DC Circuits: EMF, Terminal Voltage, and Series/Parallel Components

Chapter 19: DC Circuits and Circuit Analysis

Introduction to DC Circuits

Direct current (DC) circuits are fundamental in physics and engineering, involving the flow of electric charge in a single direction. This chapter covers the behavior of resistors, capacitors, and sources of electromotive force (EMF) in series and parallel arrangements, as well as the analysis of complex circuits using Kirchhoff's Rules.

Electromotive Force (EMF) and Terminal Voltage

Definition of EMF

Electromotive force (EMF) is the energy provided by a battery or generator per coulomb of charge. It is not a force, but a potential difference (voltage) that drives current in a circuit.

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

• Measured in volts (V).

Example: A 1.5 V battery has an EMF of 1.5 V.

Terminal Voltage

The terminal voltage is the actual voltage measured across the terminals of a battery when it is supplying current. It is typically less than the EMF due to the battery's internal resistance.

• Internal resistance (r): The small resistance inside the battery that causes energy loss.

• When current (I) flows, the terminal voltage (\( V_{ab} \)) is:

$V_{ab} = \mathcal{E} - Ir$

Example: If a battery with \( \mathcal{E} = 12\,\text{V} \) and \( r = 1\,\Omega \) supplies \( I = 2\,\text{A} \), then \( V_{ab} = 12 - 2 \times 1 = 10\,\text{V} \).

Resistors in Series and in Parallel

Series Connection

Resistors are in series if they are connected end-to-end, providing a single path for current.

• The total resistance is the sum of individual resistances:

$R_{\text{eq}} = R_1 + R_2 + R_3 + \ldots$

Key Point: The same current flows through each resistor.

Example: Three resistors of 2, 3, and 5 \( \Omega \) in series have \( R_{\text{eq}} = 10\,\Omega \).

Parallel Connection

Resistors are in parallel if they are connected across the same two points, providing multiple paths for current.

• The reciprocal of the total resistance is the sum of reciprocals of individual resistances:

$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$

Key Point: The same voltage is across each resistor.

Example: Two resistors of 4 and 6 \( \Omega \) in parallel have \( R_{\text{eq}} = 2.4\,\Omega \).

Capacitors in Series and in Parallel

Series Connection

Capacitors in series have the same charge but different voltages across each capacitor.

• The reciprocal of the equivalent capacitance is the sum of reciprocals:

$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots$

Parallel Connection

Capacitors in parallel have the same voltage across each but can store different charges.

• The total capacitance is the sum of individual capacitances:

$C_{\text{eq}} = C_1 + C_2 + C_3 + \ldots$

EMFs in Series and in Parallel; Charging a Battery

Multiple EMF sources (batteries) can be connected in series or parallel to increase voltage or current capacity. Charging a battery involves applying a voltage greater than its EMF, causing current to flow in the reverse direction.

• Series: Voltages add up.

• Parallel: Current capacity increases, but voltage remains the same.

Kirchhoff's Rules

For complex circuits, Kirchhoff's Rules are used to analyze current and voltage:

• Junction Rule (Current Law): The sum of currents entering a junction equals the sum leaving it.

• Loop Rule (Voltage Law): The sum of potential differences around any closed loop is zero.

Example: In a circuit with two loops and three resistors, set up equations for each loop and junction to solve for unknown currents.

Ammeter and Voltmeter Effects

Measuring devices affect the circuit:

• Ammeter: Measures current; should have very low resistance to minimize effect on the circuit.

• Voltmeter: Measures voltage; should have very high resistance to minimize current draw.

Summary Table: Series vs. Parallel Connections

Property	Series	Parallel
Resistors	$R_{\text{eq}} = R_1 + R_2 + \ldots$	$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$
Capacitors	$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$	$C_{\text{eq}} = C_1 + C_2 + \ldots$
Current	Same through all elements	Divides among branches
Voltage	Divides among elements	Same across all branches

Practice Question Example

What can make a charge move?

• An electric field turned on in the region of the charge will make it move.

• A potential difference in the region of the charge will make it move.

• If it can lower the potential energy of the system, it will move.

• All of the above.

Correct answer: All of the above.

Additional info:

Some sections such as RC circuits and electric hazards are marked as skipped in the source material and are not included in these notes.