A \(14.9-\mu F\) capacitor, a \(24.3-\mathrm{k} \Omega\) resistor, a switch, and a 25.-V battery are connected in series. What is the rate of change of the electric field between the plates of the capacitor at \(t=0.3621 \mathrm{~s}\) after the switch is closed? The area of the plates is \(1.00 \mathrm{~cm}^{2}\) .

Understanding the principles of an RC circuit charging is fundamental to analyzing how electrical circuits work, particularly when involving resistors (R) and capacitors (C). When you close the switch in a circuit consisting of a resistor, capacitor, and a voltage source, the capacitor doesn't charge instantaneously. Instead, it undergoes a fascinating process whereby its voltage gradually increases and approaches the battery voltage with time.

As current flows through the resistor, a voltage drop occurs across it (following Ohm's Law), which thereby reduces the initial current over time. Consequently, the charging rate of the capacitor is not uniform; it follows an exponential function. Initially, the charging happens rapidly but slows down as the capacitor's voltage equals the battery's. This important concept of RC circuit charging helps us understand the dynamic relationship of current, voltage, and time as they influence one another within the circuit.

When electric charge accumulates on a capacitor's plates, it creates an electric field in the region between them. This field is a vector quantity pointing from the positive to the negative plate and its strength, or magnitude, is proportional to the charge (Q) on the plates. We use the concept of an electric field (E) to describe the influence that a charged object exerts on other charges in the vicinity without making contact.

Mathematically, the electric field due to a capacitor can be determined by dividing the voltage across the capacitor (V) by the distance between the capacitor plates (d), typically given by the equation:
\(E = \frac{V_C}{d}\)
The electric field can also be expressed in terms of the charge on the capacitor and the area of the plates (A), using the constant of vacuum permittivity \(\rho_0\). This relationship is crucial as it lets us work out the electric field's strength at any given point in time for a charging or discharging capacitor.

An exponential decay pattern is commonly observed in the processes of charging and discharging in RC circuits. It shows how a quantity decreases at a rate proportional to its current value, which is exactly what happens to the voltage across a charging or discharging capacitor in an RC circuit.

Through the lens of mathematics, this behavior is described by the equation:
\(V(t) = V_0 e^{-\frac{t}{\tau}}\),
where \(V(t)\) is the voltage at time t, \(V_0\) is the initial voltage, \(e\) is the base of the natural logarithm, t is time, and \(\tau\) is the time constant of the circuit (the product of R and C). This concept of exponential decay is invaluable since it provides a predictive tool for determining not just when a capacitor will be fully charged but also how it discharges over time, and thus, the rate of change of various quantities, including voltage and electric field, over time.

The field of differential calculus is a powerful tool employed in physics to describe how physical quantities change over time. When we look at the rate of change of the electric field in a capacitor, we are delving into a real-world application of differential calculus. To ascertain how quickly the electric field between the plates of a capacitor is changing at any point in time, we differentiate the expression that relates electric field strength to charge and time.

Here, the rate of change of the electric field (\(\frac{dE}{dt}\)) is conceptually the same as finding the derivative of the electric field with respect to time, which in this case involves applying the chain rule since the electric field depends on the charge Q, which itself changes over time as the capacitor charges. In essence, the tools of differential calculus not only allow us to analyze the instantaneous rates of change in physical systems but also enhance our understanding and control over these systems by predicting their behavior over time.