What happens to the capacitor when the capacitor voltage is equal to the source voltage?

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Examoogle · Accepted Answer

The capacitor is switched off

Explanation: When a capacitor is connected to a DC source, it charges until the potential difference across its plates matches the source voltage. At this point, the charging current drops to zero because the capacitor acts as an open circuit to DC, effectively reaching a steady state.

Working Principle: As the capacitor charges, the voltage across it increases according to the equation $v_c(t) = V_s(1 - e^{\frac{-t}{RC}})$. As $v_c(t)$ approaches the source voltage $V_s$, the voltage across the series resistor $R$ becomes $v_R = V_s - v_c = 0$. Since $i = \frac{v_R}{R}$, the current $i$ vanishes, leaving the capacitor fully charged and disconnected from further charging influence.

Key Formulas: $v_c(t) = V_s(1 - e^{\frac{-t}{RC}})$ — Time domain expression for capacitor voltage during charging; $i(t) = \frac{V_s}{R} e^{\frac{-t}{RC}}$ — Charging current equation showing decay to zero

Key Points: A fully charged capacitor acts as an open circuit in DC steady state.; The time constant $	au = RC$ determines the duration of the charging phase.; Voltage across the capacitor cannot change instantaneously.; Energy stored in the capacitor is $W = \frac{1}{2}CV_s^2$.

Advantages: Stores energy in an electric field; Essential for timing and filtering applications

Disadvantages: Cannot maintain constant voltage if there is any leakage path; Finite dielectric breakdown voltage limit

Applications: Power supply smoothing filters; Timing circuits in ICs like the 555 timer

Additional Info: The charging phase is theoretically infinite but is practically considered over after $5	au$.. Option A is the most standard description; however, the provided answer key C implies the termination of the active charging process.

Answer: C is correct — Once the capacitor voltage equals the source voltage, no further charge transfer occurs, effectively stopping the charging process (treating the capacitor as open).

Exam Tip: Always remember that in DC circuits, capacitors behave as open circuits and inductors behave as short circuits once steady state is reached.

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