(II) In Fig. 26-58 (same as Fig. 26-17 a), the total...

(II) In Fig. $26-58$ (same as Fig. $26-17$ 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 $24.0 \mu \mathrm{s}$, 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 Circuits

An RC circuit is a system comprised of resistors and capacitors, where the interplay between these components governs the charging and discharging behavior of the capacitor. This concept is fundamental in understanding how energy is stored and released over time in various electrical systems.

Time Constant

The time constant, denoted by ?, is a key parameter in RC circuits defined as the product of resistance (R) and capacitance (C). It represents the characteristic time over which a capacitor charges or discharges, serving as a measure of the speed at which the circuit responds to changes in voltage.

Exponential Behavior in RC Circuits

During the charging or discharging processes in an RC circuit, both voltage and current change exponentially with time. For instance, the voltage across a charging capacitor approaches the supply voltage exponentially while the voltage across the resistor decays exponentially. This behavior is mathematically described by exponential functions that include the time constant.

Logarithmic Analysis for Time Calculation

When determining the time required for an RC circuit to reach a certain voltage level, logarithmic functions are used to invert the exponential relationship. By applying the natural logarithm to the exponential decay or growth formulas, one can solve for the specific time corresponding to a targeted voltage value.

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