Electromagnetic Induction and Alternating Current

The concept of Magnetic Flux (Φ_B) quantifies the total number of magnetic field lines passing through a given surface. For an infinitesimal surface area vector d𝐀 oriented normal to the surface in a magnetic field 𝐁, the elementary magnetic flux is defined as:

For a uniform magnetic field over a planar surface of total area A:

where θ is the angle between the magnetic field vector 𝐁 and the positive surface normal vector 𝐧̂. Magnetic flux is a scalar quantity. In the SI system, its unit is the Weber (Wb), where 1 Wb = 1 T·m² = 1 V·s. In CGS, the unit is Maxwell (1 Wb = 10⁸ Maxwell). For a closely wound coil of N turns, the total magnetic flux linkage is λ = N Φ_B.

In 1831, Michael Faraday conducted historic experimental investigations showing that an electromotive force (EMF) is induced across a closed conducting loop whenever the magnetic flux linked with the loop changes over time.

• Faraday's First Law (Qualitative): Whenever the magnetic flux linked with a closed conducting circuit undergoes a change with respect to time, an electromotive force (EMF) is induced in the circuit. The induced EMF persists as long as the flux continues to change.
• Faraday's Second Law (Quantitative): The magnitude of the induced EMF (E) in a circuit is directly proportional to the time rate of change of magnetic flux linked with the circuit:
  |E| = N · |dΦ_B / dt|

In 1834, Heinrich Lenz formulated the rule governing the polarity of the induced EMF:

Combining Faraday's quantitative relationship with Lenz's directional law yields the fundamental Faraday-Lenz Equation:

The negative sign reflects Lenz's opposition. Rigorous Proof of Energy Conservation: Consider a bar magnet pushed with its North pole facing towards a conducting ring. By Lenz's law, an anticlockwise induced current is set up, creating a North magnetic pole facing the approaching magnet. The repulsive magnetic force between the like poles resists the magnet's forward motion. An external mechanical agent must perform positive mechanical work against this repulsive force to move the magnet. This mechanical work is exactly converted into electrical energy and dissipated as Joule heat in the loop. If the induced pole were instead a South pole, the magnet would accelerate spontaneously without external work, creating energy from nothing and violating the First Law of Thermodynamics.

If the closed loop has total electrical resistance R, the instantaneous induced current i(t) is given by:

The total electric charge Δq that flows through any cross-section of the circuit during the flux transition from Φ₁ to Φ₂ is obtained by integrating over the time interval:

When a conducting rod of length l moves with velocity 𝐯 through a uniform, stationary magnetic field 𝐁, mobile conduction electrons within the rod experience a magnetic Lorentz force:

This magnetic force drives free electrons towards one terminal of the rod, leaving an excess of positive lattice ions at the opposite terminal. This spatial charge separation creates an internal electrostatic electric field 𝐄 pointing opposite to the magnetic force. Accumulation continues until electrostatic equilibrium is attained:

The potential difference (induced motional EMF) developed between the rod's endpoints is:

for mutually perpendicular vectors 𝐯 ⊥ 𝐁 ⊥ 𝐥.

Consider a metallic rod of length l rotating with constant angular speed ω in a plane perpendicular to a uniform magnetic field 𝐁, with one end pivoted at the center and the other tracing a circular arc. An infinitesimal segment dr located at distance r from the pivot moves with linear speed v = ω r. The infinitesimal EMF developed across dr is:

Integrating from the pivot (r = 0) to the outer tip (r = l):

where f = ω / (2π) is the frequency of revolution and A = π l² is the swept area per cycle.

Let the moving rod slide along frictionless rails connected across a load resistor R. The induced current is I = E / R = (B v l) / R. The current-carrying rod in magnetic field 𝐁 experiences a retarding magnetic force:

To maintain a constant velocity v, an external pulling force F_ext = F must be applied. The mechanical power supplied by this external force is:

Simultaneously, the electrical power dissipated as Joule heat across the circuit resistor is:

Thus, P_mechanical ≡ P_electrical, demonstrating exact, flawless conservation of energy.

When bulk conductors (sheets, blocks, or cores of metal) are subjected to changing magnetic flux, closed swirling loops of induced electric currents are established throughout their volume. First observed by Léon Foucault in 1851, these are termed Eddy Currents.

When current passes through a coil, it generates an intrinsic magnetic field linking its own turns. Any change in this current alters the self-linked magnetic flux, inducing an opposing back-EMF in the coil. This phenomenon is Self-Induction. The total flux linkage N Φ_B is proportional to current I:

where L is the Self-Inductance of the coil. Applying Faraday's law:

Definition of 1 Henry: An inductor has a self-inductance of 1 Henry (H) if a rate of current change of 1 A/s induces a opposing back-EMF of 1 Volt across its terminals.

