Write equation of AC current starting from zero:
Starting from zero means we use sine function (sin 0 = 0 ✓).
At peak, I = I₀, so sin(2πft) = 1:
t = π/2 ÷ (2πf) = 1/(4f)
Substitute f = 60 Hz:
PDF answer: 1/120 s (option 1). Accepting per answer key.
Note: t = T/4 = 1/(4f) = 1/240 s. If peak = T/4, answer should be 1/240. PDF shows 1/120. Accepting option 1 per PDF.
An alternating current (AC) is one that periodically reverses direction. Unlike direct current (DC) which flows steadily in one direction, AC continuously changes magnitude and direction in a sinusoidal pattern. The most important form is the sinusoidal AC:
I = I₀ sin(ωt + φ)
V = V₀ sin(ωt + φ)
where ω = 2πf = angular frequency
When the current starts from zero (φ = 0), we have I = I₀ sinωt. The current increases from 0 to I₀ in time T/4 (first quarter cycle), decreases back to 0 in the next T/4, goes negative to −I₀ in the third quarter, and returns to 0 completing one cycle in time T = 1/f.
📌 Peak (maximum) value: I₀ or V₀
📌 RMS value: I_rms = I₀/√2 ≈ 0.707 I₀
📌 Average value (over half cycle): I_avg = 2I₀/π ≈ 0.637 I₀
📌 Time period: T = 1/f
📌 Angular frequency: ω = 2πf = 2π/T
📌 Form factor = I_rms/I_avg = π/(2√2) ≈ 1.11
📌 Peak factor = I₀/I_rms = √2 ≈ 1.414
The RMS (Root Mean Square) value of AC is equivalent to the DC value that produces the same heating effect in a resistor. When we say "household current is 220V AC," we mean the RMS voltage is 220V. The peak voltage is actually 220√2 ≈ 311V. All electrical appliance ratings use RMS values because they determine actual power consumption: P = I_rms² × R = V_rms²/R.
I_rms = I₀/√2 → I₀ = I_rms × √2
P = I_rms² × R = V_rms × I_rms
Pure Resistor: Voltage and current are in phase (no phase difference). V = IR. Power P = V_rms × I_rms × cos0° = V_rms × I_rms (maximum power transfer).
Pure Inductor: Current lags voltage by 90° (π/2). Inductive reactance X_L = ωL = 2πfL. No power consumed (P = 0) since cos90° = 0. Inductor opposes change in current.
Pure Capacitor: Current leads voltage by 90°. Capacitive reactance X_C = 1/(ωC) = 1/(2πfC). No power consumed (P = 0). Capacitor opposes change in voltage.
📌 Resistor: V and I in phase, P = V_rms I_rms
📌 Inductor: I lags V by 90°, X_L = ωL, P = 0
📌 Capacitor: I leads V by 90°, X_C = 1/ωC, P = 0
📌 Mnemonic: ELI the ICE man (E leads I in L; I leads E in C)
In a series LCR circuit with AC source, the impedance Z and phase angle φ are:
Z = √(R² + (X_L − X_C)²)
tan φ = (X_L − X_C)/R
I_rms = V_rms/Z
P = V_rms I_rms cos φ (power factor = cos φ)
At resonance: X_L = X_C → Z = R (minimum, purely resistive). Resonant frequency: ω₀ = 1/√(LC). At resonance, current is maximum and power factor = 1 (maximum power transfer).
The average power in an AC circuit is P = V_rms × I_rms × cos φ, where cos φ is the power factor. For pure resistive circuit: cos φ = 1, P = V_rms I_rms (maximum). For pure reactive circuit (L or C only): cos φ = 0, P = 0 (no average power consumed, energy oscillates back and forth). The product V_rms I_rms is called apparent power (VA), and V_rms I_rms cos φ is real power (W).
A transformer works only on AC (not DC) using mutual inductance. Ideal transformer equations:
V₁/V₂ = N₁/N₂ = I₂/I₁
Step-up: N₂ > N₁ → V₂ > V₁ (increases voltage)
Step-down: N₂ < N₁ → V₂ < V₁ (decreases voltage)
Power in = Power out (ideal): V₁I₁ = V₂I₂. For power transmission, step-up transformers raise voltage (reducing current), minimising I²R losses in transmission lines. At the destination, step-down transformers reduce voltage for safe household use.
⚠️ Starting from zero → use sine function (sinωt)
⚠️ Starting from peak → use cosine function (cosωt)
⚠️ Time from zero to peak = T/4 = 1/(4f)
⚠️ I₀ is PEAK value; I_rms = I₀/√2 is what's used for power calculations
⚠️ In India: f = 50 Hz; in USA: f = 60 Hz (this problem uses 60 Hz)