Q.A pure capacitor of capacitance $15\ \mu\text{F}$ is connected across a $230\ \text{V}$ (rms), $50\ \text{Hz}$ AC supply. Calculate

Capacitive Reactance: The AC Resistance of a Capacitor

When you first meet a capacitor in a DC circuit, it behaves like a break in the wire once it's fully charged — no current flows. But in an AC circuit, something entirely different happens. The voltage keeps reversing, so the capacitor never finishes charging. It's constantly being filled, emptied, refilled, and re-emptied. This continuous back-and-forth means current does flow, but the capacitor resists that flow in a frequency-dependent way. That resistance is called capacitive reactance.

The Intuition: Why Frequency Matters

Imagine a water pipe with a flexible rubber membrane stretched across it (a crude capacitor). If you push water slowly from one side, the membrane bulges and eventually stops the flow — that's DC. But if you push and pull the water rapidly (AC), the membrane just vibrates, and water sloshes back and forth through the pipe. The faster you push-pull (higher frequency), the less the membrane impedes the flow. At very high frequencies, it's almost like the membrane isn't there.

In a capacitor, the "membrane" is the electric field between the plates. Higher frequency means the voltage changes faster, so the capacitor has less time to oppose the current. The result: capacitive reactance decreases as frequency increases.

The Precise Statement

Capacitive reactance $X_C$ is the opposition a capacitor offers to alternating current. It is measured in ohms ($\Omega$), just like resistance. The formula is:

Where:

• $X_C$ = capacitive reactance (ohms)
• $f$ = frequency of the AC signal (hertz)
• $C$ = capacitance (farads)

What the Formula Tells You

Three key relationships jump out:

1. Inverse with frequency: Double the frequency, halve the reactance. At DC ($f = 0$), $X_C$ becomes infinite — the capacitor blocks DC completely.
2. Inverse with capacitance: A larger capacitor (more farads) offers less opposition. It can store more charge per volt, so it "gives way" more easily.
3. No power dissipation: Unlike a resistor, a pure capacitor doesn't convert electrical energy to heat. Reactance is a reactive opposition — energy is stored and returned, not lost.

Phase: The Hidden Twist

There's a critical detail that separates reactance from resistance. In a purely resistive circuit, voltage and current peak at the same time — they are in phase. In a purely capacitive circuit, current leads voltage by $90^\circ$ (or $\pi/2$ radians).

Why? Because current is the rate of change of charge: $I = C \frac{dV}{dt}$. When the voltage is at its peak (not changing), the current is zero. When the voltage is crossing zero (changing fastest), the current is maximum. This quarter-cycle shift is baked into the definition of reactance.

Worked Example

A $10\ \mu\text{F}$ capacitor is connected to a $50\ \text{Hz}$ mains supply. Find its reactance.

At $500\ \text{Hz}$, the same capacitor gives $X_C \approx 31.8\ \Omega$ — ten times smaller for ten times the frequency.

Summary for Exams

• Capacitive reactance $X_C = \frac{1}{2\pi f C}$ (ohms)
• It decreases with increasing frequency and capacitance
• Current leads voltage by $90^\circ$ in a pure capacitor
• No power is dissipated (ideal case)
• At DC ($f = 0$), $X_C = \infty$ — the capacitor blocks steady current

Final answer: $X_C = \frac{1}{2\pi f C}$

Capacitive reactance, X_C = 1/(2πfC), and its inverse relationship with frequency is a key part of the NCERT Class 12 Physics chapter on alternating current, tested through numericals in CBSE boards and JEE Main. Anyone searching "capacitive reactance formula and phase difference class 12 physics" will find this current-leads-voltage explanation matches the standard NCERT derivation.