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UY1: Resistors, Inductors & Capacitors In A.C. Circuits

Compare pure R, L, and C behavior in AC: phase shifts, reactances, and average power flow.

  • University Physics Year 1
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Learning objectives

  • Analyse capacitance, resistance, energy transfer, and transient circuit behaviour.
  • Analyse magnetic forces, induction, inductance, and alternating-current systems with consistent signs.
Why this matters + quick links

This page gives the UY1 working model/result for Resistors, Inductors & Capacitors In A.C. Circuits. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

1) At a glance

  • Pure resistor: voltage and current in phase.
  • Pure inductor: voltage leads current by 90°.
  • Pure capacitor: current leads voltage by 90°.
  • Reactances:
X_L = ω L, X_C = 1/(ω C).
  • Impedances in phasor form (cosine reference):
Z_R = R, Z_L = iω L, Z_C = 1/(iω C).

Prerequisites: Phasors & Alternating Currents, Complex Numbers
Next uses: L-R-C Series Circuit With A.C.

2) Setup

Assume sinusoidal current:

i(t) = I₀ cos ω t.

Analyze each ideal element separately with sign conventions consistent with passive element behavior.

Common traps (lead/lag + power)
  • Memorize one fact and derive the rest: for an inductor v = L di/dt, so v leads i by 90°.
  • For a capacitor i = C dv/dt, so i leads v by 90°.
  • Ideal L and C have zero average power over a full cycle (they store then return energy). Only R dissipates average power.
  • Reactance X is a magnitude; impedance Z carries the sign/phase via i.

3) Core derivation/explanation

Resistor

v_R = iR = I₀R cos ω t.

No phase shift. Average power:

Pₐᵥ = IᵣₘₛVᵣₘₛ = Iᵣₘₛ²R.

Inductor

v_L = Ldi/dt = ω LI₀ cos(ω t + 90°).

So v leads i by 90°. Amplitude relation:

V_L0 = I₀X_L, X_L = ω L.

Average power over one full cycle is zero (energy stored then returned).

Capacitor

Using i = C dv_C/dt:

v_C = I₀/(ω C) cos(ω t-90°).

So i leads v by 90°. Amplitude relation:

V_C0 = I₀X_C, X_C = 1/(ω C).

Average power over one cycle is zero.

Frequency behavior:

  • X_L increases with ω.
  • X_C decreases with ω.

Checks (sanity)

  • Low frequency: X_C large (capacitor blocks), X_L small (inductor passes).
  • High frequency: X_L large (inductor blocks), X_C small (capacitor passes).

4) Worked example(s)

A pure inductor carries current amplitude I₀ = 250 μA at f = 1.60 MHz with voltage amplitude V₀ = 3.60 V.

X_L = V₀/I₀ = 3.60/(250 × 10⁻⁶) = 1.44 × 10⁴ Ω.
L = X_L/ω = (1.44 × 10⁴)/(2π(1.60 × 10⁶)) ≈ 1.43 × 10⁻³ H.

5) Practice set (with hints + answers)

  1. A pure resistor has R = 50 Ω and Iᵣₘₛ = 0.40 A. Find average power. Hint: P = Iᵣₘₛ²R. Answer: 8.0 W.

  2. For a capacitor C = 10 μF at f = 500 Hz, find X_C. Hint: ω = 2π f. Answer: 31.8 Ω.

  3. At very high frequency, which ideal element tends to block more: inductor or capacitor? Hint: compare X_L and X_C trends. Answer: inductor (large X_L), while capacitor tends to pass more (small X_C).

6) Summary + next steps

  • R dissipates average power; ideal L and C exchange energy with the source but net zero over a cycle.
  • Reactance and phase shifts are the building blocks of AC impedance.
  • Next: combine R, L, and C in one driven series circuit.

Next: L-R-C Series Circuit With A.C. Previous: Phasors & Alternating Currents Back To Electromagnetism (UY1)