Resistors, Capacitors & Inductors on AC

Each element limits and shifts the current differently.

10 minUpdated 22 October 2026capacitive reactanceinductive reactanceimpedancephase

Lesson at a glanceShow the key points

R: in phase, i₀ = ℰ₀/R (39.4–39.5).
C: i leads by π/2, X_C = 1/ωC (39.6–39.7); blocks DC, passes AC.
L: i lags by π/2, X_L = ωL (39.8–39.9).
i₀ = ℰ₀/Z (39.10).

Put a resistor, a capacitor or an inductor alone across an AC source. Each limits the current in its own way, and each shifts the current’s timing relative to the emf differently.

Capacitor only

With no resistance, the capacitor’s voltage equals the source’s: q/C = ℰ₀ sin ωt, so q = Cℰ₀ sin ωt and

i = \frac{dq}{dt} = C\mathcal E_0\omega\cos\omega t

Purely capacitive(39.6–39.7)

i = i_0\sin(\omega t + \pi/2), \qquad i_0 = \frac{\mathcal E_0}{X_C}, \qquad X_C = \frac{1}{\omega C}

the current leads the emf by π/2

On a battery a capacitor takes current only briefly while charging. On AC the charge must keep changing, so current flows all the time: a capacitor blocks DC but passes AC.
X_C, the capacitive reactance, acts like a resistance (in ohms). It is large at low frequency and small at high frequency; for DC (ω = 0) it is infinite.
When the emf is zero the current is at its largest, and vice versa.

Worked example

Reactance at three frequencies

Find the reactance of a 50 µF capacitor at 20 Hz, 50 Hz and 1000 Hz.

Inductor only

For a pure inductor, ℰ₀ sin ωt − L di/dt = 0, so di/dt = (ℰ₀/L) sin ωt and i = −(ℰ₀/ωL) cos ωt + c. A sinusoidal current must average to zero, so c = 0:

Purely inductive(39.8–39.9)

i = i_0\sin(\omega t - \pi/2), \qquad i_0 = \frac{\mathcal E_0}{X_L}, \qquad X_L = \omega L

the current lags the emf by π/2

The inductive reactance X_L is zero for DC and grows with frequency.

Interactive

Resistor, capacitor or inductor on AC

A 10 V peak source drives a single element. A resistor keeps current in step with the emf. A capacitor passes AC but its current leads the emf by 90°, and it lets more through at higher frequency (X_C = 1/ωC). An inductor makes the current lag by 90° and lets less through at higher frequency (X_L = ωL).

reactance X_C = 1/ωC31.8Ω

i_0 = \mathcal E_0/X

314.2mA

current relative to emfleads by 90°

Element

C100µF

Frequency

\nu

50Hz

Worked example

Inductor on the mains

A 0.30 H inductor is connected to a source of peak emf 150 V at 50 Hz. Find the peak current, and the source voltage at the instant the current is at its peak.

Impedance and phase

In all three cases the peak current is the peak emf divided by an effective resistance, the impedance Z:

Impedance(39.10)

i_0 = \frac{\mathcal E_0}{Z}

Summary

Element	Z	Phase φ of current (i = i₀ sin(ωt + φ))	At high frequency
Resistor	R	0 (in phase)	unchanged
Capacitor	1/ωC	+π/2 (leads)	passes more
Inductor	ωL	−π/2 (lags)	passes less

What to remember

1R: in phase, i₀ = ℰ₀/R.
2C: current leads by 90°, X_C = 1/ωC; blocks DC, passes AC.
3L: current lags by 90°, X_L = ωL; zero for DC.
4i₀ = ℰ₀/Z (39.10); φ is the phase factor.

Check yourself

Raising the frequency of the supply to a capacitor makes the current…

In a purely inductive circuit, when the current is zero the emf is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadSimple AC circuitsOpenStax University Physics Vol. 2 §15.2 ReadElectrical reactanceWikipedia

PreviousRMS Values Next LCR Circuits & Resonance

Resistors, Capacitors & Inductors on AC

Each element limits and shifts the current differently.

