AC voltage applied to a capacitor

What happens when an ac voltage is applied to a capacitor?

In plain words: no charge actually crosses the gap between the plates, but charge flows back and forth in the wires as the capacitor fills and empties. The current is largest when the voltage is changing fastest, which is a quarter cycle before the voltage peaks. The faster the voltage alternates, the more charge sloshes per second, so a capacitor lets high frequencies through more easily.

Model used throughout: an ideal, lossless, linear capacitor of constant C. The sinusoidal formulas describe operation below frequencies where stray inductance and other real-component effects matter.

Current leads the voltage by π/2

• At any instant the voltage across the capacitor is v = q/C (NCERT Eq. (7.15), page 184), and by Kirchhoff’s loop rule it equals the source voltage, so q = Cv_m sin ωt.
• The current is i = dq/dt = ωCv_m cos ωt. Using cos ωt = sin(ωt + π/2) (NCERT Eq. (7.16), page 184):

• So the current leads the voltage by π/2. In the phasor diagram the current phasor I is π/2 ahead of the voltage phasor V as they rotate anticlockwise, and the current reaches its maximum a quarter period earlier than the voltage (NCERT page 185, Figure 7.8).

Capacitive reactance

• The current amplitude is i_m = ωCv_m = v_m/(1/(ωC)). The quantity 1/(ωC) plays the part of a resistance and is called the capacitive reactance (NCERT Eqs. (7.17)–(7.18), page 184):

• With it, i_m = v_m/X_C; dividing both amplitudes by √2, the rms values obey I = V/X_C too (as in NCERT Example 7.4). X_C has the dimensions of resistance and its SI unit is the ohm, but it is inversely proportional to the frequency and to the capacitance (NCERT page 185).
• For steady dc (ν = 0), X_C is infinite: a charged capacitor blocks dc (reasoned from NCERT Eq. (7.17), agreeing with NCERT Example 7.3).

Zero average power in a capacitor

• The instantaneous power supplied to the capacitor is p_C = iv = i_mv_m cos ωt sin ωt = (i_mv_m/2) sin 2ωt (NCERT Eq. (7.19), page 185).
• Since sin 2ωt averages to zero over a cycle, the average power supplied to a capacitor is zero, as for an inductor (NCERT page 185).
• So a current flows but no power is used up on average. NCERT calls the current in a purely inductive or capacitive circuit a wattless current (NCERT Section 7.7, page 191).
• In one quarter cycle the capacitor stores energy in its electric field; in the next it returns it to the source (reasoned from the sign of p_C).
• NCERT sums up: in an inductor the current lags the voltage by π/2, and in a capacitor it leads by π/2 (NCERT page 185).

Solved examples

Common mistakes to avoid

Quick revision summary

• In a purely capacitive circuit, v = v_m sin ωt gives i = i_m sin(ωt + π/2): the current leads the voltage by π/2.
• Capacitive reactance X_C = 1/(ωC) = 1/(2πνC), in ohm; i_m = v_m/X_C.
• X_C falls as the frequency or the capacitance rises; a capacitor blocks steady dc.
• The average power supplied to a capacitor over a cycle is zero: the current is wattless.

Questions students ask

Why does a capacitor block DC but allow AC?

A capacitor blocks a steady direct current because its plates charge towards the supply voltage and the transient current dies away on a timescale set by the circuit resistance and capacitance, but it allows an alternating current because a voltage that keeps reversing keeps charging and discharging the plates, so charge flows to and fro in the wires (NCERT page 184). No charge crosses the gap in either case: the changing field between the plates, the displacement current of NCERT Chapter 8, links the currents on the two sides. Thus a series capacitor blocks the steady component and can couple the alternating component, with attenuation depending on frequency and the rest of the circuit (beyond the NCERT text: US Navy, NEETS Module 2, 1998, page 4-8).

What is the difference between inductive reactance and capacitive reactance?

Inductive reactance X_L = ωL rises in proportion to the frequency and comes with a current lagging the voltage by π/2, while capacitive reactance X_C = 1/(ωC) falls as the frequency rises and comes with a current leading by π/2; both are in ohm, and neither takes net energy over a cycle (NCERT pages 182–185). So they are opposite at the extremes: for steady dc a coil has no reactance and a capacitor an infinite one, and their ideal reactances approach the opposite limits as frequency rises. Real components cease to follow these simple models when parasitic effects matter. In series their effects oppose each other and cancel fully at resonance, where X_L = X_C.

Is capacitive reactance negative, and why do some books give it a minus sign?

No, capacitive reactance as NCERT defines it, X_C = 1/(ωC), is positive, in ohm, like X_L = ωL (NCERT page 184). Some books give it a minus sign for its opposite effect on phase: impedance diagrams draw X_L along +90° and X_C along −90°, so they subtract to a net reactance X = X_L − X_C (beyond the NCERT text: US Department of Energy, DOE Fundamentals Handbook: Electrical Science, 1992, Module 8). NCERT uses X_C − X_L in the impedance and tan φ (NCERT pages 187–188); its sign shows whether a circuit is capacitive or inductive.

Why is capacitive reactance measured in ohms, and what is its dimensional formula?

Capacitive reactance is measured in ohms because it is the ratio of the voltage amplitude to the current amplitude, X_C = v_m/i_m, a volt per ampere, just as a resistance is, and its dimensional formula is that of resistance, [M L^² T^⁻³ A^⁻²] (NCERT pages 184–185 and the table on page 198). The units of X_C = 1/(ωC) agree: ω is in s^⁻¹ and the farad is C V^⁻¹ = A s V^⁻¹ (NCERT Chapter 2, page 68), so 1/(s^⁻¹ × A s V^⁻¹) = V A^⁻¹ = Ω. Unlike an ohm of resistance, though, an ohm of reactance turns no energy into heat over a cycle (NCERT page 185).

What is the difference between capacitance and capacitive reactance?

Capacitance C = Q/V is a property of the capacitor itself, fixed by its plates, their separation and the dielectric and measured in farad, while capacitive reactance X_C = 1/(ωC) is its opposition to an alternating current of a given frequency, measured in ohm (NCERT Chapter 2, pages 67–68; NCERT page 184). The two vary inversely: a larger capacitance takes more charge each half cycle (q = Cv), so it passes more current and has less reactance; a 22 µF capacitor has X_C ≈ 145 Ω at 50 Hz but about 7.2 Ω at 1.0 kHz. Capacitance means something in dc circuits too, while reactance belongs only to sinusoidal ac.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. In a purely capacitive circuit, does the current lead or lag the voltage, and by how much?

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   It leads by π/2 (NCERT page 185).

2. Q2. Find the reactance of a 5.0 µF capacitor at 50 Hz.

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   X_C = 1/(2π × 50 × 5.0 × 10⁻⁶) ≈ 637 Ω (NCERT Eq. (7.17), page 184).

3. Q3. Why does a lamp in series with a capacitor not glow on a dc supply?

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   Once the capacitor is charged, no current flows in a dc circuit (NCERT Example 7.3, page 185).