(II) In Fig. 19-56 (same as Fig. 19-20 a ) , the total resistance is 15.0 k Ω, and the battery's

step 1 In the first part of this problem we will calculate capacitance of capacitor time constant of r

(II) In Fig. 19-56 (same as Fig. 19-20 a ) , the total...

(II) In Fig. $19-56$ (same as Fig. $19-20 a )$ , the total resistance is 15.0 $\mathrm{k} \Omega$ , and the battery's emf is 24.0 $\mathrm{V}$ . If the time constant is measured to be 35.0$\mu \mathrm{s}$ , calculate $(a)$ the total capacitance of the circuit and $(b)$ the time it takes for the voltage across the resistor to reach 16.0 $\mathrm{V}$ after the switch is closed.

Key Concepts

RC Time Constant

The RC time constant (denoted by ?) is a fundamental concept in circuits that involve resistors and capacitors. It is defined as the product of resistance (R) and capacitance (C) and provides a measure of how fast the circuit responds to sudden changes, such as the closing of a switch. For instance, in a charging or discharging RC circuit, the time constant indicates the time needed for the capacitor’s voltage to either reach about 63.2% of its final value or decay to about 36.8% of its initial value, characterizing the exponential nature of the process.

Exponential Behavior in RC Circuits

The charging and discharging of capacitors in an RC circuit follow exponential laws. The voltage across the capacitor during charging is given by an equation that approaches the supply voltage asymptotically, while the voltage across the resistor decays exponentially. Analyzing these exponential functions often requires the use of natural logarithms to solve for the time at which the voltage reaches a specified value, making this concept essential for time-domain analysis in circuit theory.

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