(a) In a charging R C circuit, how many time constants have...

(a) In a charging $R C$ circuit, how many time constants have elapsed when the capacitor has $99.0 \%$ of its final charge? (b) How many time constants have elapsed when the capacitor has $99.90 \%$ of its final charge? (c) How many time constants have elapsed when the current has $1.0 \%$ of its initial value?

(a) In a charging $R C$ circuit, how many time constants have elapsed when the capacitor has $99.0 \%$ of its final charge?
(b) How many time constants have elapsed when the capacitor has $99.90 \%$ of its final charge?
(c) How many time constants have elapsed when the current has $1.0 \%$ of its initial value?

Key Concepts

Exponential Charging

In an RC circuit, the charging of a capacitor follows an exponential behavior described by the equation Q(t) = Q_final*(1 - exp(-t/?)), where Q_final is the final charge and t is time. This exponential function illustrates that the capacitor’s charge gradually approaches its maximum value, never reaching it instantaneously, and allows for the determination of the elapsed time (in multiples of ?) needed to reach a specific percentage of Q_final.

Exponential Decay of Current

When a capacitor in an RC circuit begins to charge, the current flowing into the capacitor decays exponentially according to the relation I(t) = I_initial*exp(-t/?). Here, I_initial is the initial current at the moment charging commences. This decay relationship means that the elapsed time required for the current to drop to a given percentage of its initial value can be derived using natural logarithms, reflecting the exponential reduction of current over time.

RC Circuit

An RC circuit is an electrical network that comprises a resistor (R) and a capacitor (C) connected in series or parallel. The behavior of the circuit is characterized by the interplay between resistance and capacitance, which affects how electrical quantities such as voltage, current, and charge evolve over time. This dynamic is essential for understanding how capacitors charge and discharge in response to applied voltages.

Time Constant

The time constant, denoted by ? (tau), is a key parameter in RC circuits and is calculated as the product of the resistance and capacitance (? = RC). It represents the characteristic time scale over which the capacitor charges or discharges. For example, after one time constant, a charging capacitor reaches approximately 63.2% of its final charge, and the circuit's response can be expressed in terms of multiples of ?.

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