While charging, what happens to the voltage in the capacitive circuit?

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Examoogle · Accepted Answer

Increases exponentially

Explanation: When a capacitor is connected to a DC voltage source through a resistor, the voltage across the capacitor follows an exponential growth curve. This is due to the inverse relationship between the charging current and the accumulated charge, which results in a time-dependent voltage buildup described by a first-order differential equation.

Working Principle: As the capacitor charges, the charging current $i(t)$ decreases exponentially as the potential difference between the source and the capacitor reduces. The capacitor voltage $v_c(t)$ approaches the source voltage $V_s$ asymptotically. The rate of charging is governed by the time constant $	au = RC$, where a larger time constant results in a slower, yet still exponential, charging process.

Key Formulas: $v_c(t) = V_s(1 - e^{-\frac{t}{RC}})$ — expression for instantaneous capacitor voltage during charging; $	au = RC$ — time constant of the RC circuit defining the charging rate

Key Points: The charging process is characterized by a first-order transient response.; The capacitor is considered fully charged after approximately $5	au$ (5 time constants).; The charging current $i(t) = \frac{V_s}{R} e^{-\frac{t}{RC}}$ also decreases exponentially over time.

Advantages: Predictable transient behavior based on RC time constant.; Essential for timing circuits, filters, and oscillators.

Disadvantages: Voltage does not reach full source potential instantaneously.; Requires high precision components to maintain stable timing in sensitive applications.

Applications: Timer circuits (e.g., NE555 IC circuits).; Pulse generation and signal filtering.; Smoothing circuits in DC power supplies.

Additional Info: The exponential nature arises from the fact that as $v_c$ increases, the voltage drop across the resistor decreases, leading to a reduction in current flow.. Linear charging only occurs if the capacitor is charged by a constant current source, not a constant voltage source through a resistor.

Answer: D is correct — the capacitor voltage increases exponentially following the equation $v_c(t) = V_s(1 - e^{-\frac{t}{RC}})$.

Exam Tip: Remember that in an RC circuit, the current decreases exponentially, while the voltage increases exponentially; they are complementary processes.

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