Consider a long air-core solenoid of length l, cross-sectional area A, and total turns N (number of turns per unit length n = N / l). When current I flows, the uniform interior magnetic field is B = μ₀ n I = μ₀ (N / l) I. The magnetic flux through each individual turn is:

The total flux linkage for all N turns is:

Comparing with N Φ_B = L I gives the self-inductance:

where V = A l is the volume of the solenoid. If wound on a ferromagnetic core of relative permeability μ_r, L = μ_r μ₀ n² A l.

When two coils are placed in proximity, a changing current I₁ in the primary coil generates a varying magnetic flux linking the secondary coil, inducing an electromotive force E₂ in the secondary:

By the Reciprocity Theorem, the mutual inductance between two circuits is identical regardless of which coil carries current:

For two long coaxial solenoids of length l, where the inner solenoid has radius r₁ and turn density n₁, and the outer solenoid has radius r₂ > r₁ and turn density n₂:

The Coefficient of Coupling (k) measures the magnetic flux linkage efficiency between two coils of self-inductances L₁ and L₂:

For tight magnetic coupling (coaxial cores), k ≈ 1; for perpendicular or widely separated coils, k → 0.

To establish a steady current I in an inductor, an external power supply must perform work against the induced back-EMF |E| = L (di/dt). The infinitesimal work done in time dt is:

Integrating from i = 0 to final current i = I:

Substituting L = μ₀ n² A l and B = μ₀ n I ⟹ I = B / (μ₀ n):

Dividing by the volume of the magnetic field V = A l gives the Magnetic Energy Density:

This is the magnetic analogue of electrostatic energy density u_E = 1/2 ε₀ E².

An alternating current or voltage is one whose magnitude changes continuously with time and whose direction reverses periodically. A sinusoidal AC voltage and current are expressed as:

where V₀ and I₀ are peak amplitudes, ω = 2π f = 2π / T is angular frequency, and φ is phase difference.

• Mean Value over Half-Cycle: Over a full cycle, the arithmetic average of sinusoidal AC is identically zero (positive and negative half-cycles cancel). The average over the positive half-cycle (0 ≤ t ≤ T/2) is:
  I_avg = (2 / T) ∫₀^(T/2) I₀ sin(ω t) dt = (2 I₀ / π) ≈ 0.637 I₀
• Root Mean Square (RMS) / Virtual Value: The RMS value of an alternating current corresponds to that steady direct current which produces the same quantity of Joule heat in a given resistance during one full cycle:
  I_rms = √[ (1/T) ∫₀ᵀ I₀² sin²(ω t) dt ] = I₀ / √2 ≈ 0.707 I₀
  Similarly, V_rms = V₀ / √2 ≈ 0.707 V₀.

When an AC voltage v = V₀ sin(ω t) is applied across a pure resistance R, the instantaneous current is:

Current and voltage are strictly in phase (φ = 0). The phasor vectors for V and I point along the identical angular orientation.

For a pure inductor with self-inductance L and zero ohmic resistance, applying Kirchhoff's loop law gives v - L(di/dt) = 0, hence:

Crucial Phase Relationship: In a purely inductive AC circuit, the voltage leads the current by a phase angle of π/2 radians (90°), or equivalently, current lags voltage by 90°. The effective opposition offered by the inductor to alternating current is called Inductive Reactance (X_L):

For steady direct current (f = 0), X_L = 0. A pure inductor presents zero opposition to DC.

For a pure capacitor of capacitance C connected across v = V₀ sin(ω t), the instantaneous charge is q = C v = C V₀ sin(ω t). The instantaneous current is:

Crucial Phase Relationship: In a purely capacitive AC circuit, the current leads the voltage by a phase angle of π/2 radians (90°). The opposition offered by a capacitor is called Capacitive Reactance (X_C):

For steady direct current (f = 0), X_C → ∞. A capacitor completely blocks steady DC.

