When will be capacitors fully charged?

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Examoogle · Accepted Answer

When the supply voltage is equal to the capacitor voltage

Explanation: A capacitor is fully charged when the potential difference across its plates becomes equal to the applied electromotive force (EMF) or supply voltage. At this point, the current flowing through the circuit ceases as the electric field between the plates perfectly opposes the external source.

Working Principle: As the capacitor charges, the voltage across it increases according to the equation $v_c(t) = V_s(1 - e^{-t/RC})$. As $t 	o \infty$, the exponential term approaches zero, resulting in $v_c(t) = V_s$. At this equilibrium state, the charging current $i(t) = \frac{V_s}{R} e^{-t/RC}$ decays to zero, effectively making the capacitor behave as an open circuit in a DC system.

Key Formulas: $v_c(t) = V_s(1 - e^{-t/RC})$ — Instantaneous capacitor voltage during charging; $i(t) = \frac{V_s}{R} e^{-t/RC}$ — Charging current as a function of time; $	au = RC$ — Time constant of the RC circuit

Key Points: Steady-state is achieved after approximately $5	au$ (five time constants).; At full charge, the capacitor acts as an open circuit to DC current.; The energy stored at full charge is given by $E = \frac{1}{2}CV_s^2$.; The charging process is exponential, not linear.

Advantages: Energy is stored in an electric field without chemical degradation.; Fast charge and discharge cycles compared to batteries.

Disadvantages: Lower energy density compared to electrochemical batteries.; Self-discharge over time due to leakage resistance.

Applications: Smoothing filters in DC power supplies.; Timing circuits in oscillators and pulse generators.

Additional Info: The time constant $	au = RC$ determines the rate of charging; higher resistance or capacitance increases the time to reach full charge.. Option A is incorrect as voltage zero implies the initial uncharged state. Option C is incorrect as infinity is a mathematical limit, not a physical voltage value.

Answer: B is correct — The capacitor reaches steady state and stops charging when the voltage across its plates exactly matches the external supply voltage.

Exam Tip: In DC circuits, always remember that a capacitor acts as a short circuit at $t=0$ and as an open circuit at $t=\infty$.

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