The pair of capacitors in Figure P 28.59 are fully charged by a 12.0-V battery. The battery is d

step 1 In this question, the pair of capacitors are fully charged by a travel battery. Then the battery

The pair of capacitors in Figure P 28.59 are fully charged...

The pair of capacitors in Figure $\mathrm{P} 28.59$ are fully charged by a $12.0-\mathrm{V}$ battery. The battery is disconnected, and the switch is then closed. After 1.00 $\mathrm{ms}$ has elapsed, (a) how much charge remains on the $3.00-\mu \mathrm{F}$ capacitor? (b) How much charge remains on the $2.00-\mu \mathrm{F}$ capacitor? (c) What is the current in the resistor at this time?

Key Concepts

Charge Redistribution and Conservation

When multiple capacitors are involved, especially after disconnecting a charging source like a battery, the total charge in the system is conserved and will redistribute among the capacitors to eventually reach an equilibrium state. Understanding this redistribution is key to analyzing how the initial charges on the capacitors evolve over time once they are connected by a resistor.

Kirchhoff’s Circuit Laws

These laws, particularly Kirchhoff’s Voltage Law, are essential for analyzing the distribution of voltage in circuits. In transient analysis, they help set up the necessary equations that relate the voltage drops across the resistor and the capacitor, allowing one to derive the time-dependent behavior of the circuit.

RC Circuit Analysis

This concept involves studying circuits that include both resistors and capacitors, focusing on how voltages, currents, and charges change over time. The underlying behavior in such circuits is typically governed by differential equations that describe the rate at which capacitors charge or discharge through resistors, leading to exponential time-dependent responses.

Time Constant

The time constant, usually denoted as ? (tau), is defined as the product of resistance and capacitance (? = R×C) and represents the characteristic time over which the exponential charging or discharging occurs in an RC circuit. It indicates how quickly the circuit reaches a significant fraction of its final steady state value.

Exponential Decay

Exponential decay describes the mathematical behavior of a capacitor discharging through a resistor, where quantities like charge or voltage decrease exponentially over time. The decay is typically modeled by an equation of the form Q(t) = Q_initial × exp(–t/?), where Q_initial is the initial charge and ? is the time constant.

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