Series and Parallel Circuits: Resistors, Capacitors, and Grounding in Physics

Series and Parallel Circuits

Introduction

Understanding how resistors and capacitors combine in series and parallel arrangements is fundamental in circuit analysis. This section reviews the rules for calculating equivalent resistance and capacitance, and explores the concept of electrical ground in circuits.

Resistors in Series and Parallel

Resistors are components that oppose the flow of electric current, and their arrangement in a circuit affects the total (equivalent) resistance.

• Series Connection: Resistors are connected end-to-end, so the same current flows through each resistor.

• Parallel Connection: Resistors are connected across the same two points, so the voltage across each resistor is the same.

Formulas

• Series: The equivalent resistance is the sum of the individual resistances: $R_{eq} = R_1 + R_2$ For more resistors: $R_{eq} = R_1 + R_2 + R_3 + \ldots$

• Parallel: The reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$ For more resistors: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$

Example

Two resistors, $R_1 = 2\,\Omega$ and $R_2 = 3\,\Omega$, in series: $R_{eq} = 2 + 3 = 5\,\Omega$.

Capacitors in Series and Parallel

Introduction

Capacitors store electric charge and energy in an electric field. Their arrangement in a circuit determines the total (equivalent) capacitance.

• Series Connection: Capacitors are connected end-to-end, so the same charge passes through each capacitor.

• Parallel Connection: Capacitors are connected across the same two points, so the voltage across each capacitor is the same.

Formulas

• Series: The reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2}$ Generally: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots$

• Parallel: The equivalent capacitance is the sum of the individual capacitances: $C_{eq} = C_1 + C_2$ Generally: $C_{eq} = C_1 + C_2 + C_3 + \ldots$

Example

Two capacitors, $C_1 = 2\,\mu F$ and $C_2 = 2\,\mu F$, in series: $\frac{1}{C_{eq}} = \frac{1}{2} + \frac{1}{2} = 1 \implies C_{eq} = 1\,\mu F$.

Comparing Equivalent Capacitance in Circuits

Which Circuit Has a Larger Equivalent Capacitance?

Given two circuits, one with capacitors in series and one with capacitors in parallel, the parallel arrangement always yields a larger equivalent capacitance.

• Series: $C_{eq} = \frac{C}{2}$ (for two identical capacitors)

• Parallel: $C_{eq} = C + C = 2C$

• Conclusion: $C_{eq,\,parallel} > C_{eq,\,series}$

Potential Difference and Charge in Capacitor Circuits

Potential Difference Across Capacitors

• In series, capacitors have the same charge, but the voltage divides according to their capacitances: $V_1 = \frac{Q}{C_1}$, $V_2 = \frac{Q}{C_2}$.

• In parallel, each capacitor has the same voltage as the battery: $V_1 = V_2 = V_B$.

• Capacitor with the smallest capacitance in series has the largest voltage across it.

Charge on Capacitors

• In series, all capacitors have the same charge: $Q = C_{eq} V_B$.

• In parallel, each capacitor's charge is $Q_i = C_i V_B$.

Example

For $C_1 = 1.0\,\mu F$, $C_2 = 2.0\,\mu F$ in series with $V_B$ applied: $\frac{1}{C_{eq}} = \frac{1}{1.0} + \frac{1}{2.0} = 1.5 \implies C_{eq} = 0.67\,\mu F$ Each capacitor has charge $Q = C_{eq} V_B$.

Complex Capacitor Circuits

Calculating Equivalent Capacitance and Voltage

For a circuit with $C_0 = 2.0\,nF$, $C_1 = 4.0\,nF$, $C_2 = 6.0\,nF$, and $V_B = 10\,V$:

• First, combine $C_1$ and $C_2$ in parallel: $C_p = C_1 + C_2 = 10.0\,nF$

• Then, combine $C_0$ and $C_p$ in series: $\frac{1}{C_{eq}} = \frac{1}{C_0} + \frac{1}{C_p}$ $\frac{1}{C_{eq}} = \frac{1}{2.0} + \frac{1}{10.0} = 0.6 \implies C_{eq} = 1.67\,nF$

• The charge on $C_0$ is $Q = C_{eq} V_B$.

• The voltage across $C_0$ is $V_0 = Q / C_0$.

Ground in Electrical Circuits

Definition and Importance

• Ground is a reference point in an electrical circuit from which voltages are measured. It is often connected to the Earth to provide a common return path for electric current.

• Grounding is crucial for safety, preventing electric shock and protecting equipment from voltage surges.

• In circuit diagrams, ground is typically represented by a symbol and is considered to have zero voltage.

Example

In household wiring, the ground wire connects exposed metal parts to the Earth, reducing the risk of electric shock.

Voltmeters and Ammeters

Measuring Voltage and Current

• Ammeter: Measures current at a single point in a circuit. It must be connected in series with the component whose current is to be measured.

• Voltmeter: Measures the potential difference (voltage) between two points in a circuit. It must be connected in parallel with the component across which the voltage is to be measured.

• In an ideal wire (zero resistance), the voltage difference between two points is zero.

Example

To measure the current through a resistor $R_1$, place the ammeter in series with $R_1$. To measure the voltage across $R_1$, place the voltmeter in parallel with $R_1$.

Summary Table: Series vs. Parallel (Resistors and Capacitors)

Component	Series	Parallel
Resistor	$R_{eq} = R_1 + R_2 + \ldots$	$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$
Capacitor	$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$	$C_{eq} = C_1 + C_2 + \ldots$