Why does a capacitor act as an open circuit to DC after a long time?

Why does a capacitor act as an open circuit to DC after a long time?

• A. It behaves as a short circuit when DC is applied.
• B. It stores energy and thus conducts DC indefinitely.
• C. Its reactance equals zero at DC.
• D. Its impedance Z_C = 1/(jωC) becomes infinite as ω → 0 (DC), so no DC current flows through a capacitor.

Answer: (D)

Capacitors block steady current because the current through a capacitor depends on how quickly the voltage across it changes. The relation i = C dv/dt means that for DC, once the voltage across the capacitor stops changing (dv/dt = 0), the current becomes zero. After a long time, the capacitor is charged to the supply voltage and no further DC current can flow, so it behaves like an open circuit. From an impedance viewpoint, the capacitor has Z_C = 1/(jωC). In DC, the angular frequency ω is zero, so Z_C grows without bound (infinite). An infinite impedance means no DC current can pass through. In real circuits there’s a tiny leakage, but in the ideal model the capacitor presents an open circuit to DC. So the best explanation is that DC causes the capacitor to charge up and then block further current, which is captured both by i = C dv/dt and by the infinite impedance at zero frequency.

Capacitors block steady current because the current through a capacitor depends on how quickly the voltage across it changes. The relation i = C dv/dt means that for DC, once the voltage across the capacitor stops changing (dv/dt = 0), the current becomes zero. After a long time, the capacitor is charged to the supply voltage and no further DC current can flow, so it behaves like an open circuit.

From an impedance viewpoint, the capacitor has Z_C = 1/(jωC). In DC, the angular frequency ω is zero, so Z_C grows without bound (infinite). An infinite impedance means no DC current can pass through. In real circuits there’s a tiny leakage, but in the ideal model the capacitor presents an open circuit to DC.

So the best explanation is that DC causes the capacitor to charge up and then block further current, which is captured both by i = C dv/dt and by the infinite impedance at zero frequency.