Capacitive Reactance⚠ unverified
Electrical / Basic · Compute the reactance of a capacitor at a given frequency
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| capacitance | capacitance | F | 1.0 | Capacitance of the capacitor |
| frequency | frequency | Hz | 1.0 | Operating frequency |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Xc | 1 / farad / hertz | Capacitive reactance, in ohms (Ohm) |
The science & history
Understanding the Parameters
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Capacitance $C$ — how much charge the device stores per volt ($Q = CV$); larger $C$ passes AC more easily (lower reactance).
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Frequency $f$ — cycles per second of the source; higher $f$ means the voltage changes faster, so more current flows for the same voltage amplitude (lower reactance).
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$X_C$ — the ratio of voltage amplitude to current amplitude, in ohms. It is the magnitude of the capacitor's impedance $Z_C = 1/(j\omega C) = -j X_C$; the minus‑$j$ encodes a $90^\circ$ phase shift.
Derivation (Approaching a Proof)
The defining relation of a capacitor is $i = C\,\dfrac{dv}{dt}$ (current is proportional to the rate of change of voltage). Drive it with a sinusoid $v(t) = V_m \sin(\omega t)$, where $\omega = 2\pi f$:
$$i(t) = C\,\frac{dv}{dt} = \omega C\, V_m \cos(\omega t) = \omega C\, V_m \sin\!\big(\omega t + \tfrac{\pi}{2}\big).$$
The current is sinusoidal at the same frequency with amplitude $I_m = \omega C\, V_m$, and it leads the voltage by $90^\circ$. The reactance is the ratio of amplitudes (analogous to resistance):
$$X_C = \frac{V_m}{I_m} = \frac{1}{\omega C} = \frac{1}{2\pi f C}.$$
Units. $\dfrac{1}{(\text{1/s})(\text{F})} = \dfrac{1}{(\text{1/s})(\text{C/V})} = \dfrac{\text{V}\cdot\text{s}}{\text{C}} = \dfrac{\text{V}}{\text{A}} = \Omega$ — reactance is genuinely in ohms. (Note: the live calculator currently mislabels the output unit as "1/farad/hertz"; the physical unit is the ohm — flagged for calculator verification.)
History
Capacitive reactance emerged from the AC theory of the 1880s–1890s. Charles Proteus Steinmetz's phasor / complex‑impedance method (1893) made $Z_C = 1/(j\omega C)$ the compact way to treat capacitors in AC circuits, turning differential equations into algebra and giving reactance its modern ohmic footing alongside resistance.
Related Concepts: Inductive Reactance, Impedance, Capacitor Energy, Power Factor
Notes: Registry calculator capacitive-reactance (unverified). Reactance is frequency‑dependent;
combine with resistance as impedance $Z = R + jX$ for full AC analysis.