In the circuit shown in the figure, a 10.0-μF capacitor is charged by a 9.00-V battery with the

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In the circuit shown in the figure, a 10.0-μF capacitor is...

In the circuit shown in the figure, a $10.0-\mu \mathrm{F}$ capacitor is charged by a $9.00-\mathrm{V}$ battery with the two-way switch kept in position $X$ for a long time. Then the switch is suddenly flicked to position $\mathrm{Y}$. What current flows through the $40.0-\Omega$ resistor a) immediately after the switch moves to position Y? b) $1.00 \mathrm{~ms}$ after the switch moves to position Y?

In the circuit shown in the figure, a $10.0-\mu \mathrm{F}$ capacitor is charged by a $9.00-\mathrm{V}$ battery with the two-way switch kept in position $X$ for a long time. Then the switch is suddenly flicked to position $\mathrm{Y}$. What current flows through the $40.0-\Omega$ resistor
a) immediately after the switch moves to position Y?
b) $1.00 \mathrm{~ms}$ after the switch moves to position Y?

Key Concepts

Capacitor Charging and Discharging

This concept describes how capacitors store and release energy in electrical circuits. When connected to a battery, a capacitor gradually charges up to the battery’s voltage, and when the circuit configuration is changed, it begins to discharge through the available resistive path. The dynamics of charging and discharging are foundational in understanding energy storage and transient behavior in circuits.

RC Circuit Time Constant

The RC time constant is a key parameter defined as the product of the resistance (R) and capacitance (C) in the circuit. It characterizes the rate at which the capacitor charges or discharges, indicating the time needed for the capacitor to charge up to about 63% of the battery voltage or, similarly, to discharge to about 37% of its initial voltage. This parameter is crucial in determining the transient response of the circuit.

Transient Analysis

Transient analysis refers to studying the circuit's response immediately after a sudden change, such as a switch operation, before the circuit reaches a steady state. It involves calculating time-dependent currents and voltages, understanding how the capacitor’s voltage does not change instantaneously, and observing exponential charging or discharging curves characteristic of RC circuits.

Exponential Decay in RC Circuits

In an RC circuit undergoing discharge, the current and voltage decrease exponentially over time. This behavior is described mathematically by exponential functions, where the initial values decay at a rate determined by the RC time constant. This concept is essential for predicting the circuit behavior at any instant following a transient event.

Ohm’s Law in Transient Conditions

Ohm’s Law, which states that the current through a resistor is equal to the voltage across it divided by the resistance, is applied in transient analysis to calculate initial currents immediately following a switch change. Although the capacitor’s voltage can persist immediately after a change, Ohm’s Law allows for determining the corresponding instantaneous current through any resistive component in the circuit.

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