A capacitor of capacitance \(C_{1}\) has a charge \(Q_{0}\) on its plates. It is connected at time \(t=0\) in series with a capacitor of capacitance \(C_{2}\) and a resistor \(R\) by the closing of a switeh. Find the time constant of the ensuing discharge.

When capacitors are connected in series, the total capacitance changes in such a way that it differs from simply adding their individual capacitances together. The key to understanding series capacitance is remembering that the charge stored (\( Q \) in coulombs) on each capacitor must be the same, because the capacitors are connected end to end and electrons cannot accumulate at any point.

To calculate the equivalent capacitance (\( C_{eq} \) in farads) in series, you use the reciprocal formula: \[ \frac{1}{C_{eq}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} \,..., \frac{1}{C_{n}} \. \] Essentially, you are adding the reciprocals of the individual capacitances and then taking the reciprocal of that sum. The effect is that the overall capacitance is less than the smallest individual capacitor in the series.

Equivalent capacitance represents a simplification of a complex network of capacitors into a single capacitor that has the same effect on the circuit as the entire network. This concept is particularly helpful in analyzing and designing electrical circuits because it allows for a more manageable understanding of the circuit's potential behavior.

For series connections, the equivalent capacitance is lower than any single capacitor's value in the series. For parallel connections, by contrast, you would add the capacitances directly to find the equivalent. Understanding how to calculate equivalent capacitance clarifies how capacitors influence the storage and discharge of electric charge in a circuit.

A resistor-capacitor (RC) circuit is one of the simplest analog electronic circuits which includes a resistor (\( R \) in ohms) and a capacitor (\( C \) in farads). The primary feature of an RC circuit is its ability to charge and discharge the capacitor through the resistor. The time required for the charge on the capacitor to rise to within a certain percentage of its maximum value or to fall to a certain percentage of its prior value during discharging is known as the 'time constant' (\( \tau \) in seconds).

Mathematically, the time constant for a simple series RC circuit is given by the product of the resistance and the equivalent capacitance: \[ \tau = RC_{eq} \]. This time constant is a measure of the rate at which charge flows in or out of the capacitor. A large time constant means a slow charge or discharge process and vice versa.

Exponential discharge in an RC circuit describes how the voltage across a capacitor decreases over time as the capacitor releases its stored energy through the resistor. The voltage decreases exponentially, meaning that it reduces by a fixed percentage in each equal time period.

The voltage across the capacitor at any time (\( t \) in seconds) during the discharge can be calculated by the equation: \[ V(t) = V_0 e^{-t/\tau} \], where \( V_0 \) is the initial voltage at time \( t = 0 \) and \( \tau \) is the time constant for the circuit. This formula reflects the characteristic half-life behavior of an RC circuit, where \( \tau \) represents the time it takes for the voltage to drop to half of its original value. Through this, students can visualize and better understand the dynamics of charge movement within the circuit.