• In the previous chapter we studied the dynamo.
• The current it produces, an alternating current (AC), changes in intensity and direction periodically: it rises from zero to a maximum, falls to zero during a half cycle, reverses, rises to a maximum in the opposite direction and falls to zero again in the second half cycle.
• This repeats every cycle.

• When AC flows in a coil, the current changes continuously, so a back emf −L ΔI/Δt is induced by self-induction; it opposes the source emf and has the same frequency.
• Applying Kirchhoff’s second law for a coil with negligible resistance, the source voltage balances the induced emf: V = L ΔI/Δt.

• The voltage is proportional to the rate of change of current (the slope of the I–t curve): it is maximum when the current passes through zero (steepest slope) and zero when the current is at maximum (zero slope); it is negative when the current is decreasing.
• Plotting the slope gives a curve shifted by a quarter of a cycle.
• Therefore:
• In an inductor the voltage leads the current by 90° (equivalently, the current lags the voltage by 90°).

1. The opposition of a capacitor to AC, due to its capacitance:
2. (ohm)

• Hot:
• Wire Ammeter An electrical instrument designed to measure both DC and the effective (RMS) value of alternating current (Ieff), operating on the thermal effect of current (\(Q = I2 R t\)).
• Inductive Reactance ($X_L$):
• The opposition encountered by alternating current when passing through an induction coil due to its self-inductance, measured in ohms (Ω).
• Capacitive Reactance ($X_C$):
• The opposition encountered by alternating current when flowing through a capacitor due to its capacitance, measured in ohms (Ω).
• Effective Value of AC (I_eff):
• The steady DC current value that generates the same amount of thermal energy in a given resistor in the same time interval as the alternating current.

• Phase Angle (φ):
• The angular lead or lag between the alternating voltage vector and alternating current vector in an AC circuit.
• ⚖️ Governing Scientific Laws, Theorems & Mathematical Models:
• Thermal Energy Formula:
• [ Formula Q = I2 R t = Ieff2 R t (Thermal power is proportional to the square of effective current, resulting in non-uniform scale spacing on hot-wire ammeters). ]
• Inductive Reactance Law:
• [ Formula XL = 2π f L = ω L (Where \(XL\) is directly proportional to frequency f and self-inductance L). ]
• Capacitive Reactance Law:
• [ Formula XC = 1 / (2π f C) = 1 / (ω C) (Where \(XC\) is inversely proportional to frequency f and capacitance C). ]

• Phase Relations:
• [ Formula Pure Resistor: V and I are in phase (φ = 0). Pure Inductor: V leads I by 90° (π/2). Pure Capacitor: I leads V by 90° (π/2). ]
• 📌 Exhaustive In:
• Depth Pedagogical Breakdown:
• ◄ I. Hot:
• Wire Ammeter Physics, Thermal Expansion, and Non-Uniform Scaling
• Unlike moving:
• coil galvanometers which rely on magnetic torque and can only deflect in one direction for DC, the Hot-Wire Ammeter depends entirely on the thermal effect of electric current.
• When alternating or direct current flows through an Ir-Pt (Platinum-Iridium) alloy wire, electrical energy is dissipated as heat, raising the wire temperature and causing it to expand.
• A silk thread connected to a spring pulls the wire down over a light pulley, rotating an indicator pointer across a calibrated scale until thermal equilibrium is established.

• The scale is non:
• uniform because thermal power is proportional to the square of current (P ∝ I2).
• Thus, doubling current quadruples pointer deflection (\(22 = 4\)), and tripling current increases deflection by nine times (\(32 = 9\)).
• An essential drawback is slow pointer movement and thermal zero error caused by ambient room temperature fluctuations, resolved by mounting the alloy wire on a base plate with an identical thermal expansion coefficient.
• Figure (hot_wire_ammeter_and_ac_phasors.png):
• Figure (1) Internal structure of Hot-Wire Ammeter and phasor diagrams for R, L, and C circuits
• ◄ II. AC Passing Through an Ohmic Resistor (R Circuit)

