Capacitors charge and discharge in ____________ manner.

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Exponential

Explanation: Capacitors charge and discharge following an exponential curve rather than a linear one because the charging current decreases as the voltage across the capacitor increases. As the potential difference across the capacitor approaches the source voltage, the driving force (voltage difference) diminishes, leading to a rate of change that slows down over time. Working Principle: The charging process is governed by the time constant $ au = RC$. The capacitor voltage at any time $t$ during charging follows $v(t) = V_s(1 - e^{-t/ au})$. During discharging, the voltage follows $v(t) = V_0e^{-t/ au}$. Because the rate of change $\frac{dv}{dt}$ is proportional to the difference between the supply voltage and the current capacitor voltage, the growth/decay is logarithmic in nature, resulting in an exponential function. Key Formulas: $v(t) = V_s(1 - e^{-t/RC})$ — Charging voltage of a capacitor; $i(t) = \frac{V_s}{R} e^{-t/RC}$ — Charging current of a capacitor; $ au = RC$ — Time constant of an RC circuit Key Points: The time constant $ au = RC$ represents the time taken for the capacitor to charge to approximately 63.2% of its maximum voltage.; A capacitor is considered fully charged after approximately 5 time constants ($5 au$).; The exponential nature is a fundamental property of first-order linear time-invariant systems involving energy storage elements. Advantages: Provides predictable timing in RC oscillator circuits.; Useful for signal smoothing and filtering applications. Disadvantages: Non-linear output can be a limitation in high-precision linear signal processing.; Requires complex components to linearize the output if a ramp signal is needed. Applications: Timing circuits (e.g., 555 timers); Smoothing filters in DC power supplies; Signal coupling and decoupling Additional Info: Option A (Linear) is incorrect as it would require a constant charging current source, which is not provided by a simple resistor.. Option B (Constant) is incorrect as current and voltage vary with time.. Option C (Square) is incorrect as square waves are typically associated with switched waveforms rather than transient charge/discharge curves. Answer: D is correct — The charge and discharge process in a capacitor follows an exponential mathematical model due to the decaying current as the potential difference across the plates reaches the source voltage. Exam Tip: Always remember that $5 au$ is the rule of thumb for steady-state conditions in DC transient analysis for both RC and RL circuits.

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