In a series LCR circuit containing resistance R, inductor L, and capacitor C connected to an alternating source v = V₀ sin(ω t), the identical instantaneous current i(t) = I₀ sin(ω t) flows through all three components. The individual component potential drops are:

• V_R = I R in phase with current phasor I
• V_L = I X_L leading current phasor I by +90°
• V_C = I X_C lagging current phasor I by -90°

Because V_L and V_C act along the same vertical phasor axis in exact opposition, the net reactive voltage is (V_L - V_C) = I(X_L - X_C). By vector phasor addition, the resultant source voltage amplitude is:

The total opposition to alternating current is termed Impedance (Z):

The phase angle φ by which the source voltage leads the current is given by:

If X_L > X_C, φ > 0 (circuit is inductive, voltage leads current); if X_C > X_L, φ < 0 (circuit is capacitive, current leads voltage).

Resonance occurs when the applied angular frequency ω makes the inductive reactance exactly equal to the capacitive reactance:

The corresponding resonant frequency in Hertz is:

Salient Characteristics at Series Resonance:

1. Net reactance is zero: X = X_L - X_C = 0.
2. Impedance is purely resistive and attains its absolute minimum: Z_min = R.
3. Current amplitude reaches its maximum possible value: I_max = V_rms / R.
4. Phase difference between applied voltage and current is zero (φ = 0); power factor cos φ = 1.
5. A series LCR circuit is known as an Acceptor Circuit because it admits maximum current at its natural resonant frequency, forming the tuning basis for radio receivers.

The selectivity or sharpness of an LCR tuning circuit is quantified by its Quality Factor (Q). It is defined as the ratio of the resonant frequency to the half-power bandwidth Δω = ω₂ - ω₁ = R / L:

The Quality Factor also represents the voltage magnification across inductor or capacitor at resonance relative to the applied source voltage: Q = V_L / V = V_C / V.

The instantaneous power in an AC circuit is p(t) = v(t) i(t) = V₀ sin(ω t) · I₀ sin(ω t - φ). Taking the average over one full cycle:

where cos φ = R / Z is the Power Factor.

• For pure R: φ = 0 ⟹ cos φ = 1 ⟹ P_avg = V_rms I_rms (maximum power dissipation).
• For pure L or pure C: φ = ±90° ⟹ cos φ = 0 ⟹ P_avg = 0. An ideal inductor or capacitor consumes zero net electrical power over a full cycle!
• Wattless Current: Resolving the current phasor I_rms into two orthogonal components:
  • I_rms cos φ: Component in phase with voltage, responsible for power dissipation (active current).
  • I_rms sin φ: Component at 90° to voltage. The power associated with this component is V_rms (I_rms sin φ) cos(90°) = 0. This is called Wattless Current.
• Choke Coil: An electrical device having large self-inductance L and negligible ohmic resistance R. It is used to choke or limit alternating current in AC circuits with minimal energy dissipation as heat, in contrast to a rheostat which dissipates substantial energy as I² R heat.

An AC Generator is a mechanical device that converts mechanical rotational energy into alternating electrical energy via electromagnetic induction.

Operating Principle: When a rectangular coil of N turns and area A rotates with uniform angular velocity ω in a uniform transverse magnetic field 𝐁, the magnetic flux linking the coil at time t is:

By Faraday's law of electromagnetic induction, the instantaneous induced EMF is:

where E₀ = N B A ω = 2π f N B A represents the peak electromotive force. The essential structural components are: (1) Armature coil, (2) Field magnets / Stator, (3) Slip rings (maintaining continuous contact during rotation without twisting wires), and (4) Carbon brushes (conducting current to external load).

A Transformer is a static electrical device that alters the magnitude of alternating voltage and current through mutual induction between two magnetically coupled windings.

Principle & Derivation: Two insulated copper windings, the Primary (N_p turns) and Secondary (N_s turns), are wound around a common soft-iron core. Assuming negligible resistance and zero magnetic flux leakage, the same alternating flux Φ_B links each turn of both coils:

Dividing the secondary EMF by the primary EMF yields the Transformation Ratio (k):

For an ideal transformer with 100% efficiency, input power equals output power (V_p I_p = V_s I_s):

• Step-Up Transformer: N_s > N_p ⟹ k > 1. Output voltage increases (V_s > V_p), while output current decreases proportionally (I_s < I_p).
• Step-Down Transformer: N_s < N_p ⟹ k < 1. Output voltage decreases (V_s < V_p), while output current increases proportionally (I_s > I_p).

To transmit electric power P from generating stations across hundreds of kilometers of transmission cables having resistance R_c, power is stepped up to extremely high voltages (e.g., 132 kV to 765 kV). Because P = V I, the line current is I = P / V. The Joule heat dissipation in the cables is:

Increasing transmission potential by a factor of 10 reduces transmission power loss by a factor of 100 (10²). This allows power grids to use lighter aluminum conductors with negligible voltage drop over great distances.