(II) The R C circuit of Fig. 19-57 (same as Fig. 19-21 a ) has R=6.7 k Ωand C=3.0 μF . The capac

step 1 At t equal to 0, voltage across capacitor is equal to v0 and we want to find the time at which v

(II) The R C circuit of Fig. 19-57 (same as Fig. 19-21 a )...

(II) The $R C$ circuit of Fig. $19-57$ (same as Fig. $19-21 \mathrm{a} )$ has $R=6.7 \mathrm{k} \Omega$ and $C=3.0 \mu \mathrm{F}$ . The capacitor is at voltage $V_{0}$ at $t=0,$ when the switch is closed. How long does it take the capacitor to discharge to 1.0$\%$ of its initial voltage?

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Key Concepts

RC Circuit

An RC circuit is an electrical circuit consisting of a resistor and a capacitor. In such circuits, the resistor controls the rate at which the capacitor charges or discharges, resulting in characteristic exponential behavior over time. This basic configuration is fundamental in understanding time-dependent voltage and current changes in many electronic applications.

Time Constant

The time constant (denoted as ?) in an RC circuit is defined as the product of resistance (R) and capacitance (C). It provides a measure of how quickly the circuit responds, with the voltage across the capacitor decaying to about 37% of its initial value after one time constant. The time constant is a key parameter in analyzing the exponential charging and discharging processes in RC circuits.

Exponential Decay

Exponential decay refers to the process by which the voltage across a capacitor in a discharging RC circuit decreases exponentially over time. The decay is modeled by the equation V(t) = V?e^(?t/?), where V? is the initial voltage. By taking the natural logarithm of both sides, one can solve for the time required for the voltage to reach a specified fraction of its initial value, making this concept essential for problems involving capacitor discharge.

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