A 4.60-μF capacitor that is initially uncharged is connected...

A $4.60-\mu F$ capacitor that is initially uncharged is connected in series with a $7.50-\mathrm{k} \Omega$ resistor and an emf source with $\mathcal{E}=245 \mathrm{V}$ and negligible internal resistance. Just after the circuit is completed, what are (a) the voltage drop across the capacitor: (b) the voltage drop across the resistor: (c) the charge on the capacitor: (d) the current through the resistor'? (e) A long time after the circuit is completed (after many time constants) what are the values of the quantities in parts $(a)-(d) ?$

Key Concepts

Kirchhoff’s Voltage Law (KVL) in RC Circuits

Kirchhoff’s Voltage Law states that the sum of the potential differences (voltage drops) around any closed loop in a circuit must equal the total applied electromotive force. In the context of RC circuits, KVL is used to relate the voltage drop across the resistor and the voltage across the capacitor to the source voltage at any point in time. This principle is fundamental in deriving the equations governing the charging and discharging processes.

Transient Versus Steady State Response

In an RC circuit, the transient response refers to the behavior immediately after the circuit is completed, where voltages and currents change over time. The steady state is the long-time behavior after the transients have died out, with the capacitor fully charged and no current flowing through the resistor. Understanding the difference between these responses is essential for analyzing the performance of circuits over time.

Capacitor Charging Dynamics

When a capacitor is connected to a voltage source through a resistor, it does not charge instantaneously. Instead, the charge (and thus the voltage across the capacitor) increases gradually according to an exponential function. Initially, the capacitor acts like a short circuit, meaning the voltage across it is zero, and most of the voltage appears across the resistor. Over time, the capacitor voltage increases until it equals the source voltage.

RC Circuit Analysis

An RC circuit is one that consists of a resistor and a capacitor connected in series or parallel to a voltage source. It is used to study the time-dependent behavior of circuits, especially how capacitors charge and discharge. The analysis of RC circuits involves understanding how voltage and current change over time due to the interplay between resistance and capacitance.

Time Constant (?)

The time constant, defined as ? = RC, is a measure of the rate at which the capacitor charges or discharges. It represents the time required for the capacitor’s voltage to reach approximately 63.2% of its final value during charging, or for the current in the circuit to drop to about 36.8% of its initial value. This concept is crucial in predicting circuit behavior at different time intervals after the circuit is activated.

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