An uncharged 10-μF capacitor and a 470-k Ωresistor are in...

An uncharged $10-\mu \mathrm{F}$ capacitor and a $470-\mathrm{k} \Omega$ resistor are in series, and $250 \mathrm{V}$ is applied across the combination. How long does it take the capacitor voltage to reach $200 \mathrm{V} ?$

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Key Concepts

Exponential Functions and Logarithms

Exponential functions appear in the behavior of RC circuits, representing the gradual change of voltage over time. In conjunction with logarithms, these functions allow for solving equations to determine time intervals, such as the time required for the capacitor voltage to reach a certain level during charging or discharging.

Capacitor Charging Equation

The capacitor charging process in an RC circuit is described by the equation V(t) = V_final (1 - e^(-t/RC)), where V(t) is the voltage across the capacitor at time t, and V_final is the applied supply voltage. This equation models the exponential approach of the capacitor voltage toward the steady state value.

RC Circuit

An RC circuit consists of a resistor and a capacitor connected in series or parallel, where the resistor controls the flow of current charging or discharging the capacitor. This circuit is fundamental in electronics for filtering, timing, and transient response applications.

Time Constant

The time constant, denoted as ? and calculated by the product of the resistance (R) and capacitance (C), characterizes the rate at which the capacitor charges or discharges. It is a measure of how quickly the circuit responds to changes, with larger time constants indicating slower responses.

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