Chapter 19: DC Circuits: Series and Parallel Circuits, Kirchhoff’s Rules, and RC Circuits

DC Circuits

Electromotive Force (EMF) and Terminal Voltage

In direct current (DC) circuits, a source of electromotive force (emf) such as a battery provides the energy needed to move charges through the circuit. The terminal voltage is the actual voltage measured across the terminals of the battery, which is less than the emf due to the battery's internal resistance.

• EMF (\(\mathcal{E}\)): The maximum potential difference a battery can provide when no current is flowing.

• Terminal Voltage (V): The voltage available to the external circuit, given by \(V = \mathcal{E} - Ir\), where \(r\) is the internal resistance and \(I\) is the current.

• Unit of Resistance: Ohm (\(\Omega\)), where \(1\,\Omega = 1\,\mathrm{V/A}\).

Ohm's Law relates the current (I), voltage (V), and resistance (R) in a circuit:

$I = \frac{V}{R}$

Resistors in Series and Parallel

Resistors in Series

When resistors are connected in series, the same current flows through each resistor, but the voltage divides among them. The total or equivalent resistance is the sum of the individual resistances.

• Current: Same through all resistors: \(I = I_1 = I_2 = I_3\)

• Voltage: Total voltage is the sum of individual voltage drops: \(V = V_1 + V_2 + V_3\)

• Equivalent Resistance: \(R_s = R_1 + R_2 + R_3 + \cdots\)

$R_s = R_1 + R_2 + R_3$

Example: For \(R_1 = 10\,\Omega\), \(R_2 = 5\,\Omega\), \(R_3 = 10\,\Omega\), and \(V = 50\,\mathrm{V}\):

• \(R_s = 10 + 5 + 10 = 25\,\Omega\)

• \(I = \frac{50\,\mathrm{V}}{25\,\Omega} = 2\,\mathrm{A}\)

• \(V_1 = IR_1 = 2 \times 10 = 20\,\mathrm{V}\)

• \(V_2 = IR_2 = 2 \times 5 = 10\,\mathrm{V}\)

• \(V_3 = IR_3 = 2 \times 10 = 20\,\mathrm{V}\)

Resistors in Parallel

When resistors are connected in parallel, the voltage across each resistor is the same, but the current divides among the branches. The reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances.

• Voltage: Same across all resistors: \(V = V_1 = V_2 = V_3\)

• Current: Total current is the sum of branch currents: \(I = I_1 + I_2 + I_3\)

• Equivalent Resistance: $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots$

Example: For \(R_1 = 2\,\Omega\), \(R_2 = 100\,\Omega\), \(R_3 = 20\,\Omega\):

• $\frac{1}{R_p} = \frac{1}{2} + \frac{1}{100} + \frac{1}{20}$

• $R_p \approx 1.8\,\Omega$ (smallest possible resistance)

Series-Parallel Circuits

Many practical circuits are combinations of series and parallel connections. To analyze these, reduce the circuit step-by-step by replacing series and parallel groups with their equivalent resistances.

Kirchhoff’s Rules

Junction Rule (First Rule)

The sum of currents entering a junction equals the sum of currents leaving the junction. This is a consequence of the conservation of charge.

• $\sum I_{in} = \sum I_{out}$

Loop Rule (Second Rule)

The sum of the potential differences around any closed loop in a circuit must be zero. This is a consequence of the conservation of energy.

• $\sum \Delta V = 0$ around a closed loop

Example: For a single loop with emf \(\mathcal{E}\), internal resistance \(r\), and two resistors \(R_1\) and \(R_2\):

$\mathcal{E} - I r - I R_1 - I R_2 = 0$

Internal Resistance and Power in Batteries

Internal Resistance

Real batteries have internal resistance, which reduces the terminal voltage and the power delivered to the external circuit.

• Terminal Voltage: $V = \mathcal{E} - Ir$

• Current: $I = \frac{\mathcal{E}}{R_{load} + r}$

• Power delivered to load: $P_{load} = I^2 R_{load}$

• Power lost in battery: $P_{r} = I^2 r$

RC Circuits (Resistor and Capacitor in Series)

Charging and Discharging a Capacitor

An RC circuit consists of a resistor and a capacitor in series. The voltage and current in the circuit change with time as the capacitor charges or discharges.

• Time Constant: $\tau = RC$ (characteristic time for charging/discharging)

• Charging: $V_C = \mathcal{E}(1 - e^{-t/RC})$

• Discharging: $V_C = V_0 e^{-t/RC}$

• At $t = \tau$, $V_C = 0.632 \mathcal{E}$ (charging), $V_C = 0.368 V_0$ (discharging)

Applications: RC circuits are used for timing devices, such as camera flashes and windshield wipers.

Summary Table: Series vs. Parallel Resistors

Property	Series	Parallel
Current	Same through all resistors	Divides among branches
Voltage	Divides among resistors	Same across all resistors
Equivalent Resistance	$R_s = R_1 + R_2 + \cdots$	$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$

Measurement Devices

• Ammeter: Measures current; connected in series with the circuit element.

• Voltmeter: Measures voltage; connected in parallel with the circuit element.

Electric Hazards

To avoid electric shocks, never allow your body to become part of a complete circuit. Even small currents can be dangerous.