Obtain an expression for the rate of increase ( \(d V / d t\) ) of the voltage across a charging capacitor in an \(R C\) circuit. Evaluate your result at time \(t=0,\) and show that if the capacitor continued charging steadily at this rate, it would reach full charge in exactly one time constant.

Differential calculus plays a pivotal role in physics, especially when examining how quantities change over time. In the context of an RC circuit, where a resistor and capacitor are connected in series to a power supply, we often need to know how quickly the voltage across the capacitor is changing. This rate of change is mathematically represented by the derivative of the voltage with respect to time, denoted as \(dV/dt\).

When the switch is closed, the capacitor begins to charge, and the voltage across it starts to increase. To calculate this increase, we apply differential calculus to find the instantaneous rate of change of the voltage. It's essential to understand that this derivative gives us a snapshot of the charging rate at any given moment, which is crucial for analyzing the circuit's behavior at various points in time.

The voltage across a capacitor in an RC circuit is not constant; it changes over time as the capacitor charges or discharges. When the capacitor is charging from a power supply with voltage \(V_0\), the voltage across the capacitor at any time \(t\) is represented by the equation \(V(t) = V_0 [1 - e^{-t/RC}]\).

This result is derived using Kirchhoff's voltage law and the fact that capacitors store electrical energy. It's important to notice that the voltage \(V(t)\) starts at zero when the capacitor is uncharged and approaches the supply voltage \(V_0\) as the capacitor becomes fully charged. The exponential term in this equation reflects the non-linear charging process of the capacitor, with the rate slowing down as it approaches full charge.

The time constant, symbolized as \(\tau = RC\), is a critical parameter in an RC circuit that gives us a measure of how quickly the circuit responds to changes, such as charging or discharging of the capacitor. It is the product of resistance \(R\) and capacitance \(C\), and it defines the time required for the voltage across the capacitor to either charge to approximately 63.2% of the power supply voltage or discharge to around 36.8% of its initial charged value.

The time constant also represents the time taken for the capacitor to nearly fully charge or discharge through the resistor. Understanding this concept is vital because it not only characterizes the transient response of the circuit but also helps in designing circuits with desired timing characteristics. For instance, in timing applications, such as in blinking LEDs or controlling the release of a camera's shutter, the time constant is a key design factor.

Exponential decay is a fundamental concept that describes the decrease of a quantity at a rate proportional to its current value. In RC circuits, the charging and discharging of a capacitor follow an exponential decay pattern. Mathematically, this is characterized by a negative exponent in the formula for voltage across the capacitor, \(V(t) = V_0 [1 - e^{-t/RC}]\), where the term \(e^{-t/RC}\) represents the exponential decay.

Exponential decay is also evident in the formula for the rate of voltage increase \(dV/dt\) as the capacitor charges. It indicates that the rate is highest at the beginning when \(t=0\) and decreases exponentially as time progresses. This behavior is intrinsic to many physical processes where the rate of change is proportional to the current state, such as in radioactive decay, cooling of hot objects, and even in financial models like compound interest.