AC in Capacitors

When a direct current (DC) flows through a capacitor, the capacitor charges up and then blocks further current entirely. Alternating current (AC) is fundamentally different: because the voltage is constantly reversing direction, the capacitor is perpetually charging and discharging, allowing current to flow continuously. The capacitor never fully blocks an AC signal — instead, it offers a frequency-dependent opposition called capacitive reactance. The most striking feature of a capacitor in an AC circuit is the phase relationship between current and voltage. In a purely capacitive circuit, the current leads the voltage by exactly 90°, or equivalently by a quarter cycle. A handy mnemonic is 'ICE' (from the broader saying 'ELI the ICE man'): in a Capacitor (C), the current I comes before (leads) the voltage E. Imagine pushing water into a balloon — the water rushes in first while the balloon pressure builds up afterward. The current flows as the charge accumulates, but the voltage across the plates only rises once charge has actually piled up. Capacitive reactance decreases as frequency increases: a capacitor is nearly transparent to high-frequency signals but strongly opposes low-frequency ones. This property makes capacitors essential in filters, tuning circuits, audio crossover networks, and power-factor correction — any application that requires frequency-selective behavior. Understanding how reactance and phase work together is the foundation for analyzing all AC circuits containing capacitors.

Key takeaways
  • The Fundamental Capacitor Equation
  • Applying a Sinusoidal Voltage
  • Defining Capacitive Reactance
  • Impedance of a Capacitor
  • Energy Storage vs. Dissipation

The Fundamental Capacitor Equation

The defining relationship for a capacitor is Q = CV, or equivalently i = C(dv/dt). This tells us that current flows whenever voltage is changing. If no voltage change is occurring (steady DC), no current flows. For an AC signal, voltage changes constantly, so current always flows.

Applying a Sinusoidal Voltage

Suppose the voltage across the capacitor is v(t) = V₀ sin(ωt), where ω = 2πf is the angular frequency. Differentiating: i(t) = C · d/dt[V₀ sin(ωt)] = CV₀ω cos(ωt). We can rewrite this as i(t) = V₀ω C · sin(ωt + 90°). The current has the same frequency but is shifted 90° ahead of the voltage — current leads voltage by a quarter cycle.

Defining Capacitive Reactance

Comparing i(t) = (V₀/Xc) sin(ωt + 90°) with the voltage, we identify the capacitive reactance Xc = 1/(ωC) = 1/(2πfC). This acts like a frequency-dependent resistance: it has units of ohms (Ω) and scales the amplitude of the current relative to the voltage amplitude. Higher frequency → smaller Xc → more current flows for the same voltage. Lower frequency → larger Xc → less current flows. At DC (f = 0), Xc → ∞, confirming total blocking.

Impedance of a Capacitor

In complex (phasor) notation, impedance generalises resistance to AC circuits. The impedance of a capacitor is Zc = −j/( ωC) = 1/(jωC), where j = √(−1). The negative imaginary sign encodes the −90° phase shift of voltage relative to current (or equivalently, +90° of current relative to voltage). The magnitude |Zc| = Xc = 1/(ωC), and the phase angle is −90°. This complex representation allows capacitors, inductors, and resistors to be combined using familiar algebraic rules (Ohm's law, series/parallel formulas) even in AC circuits.

Energy Storage vs. Dissipation

Unlike a resistor, an ideal capacitor dissipates zero average power. Energy flows into the capacitor (stored in the electric field) during one quarter-cycle and is returned to the circuit in the next. The average power P = V_rms · I_rms · cos(90°) = 0. The term cos(φ) is the power factor; for a pure capacitor φ = 90°, so the power factor is zero. In real circuits, capacitors have a small series resistance causing minor power loss, but the ideal model is an excellent approximation.

The Key Equation

XC=1ωC=12πfCX_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}
  • X_C — Capacitive Reactance (Ω (ohms)): The opposition a capacitor offers to AC current. Analogous to resistance but frequency-dependent and causes a 90° phase shift.
  • \omega — Angular Frequency (rad/s): Equal to 2πf, where f is the ordinary frequency in hertz. Describes how rapidly the AC signal oscillates.
  • f — Frequency (Hz): The number of complete AC cycles per second.
  • C — Capacitance (F (farads)): A measure of the capacitor's ability to store charge per unit voltage. Larger capacitance means lower reactance.

Worked Example

Problem. A 10 μF capacitor is connected to a 120 V (rms), 60 Hz AC supply. (a) Calculate the capacitive reactance. (b) Find the rms current through the capacitor. (c) If the frequency is doubled to 120 Hz, what happens to the current?

XC≈265 Ω,Irms≈0.453 A  (60 Hz);Irms≈0.905 A  (120 Hz)X_C \approx 265\,\Omega,\quad I_{\text{rms}} \approx 0.453\,\text{A}\;(60\,\text{Hz});\quad I_{\text{rms}} \approx 0.905\,\text{A}\;(120\,\text{Hz})

Answer. At 60 Hz: Xc ≈ 265 Ω, Irms ≈ 0.453 A. At 120 Hz: Xc ≈ 133 Ω, Irms ≈ 0.905 A.

Figure
Diagram
AC Capacitor: Voltage and Current Phasors & WaveformsPhasor DiagramVI90°I leads V by 90°Waveforms over TimetVICurrent (blue dashed) peaks ¼ cycle before Voltage (red)
Origins · 1 minMichael Faraday and how this was discovered
MF
Michael Faraday
1791–1867 · British

Faraday discovered electromagnetic induction (1831) and was the first to systematically study the properties of capacitors (then called 'condensers'), introducing the concept of the dielectric and the unit of capacitance — the farad — named in his honour. His experimental insight that charge stored depends on the medium between the plates laid the groundwork for understanding how capacitors behave in changing fields, directly informing all AC capacitor theory. Faraday had almost no formal education and taught himself science by reading books while working as a bookbinder's apprentice. He never learned advanced mathematics, yet his physical intuition for electromagnetic phenomena was so profound that James Clerk Maxwell translated his ideas into the elegant equations we use today.

  • AC in Inductors — Inductors also introduce a 90° phase shift, but current lags voltage — exactly opposite to capacitors. Comparing the two reveals how they cancel each other in LC and RLC circuits.
  • AC in Resistors — Resistors have no phase shift and frequency-independent opposition. Understanding resistors in AC provides the baseline for contrasting the reactive behaviour of capacitors.
  • Impedance and Phasors — Impedance combines resistance and reactance into a single complex quantity. Phasor analysis is the standard tool for solving circuits that mix capacitors, inductors, and resistors.
  • Resonance in LC Circuits — At resonance, capacitive and inductive reactances exactly cancel. This concept is impossible to understand without first mastering capacitive reactance.