10 minUpdated 22 October 2026capacitive reactanceinductive reactanceimpedancephase

Lesson at a glanceShow the key points

R: in phase, i₀ = ℰ₀/R (39.4–39.5).
C: i leads by π/2, X_C = 1/ωC (39.6–39.7); blocks DC, passes AC.
L: i lags by π/2, X_L = ωL (39.8–39.9).
i₀ = ℰ₀/Z (39.10).

Put a resistor, a capacitor or an inductor alone across an AC source. Each limits the current in its own way, and each shifts the current’s timing relative to the emf differently.

Resistor only

With ℰ = ℰ₀ sin ωt across R, the loop law gives ℰ₀ sin ωt = Ri:

Purely resistive(39.4–39.5)

i = i_0\sin\omega t, \qquad i_0 = \frac{\mathcal E_0}{R}

the current is in phase with the emf

Capacitor only

With no resistance, the capacitor’s voltage equals the source’s: q/C = ℰ₀ sin ωt, so q = Cℰ₀ sin ωt and

i = \frac{dq}{dt} = C\mathcal E_0\omega\cos\omega t

Purely capacitive(39.6–39.7)

i = i_0\sin(\omega t + \pi/2), \qquad i_0 = \frac{\mathcal E_0}{X_C}, \qquad X_C = \frac{1}{\omega C}

the current leads the emf by π/2

On a battery a capacitor takes current only briefly while charging. On AC the charge must keep changing, so current flows all the time: a capacitor blocks DC but passes AC.
X_C, the capacitive reactance, acts like a resistance (in ohms). It is large at low frequency and small at high frequency; for DC (ω = 0) it is infinite.
When the emf is zero the current is at its largest, and vice versa.

Worked example

Reactance at three frequencies

Find the reactance of a 50 µF capacitor at 20 Hz, 50 Hz and 1000 Hz.

Inductor only

For a pure inductor, ℰ₀ sin ωt − L di/dt = 0, so di/dt = (ℰ₀/L) sin ωt and i = −(ℰ₀/ωL) cos ωt + c. A sinusoidal current must average to zero, so c = 0:

Purely inductive(39.8–39.9)

i = i_0\sin(\omega t - \pi/2), \qquad i_0 = \frac{\mathcal E_0}{X_L}, \qquad X_L = \omega L

the current lags the emf by π/2

The inductive reactance X_L is zero for DC and grows with frequency.

Interactive

Resistor, capacitor or inductor on AC

reactance X_C = 1/ωC31.8Ω

i_0 = \mathcal E_0/X

314.2mA

current relative to emfleads by 90°

Element

C100µF

Frequency

\nu

50Hz

Worked example

Inductor on the mains

A 0.30 H inductor is connected to a source of peak emf 150 V at 50 Hz. Find the peak current, and the source voltage at the instant the current is at its peak.

Impedance and phase

In all three cases the peak current is the peak emf divided by an effective resistance, the impedance Z:

Impedance(39.10)

i_0 = \frac{\mathcal E_0}{Z}

Summary

Element	Z	Phase φ of current (i = i₀ sin(ωt + φ))	At high frequency
Resistor	R	0 (in phase)	unchanged
Capacitor	1/ωC	+π/2 (leads)	passes more
Inductor	ωL	−π/2 (lags)	passes less

What to remember

1R: in phase, i₀ = ℰ₀/R.
2C: current leads by 90°, X_C = 1/ωC; blocks DC, passes AC.
3L: current lags by 90°, X_L = ωL; zero for DC.
4i₀ = ℰ₀/Z (39.10); φ is the phase factor.

Check yourself

Raising the frequency of the supply to a capacitor makes the current…

In a purely inductive circuit, when the current is zero the emf is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadSimple AC circuitsOpenStax University Physics Vol. 2 §15.2 ReadElectrical reactanceWikipedia

PreviousRMS Values Next LCR Circuits & Resonance