s law underpins generators and transformers.
4.5 AC GENERATOR (ALTERNATOR)
4.5.1 Introduction
An AC generator converts mechanical energy to electrical energy using electromagnetic induction.
Generators are used for large-scale power generation.
4.5.2 Principle
A relative motion between a conductor and a magnetic field induces an emf; thus, AC is generated when the magnetic flux linked with a stationary coil varies due to a rotating field or rotating coil.
4.5.3 Construction
Main parts: stator (stationary) and rotor (rotating).
Stator: core + armature windings. Winding is laminated to reduce eddy-current losses.
Rotor: field magnet windings; poles magnetized to provide the rotating magnetic field.
In standard designs, armature windings are on the stator, field magnets on rotor; slip rings transfer current to/from the rotating field windings.
4.5.4 Advantages of stationary armature – rotating field system
High current and high voltage generation with fixed terminal access on stator, fewer slip rings, robust winding, and easier insulation.
4.5.5 Single-phase AC generator
A single-turn rectangular loop PQRS on the stator; a field magnet rotates; ends of field windings connect to slip rings, brushes provide DC supply to field windings. The emf in the loop is sinusoidal as the magnet rotates; the three-phase generator extends this concept to three windings spaced 120° apart.
4.5.6 Poly-phase AC generators
Generators may have multiple armature windings; two-phase or three-phase outputs with phase separation (e.g., 120° apart for three-phase).
4.5.7 Three-phase AC generator
Armature core with 6 slots spaced 60° apart; conductors form three coils separated by 120°; the stator produces three simultaneous sinusoidal emf waveforms 120° apart.
Advantages of three-phase systems include higher power output, smaller size for the same power, and cheaper transmission.
4.6 TRANSFORMER
A transformer is a stationary device used to transform AC power between circuits without changing frequency.
Basic principle: mutual induction between two coils on the same magnetic core. A changing current in the primary coil produces a changing magnetic flux, which links the secondary coil and induces emf in it.
Construction: two windings (primary with Np turns, secondary with Ns turns) on a laminated magnetic core to reduce eddy currents; windings are insulated to reduce coupling losses; cores are laminated to minimize iron losses.
Equations
Voltage transformation: \frac{Vs}{Vp} = \frac{Ns}{Np}
Current transformation (for an ideal transformer): \frac{Ip}{Is} = \frac{Ns}{Np}
Power balance (ideal): Vp Ip = Vs Is; hence the current ratio follows turn ratio.
Transformation ratio K (often used): K = \frac{Ns}{Np} = \frac{Vs}{Vp}; thus Ip = K Is and Vs = K Vp
Efficiency and losses
Efficiency η = (Output power) / (Input power), typically 96–99% in well-designed transformers.
Core (Iron) losses: hysteresis loss + eddy-current loss; minimized by using laminated steel (thin laminations, high silicon content).
Copper losses: resistive heating in windings; mitigated by using thicker conductors.
Flux leakage: incomplete linkage between windings; mitigated by winding arrangements to maximize coupling.
Transformer use in long-distance transmission
High-voltage step-up at the source, reducing current and hence I²R losses in the transmission lines; step-down at the destination for domestic use.
Example 4.16–4.17 could involve calculating turns, voltages, and currents for given transformers and loads.
4.7 ALTERNATING CURRENT (AC)
4.7.1 Introduction
Alternating voltage changes polarity with time; resulting current changes direction accordingly.
Sinusoidal AC: v(t) = Vm sin(ω t) and i(t) = Im sin(ω t + φ) where φ is the phase angle between v and i.
4.7.2 Sinusoidal Alternating Voltage
If the source is sinusoidal, the instantaneous voltage is
v(t) = V_m \sin(\omega t)The corresponding instantaneous current for a resistive circuit is in phase with the voltage.
4.7.3 AC Circuit with a Pure Resistor
Ohm’s law for AC: v(t) = i(t) R, so i(t) = (V_m/R) sin(ω t).
