Study of Capacitors in Series and Parallel Circuits

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A capacitor is a fundamental passive electronic component that stores electrical energy in the form of an electrostatic field between two conductive plates. The ability of a capacitor to store charge is known as its capacitance ($C$). It is mathematically defined as the ratio of the stored electric charge ($Q$) to the applied voltage ($V$):
$$C = \frac{Q}{V}$$
Network ConfigurationsCapacitors are frequently combined in electronic circuits to achieve a specific target capacitance. Interestingly, their mathematical behavior in networks is the exact opposite of resistors.Series Connection: When capacitors are connected end-to-end, the total equivalent capacitance ($C_{eq}$) decreases. The equivalent capacitance will always be less than the smallest individual capacitor in the chain.
$$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$$
Parallel Connection: When capacitors are connected across the same two nodes, the equivalent capacitance increases. The total is simply the direct sum of all the individual capacitors.
$$C_{eq} = C_1 + C_2 + C_3 + \dots$$
Transient Response (Charging & Discharging)In a direct current (DC) circuit containing a resistor and a capacitor (an RC circuit), the capacitor does not charge or discharge instantly. Instead, the voltage across the capacitor changes exponentially over time. The rate of this exponential change is governed by the circuit's Time Constant ($\tau$), defined as $RC$ (measured in seconds).Charging Phase: When connected to a DC voltage source ($V$) through a resistor, the voltage across the capacitor ($V_c$) at any given time $t$ builds up according to the equation:
$$V_c(t) = V(1 - e^{-t/RC})$$
Discharging Phase: When the power source is bypassed and the fully charged capacitor discharges its stored energy through the resistor, the voltage decays exponentially:
$$V_c(t) = V e^{-t/RC}$$

To verify the relationship of equivalent capacitance for capacitors connected in series and in parallel.

To measure the charging and discharging behavior of a capacitor through a resistor.

To compare experimental results with theoretical values.

DC Power Supply / Battery (6–12 V)

Capacitors (e.g., 10 µF, 22 µF, 47 µF)

Resistor (1 kΩ – 10 kΩ for charging/discharging)

Voltmeter (or Digital Multimeter)

Stopwatch (or oscilloscope for accurate time measurement)

Connecting wires, breadboard, switch

Part A: Capacitors in SeriesCircuit Assembly: Select three capacitors with known values ($C_1$, $C_2$, and $C_3$) from the component palette. Connect them end-to-end in a series configuration across the DC power supply.Voltage Measurement: Turn on the power supply. Using the virtual multimeter, measure and record the total applied voltage ($V_{total}$) and the individual voltage drops across each specific capacitor ($V_1$, $V_2$, and $V_3$).Calculation & Verification: * Verify Kirchhoff's Voltage Law by ensuring $V_{total} = V_1 + V_2 + V_3$.Calculate the theoretical equivalent capacitance ($C_{eq}$) using the series formula and compare it against the experimental data.Part B: Capacitors in ParallelCircuit Assembly: Reconfigure your workspace by connecting the same three capacitors ($C_1$, $C_2$, and $C_3$) in a parallel arrangement across the DC power supply.Voltage Measurement: Turn on the power supply and use the multimeter to measure the voltage across each individual capacitor branch. Observe and verify that the voltage across each parallel branch is identical and equals the total supply voltage ($V_1 = V_2 = V_3 = V_{total}$).Calculation & Verification: Calculate the theoretical equivalent capacitance ($C_{eq}$) using the parallel formula ($C_{eq} = C_1 + C_2 + C_3$) and compare it with the measured system capacitance.

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