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 \(200 \mathrm{V} ?\)

An RC circuit, or resistor-capacitor circuit, is a simple electric circuit where a resistor and capacitor are connected in series or parallel. This form of circuit is fundamental in electrical engineering and physics because it demonstrates how voltage and current decay over time in a system.

In our exercise, the uncharged capacitor connected in series with a resistor begins accumulating charge when a voltage is applied. The time it takes to charge (or discharge) through the resistor dictates how quickly the overall voltage across the capacitor changes. Understanding the charging process is critical in timing applications, such as setting a delay in an electronic device or filtering signals.

Exponential decay is a mathematical concept that is observed when a capacitor charges or discharges through a resistor. The voltage across the capacitor increases rapidly at first, but the rate of change slows down as it approaches the maximum value determined by the applied voltage.

This behavior is described by a formula containing an exponential function, detailed in our solution step. It's important to grasp that as time progresses, the changes become more subtle, resulting in a curve that never truly reaches the full voltage but gets tantalizingly close. Explaining this idea often requires visual aids like graphs showing the asymptotic approach to the maximum voltage.

The time constant of an RC circuit, typically denoted as \( \tau \), is a measure of the time it takes for the voltage across the capacitor to reach approximately 63.2% of its maximum value during charging. It's calculated by multiplying the resistance (in ohms) by the capacitance (in farads).In the context of our exercise, it is found by the equation \( \tau = R \cdot C \), which, with the given values, would be \( \tau = 470 \times 10^3 \cdot 10 \times 10^{-6} \). The time constant is an essential concept not just in understanding the dynamics of the specific RC circuit at hand, but in the broader realm of circuit analysis and design. It helps predict the behavior of the circuit over time and in response to different input voltages.