Phasor representation: the current phasor is in phase with the voltage phasor; no phase shift (φ = 0).
4.7.4 AC Circuit with a Pure Inductor
Inductor resists changes in current; the current lags the voltage by 90° (φ = +90° for inductive load).
Inductive reactance: X_L = \omega L. In a purely inductive circuit, V leads I by 90°.
Current expression: i(t) = I_m \sin(\omega t - \tfrac{\pi}{2})
4.7.5 AC Circuit with Resistor, Inductor, and Capacitor in Series (R-L-C)
Impedances: R, L (XL = ω L), C (XC = 1/(ω C)).
Impedance magnitude: Z = \sqrt{R^2 + (XL - XC)^2}
Current phasor: I = V / Z, with phase angle φ = arctan((XL - XC)/R).
4.7.6 AC Circuit containing all three: R, L, C in series
Impedance: Z = \sqrt{R^2 + (XL - XC)^2}. Power and phase relations depend on φ.
4.7.7 Resonance in Series RLC Circuit
At resonance, XL = XC, so the impedance is purely resistive and equals R.
Resonant angular frequency: \omegar = \frac{1}{\sqrt{LC}}; resonant frequency fr = \frac{\omega_r}{2\pi} = \frac{1}{2\pi\sqrt{LC}}.
At resonance, current is maximized: Im = Vm / R (for an idealized case).
The condition XL = XC also implies X_L C = −? (depending on sign conventions); the important consequence is a purely resistive circuit at resonance.
4.7.8 Quality Factor (Q-factor)
Q = (ratio of reactive energy to real energy per cycle or, for a series RLC at resonance, Q = ωr L / R = 1/(ωr C R) depending on form; standard expression Q = (ω_r L)/R = 1/(R)√(L/C).
Q measures sharpness of resonance; higher Q means a sharper resonance peak.
4.7.9 (Not explicitly numbered here) Related notes: At resonance, the reactive components’ voltages can be much larger than the applied voltage; Q captures this magnification.
4.8 POWER IN AC CIRCUITS
4.8.1 Introduction
Instantaneous power: P(t) = v(t) i(t); average power over a cycle is related to rms quantities and phase angle φ:
P{avg} = V{rms} I_{rms} \cos\phi.Apparent power: S = V{rms} I{rms} and real power: P = V{rms} I{rms} cosφ.
4.8.2 Wattless Current
Decomposition of Irms into components: Irms cosφ (in phase with V) and I_rms sinφ (quadrature, 90° out of phase) which does not contribute to real power but constitutes reactive power.
In purely inductive or purely capacitive circuits, the reactive component is maximum and real power is zero.
4.8.3 Power Factor
Defined as cosφ or as R/Z, where Z is the impedance.
For purely resistive circuits, φ = 0 and power factor = 1.
For purely inductive or capacitive, φ = ±90°, power factor = 0.
For series RLC, φ is given by tanφ = (XL − XC) / R; hence cosφ is the power factor.
At resonance (XL = XC), φ = 0 and cosφ = 1.
4.8.4 Advantages and Disadvantages of AC vs DC
Advantages: easier to transform voltage levels (step-up/step-down with transformers), cheaper long-distance transmission due to reduced I²R losses at higher voltage, easier generation and distribution across networks.
Disadvantages: AC is not always suitable for some processes (batteries charging, electroplating, traction), safety concerns at high voltages.
4.9 LC OSCILLATIONS
4.9.1 Energy conversion during LC oscillations
In an LC circuit, energy oscillates between the magnetic field of the inductor and the electric field of the capacitor.
In an ideal LC circuit (no losses), the total energy remains constant: Utotal = UL + U_C.
Electric energy in the capacitor: UC = q^2/(2C). Magnetic energy in the inductor: UL = (1/2) L i^2.
