The capacitor in Fig. 34-1 is initially uncharged when the...

The capacitor in Fig. $34-1$ is initially uncharged when the switch is closed. Find the current in the circuit and the charge on the capacitor five seconds later. Use $R=7.00 \mathrm{M} \Omega, \mathrm{C}=0.300 \mu \mathrm{F}$, and $\varepsilon=12.0 \mathrm{~V}$

Key Concepts

Transient Response

The transient response of a circuit refers to the short-term behavior of electrical quantities immediately after a change, such as the closing of a switch. In the context of RC circuits, it involves the analysis of how the capacitor's voltage, the current through the resistor, and the charge on the capacitor change from their initial conditions to their steady-state values. This analysis is critical for designing circuits that must meet speed or timing requirements.

RC Circuit

An RC circuit is an electrical circuit consisting of a resistor and a capacitor. It is one of the simplest circuits used in electronics to study transient phenomena where the capacitor charges or discharges over time. The analysis of such circuits is foundational for understanding how systems respond to sudden changes, such as closing a switch.

Time Constant (?)

The time constant, denoted by ?, is defined as the product of the resistance (R) and the capacitance (C) in an RC circuit. It represents the characteristic time over which the capacitor charges (or discharges), with about 63% of the final change occurring in one time constant. This concept is essential in determining the speed and dynamics of the circuit's transient response.

Exponential Charging Equation

The process of capacitor charging in an RC circuit follows an exponential behavior. The charging equation is given by Q(t) = Q_final * (1 - e^(-t/?)), where Q_final is the final charge on the capacitor and ? is the time constant. Similarly, the current decays exponentially as I(t) = (?/R) * e^(-t/?). These equations provide a mathematical model for predicting how the charge and current evolve over time during the transient state.

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