Q.The power consumed in alternating current in circuit containing only capacitor will be:

AC Through a Capacitor — From Intuition to the Exact Statement

Imagine a capacitor as a tiny, two-plate storage tank for charge. When you connect it to a DC battery, it charges up quickly and then blocks any further current — that's why a capacitor is an open circuit for steady DC. But AC is different: the voltage keeps reversing, so the capacitor never gets a chance to settle. It is constantly being charged, discharged, charged the other way, discharged again — and that motion of charge is an alternating current.

The key intuition: current flows because the voltage is changing. If the voltage were steady, no current would flow. The faster the voltage changes, the larger the current. This is the opposite of a resistor, where current depends on the voltage itself, not its rate of change.

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The Mathematical Link

For a capacitor, the charge stored is $Q = C V$. Current is the rate of flow of charge: $I = dQ/dt$. So:

This single equation is the whole story. If the applied voltage is sinusoidal, say $V = V_0 \sin(\omega t)$, then:

Now compare the two waveforms:

• Voltage: $V_0 \sin(\omega t)$ — starts at zero, rises to peak.
• Current: $C V_0 \omega \cos(\omega t)$ — starts at its maximum value, then falls.

A cosine is a sine shifted forward by $90^\circ$ (or $\pi/2$ radians). So the current reaches its peak a quarter-cycle before the voltage does. That is the famous result: in a purely capacitive circuit, current leads voltage by $90^\circ$.

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Why "Leads" and Not "Lags"?

Think physically. At the instant you first apply the AC voltage, the voltage is zero but rising fastest (the slope of $\sin$ is maximum at zero). A fast-changing voltage means a large current. So the current is already at its peak while the voltage is still near zero. That is the meaning of "leading" — the current's peak comes first.

Later, when the voltage reaches its peak, it is momentarily not changing (slope = 0), so the current drops to zero. The current is always ahead of the voltage by exactly one quarter-cycle.

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The Limiting Factor: Capacitive Reactance

From the current expression above, the peak current is:

This looks like Ohm's law if we define an effective resistance-like quantity:

This $X_C$ is called capacitive reactance. It has units of ohms, but it is not a resistance — it does not dissipate energy. It merely limits the current by the capacitor's opposition to changes in voltage.

Key points about $X_C$:

• It is inversely proportional to frequency. At high $f$, the voltage changes rapidly, so the current is large — low reactance. At low $f$, the voltage changes slowly, so the current is small — high reactance. At DC ($f = 0$), $X_C \to \infty$, which is the open-circuit behaviour you already know.
• It is also inversely proportional to capacitance $C$. A larger capacitor stores more charge per volt, so for the same voltage change it pushes more current — lower reactance.

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The Complete Picture in One Table