4.9.2 Conservation of Energy in LC Oscillations
Total energy remains constant: U_total = (1/2) C q^2 + (1/2) L i^2.
4.9.3 Analogies between LC oscillations and a mechanical spring-mass system
Charge q ↔ displacement x; current i ↔ velocity v; inductance L ↔ mass m; reciprocal of capacitance 1/C ↔ spring constant k; energy analogies:
Capacitive energy: (1/2) C q^2 ↔ Mechanical potential energy (1/2) k x^2
Inductive energy: (1/2) L i^2 ↔ Mechanical kinetic energy (1/2) m v^2
Angular frequency: \omega = \frac{1}{\sqrt{LC}}.
SUMMARY OF KEY EQUATIONS AND CONCEPTS
Magnetic flux: \PhiB = \intA \mathbf{B} \cdot d\mathbf{A}; for uniform B and area perpendicular: \PhiB = B A (or \PhiB = B A \cos\theta).
1 Weber = 1 T·m².
Flux linkage: N\PhiB; for N turns a coil experiences emf: \mathcal{E} = -\frac{d(N\PhiB)}{dt} = -N\frac{d\Phi_B}{dt}.
Faraday’s law in differential form: \mathcal{E} = -\dfrac{d\Phi_B}{dt}.
Lenz’s law (opposes flux change).
Self-inductance: N\PhiB = L i\quad\Rightarrow\quad L = \frac{N\PhiB}{i}; for a long solenoid, L = \frac{\mu0 N^2 A}{l}; with core, L = \mu0\mu_r \frac{N^2 A}{l}.
Magnetic energy in an inductor: U = \tfrac{1}{2} L i^2; \quad u = \frac{B^2}{2\mu_0}.$'
Mutual inductance: emf in coil 2 due to coil 1: \mathcal{E}2 = -M\frac{di1}{dt}; \quad M = \frac{N2\Phi{21}}{i1}.; for coaxial long solenoids: M = \frac{\mu0 N1 N2 A}{l} = \mu0\mur \frac{N1 N2 A}{l}.
Motional emf: \mathcal{E} = B l v.; current: i = \dfrac{\mathcal{E}}{R} = \dfrac{B l v}{R}.
Eddy currents: currents in layered or laminated conductors, reduced by laminations; devices include induction stoves, eddy-current brakes, testing.
AC fundamentals: v(t) = Vm sin(ω t); i(t) = Im sin(ω t + φ); rms values: V{rms} = Vm/\sqrt{2},\; I{rms} = Im/\sqrt{2}.
Impedance in R-L-C series: Z = \sqrt{R^2 + (XL - XC)^2}, \quad XL = \omega L,\; XC = \frac{1}{\omega C}.
Phase angle: \tan\phi = \frac{XL - XC}{R}; power factor: \cos\phi,
and active (P) vs reactive (Q) power; Pavg = Vrms I_rms cosφ.Resonance in series RLC: \omegar = \frac{1}{\sqrt{LC}}, \; fr = \frac{1}{2\pi\sqrt{LC}};\; Z{res} = R.; Im = V_m / R at resonance.
Q-factor: Q = \frac{\omega_r L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}.
Transformer relations: \frac{Vs}{Vp} = \frac{Ns}{Np}, \quad \frac{Ip}{Is} = \frac{Ns}{Np}, \quad P{in} \approx P{out}.
Transformer losses: core (hysteresis, eddy currents), copper, flux leakage.
Three-phase AC generators: stator windings + rotor field; 3-phase outputs separated by 120°; advantages include higher power density, reduced line losses, and smaller size for the same power.
Phasors: rotating vectors representing sinusoidal quantities; phasor magnitude equals peak value; angle represents phase; instantaneous values obtained by projection.
LC oscillations: energy oscillates between capacitor (electrical) and inductor (magnetic); total energy conserved in ideal LC circuits; angular frequency \omega = \frac{1}{\sqrt{LC}}.$$