• When an AC source is connected to an induction:
• free ohmic resistor R, both voltage and current follow sinusoidal waveforms: v = Vmax sin (ω t) and i = Imax sin (ω t).
• According to Ohm's law, the current and voltage reach their positive peaks, zeros, and negative peaks simultaneously.
• Therefore, the phase angle φ = 0, meaning voltage and current are in phase.
• The resistance R = ρe l / A is independent of frequency f.
• Electric power is continuously dissipated in the resistor as Joulean heat at a rate P = Ieff2 R = Veff2 / R.
• ◄ III. AC Passing Through a Pure Inductor (L Circuit) and Self:

Inductance

• When an alternating voltage is applied across an induction coil of negligible resistance, the continuously changing current generates a varying magnetic flux, inducing a back EMF:
• emf = -L (Δ i) / (Δ t).
• By Lenz's law, this back EMF opposes current growth.
• As a consequence, current lags behind voltage by a quarter cycle (90° or π/2 radians).

• The opposition is inductive reactance XL = 2π f L.
• Significantly, pure inductors do not consume or dissipate electrical energy; instead, energy is temporarily stored during the first quarter cycle in the surrounding magnetic field and returned to the circuit during the subsequent quarter cycle.
• At very high frequencies, XL → ∞, causing the inductor to behave as an open switch.
• ◄ IV. AC Passing Through a Pure Capacitor (C Circuit) and Charge Dynamics
• A capacitor consists of two conducting plates separated by an insulator dielectric.
• When connected to an AC generator, current flows back and forth to charge and discharge the plates.
• The instantaneous charge is q = C v.
• Current represents the rate of charge flow:
• i = (Δ q) / (Δ t) = C (Δ v) / (Δ t).

• When voltage is zero and changing at its steepest rate, current reaches its maximum peak; when voltage reaches its maximum, its rate of change is zero, and current drops to zero.
• Consequently, current leads voltage by 90°.
• The capacitive reactance XC = 1 / (2π f C) drops dramatically at high frequencies (f → ∞ ⇒ XC → 0), allowing the capacitor to act virtually as a closed switch or short circuit.
• 🧮 Exemplary Step:
• by-Step Worked Problems & Applied Exercises:
• Worked Example 1:
• Worked Problem 1 – Calculating Reactance and Current for Inductor and Capacitor
• Problem Statement:
• Problem An AC dynamo provides an RMS voltage of 220 V at frequency 50 Hz. Calculate:
• 1) The inductive reactance and RMS current if connected to an inductor of L = 0.7 H (negligible resistance).
• 2) The capacitive reactance and RMS current if connected to a capacitor of C = 14 \(\mu\)F.
• Detailed Solution:
• 1) X_L = 2π f L = 2 × 22 / 7 × 50 × 0.7 = 220 Ω.
• I_eff = V_eff / X_L = 220 / 220 = 1.0 A.
• 2) X_C = 1 / (2π f C) = 1 / (2 × 22 / 7 × 50 × (14 × 10^-6)) = 1 / (4400 × 10^-6) = 10⁶ / 4400 ≈ 227.3 Ω.
• I_eff = V_eff / X_C = 220 / 227.3 ≈ 0.968 A.

• 💡 Master Teacher Insights & Critical Exam Cautions:
• Exam Trap:
• In pure inductors and pure capacitors, the average power dissipated over a complete cycle is EXACTLY ZERO (P_av = 0) because energy is stored as magnetic/electric fields and fully returned.
• High:
• Yield Rule At very high frequencies (f → ∞), an inductor acts as an OPEN switch (XL → ∞), while a capacitor acts as a CLOSED switch (XC → 0).
• Scaling Secret:
• The Hot-Wire Ammeter scale divisions expand quadratically 1A = 1 unit, 2A = 4 units, 3A = 9 units, 4A = 16 units.
• 🌍 Practical Real:
• World, Industrial & Societal Connections:
• ✔ Radio Frequency Filters:
• LC network filters are universally employed in smartphone transceivers to separate carrier waves from audio signals based on frequency-dependent reactances.

• ✔ Power Factor Correction in Power Grids:
• Industrial facilities with heavy motor loads install capacitor banks to neutralize inductive reactance, raising the power factor to near unity.