Worksheet 6: Ohm’s Law and DC Circuits – Kirchhoff’s Laws¶

Understanding the behavior of electrical circuits is essential in physics and engineering. In this lab, we will explore key principles governing direct current (DC) circuits, including Ohm’s Law, Kirchhoff’s Current Law (KCL), and Kirchhoff’s Voltage Law (KVL). These laws form the foundation for circuit analysis and allow us to determine the relationships between voltage, current, and resistance in complex electrical networks.

If you need a review of Ohm's Law, please reference the introduction material for worksheet 4.

Resistors in Series and Parallel¶

When resistors are connected in series, the total or equivalent resistance is the sum of individual resistances:

$R_{\text{eq, series}} = R_1 + R_2 + R_3 + \dots$

In a parallel circuit, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances:

$\frac{1}{R_{\text{eq, parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$

These principles will be experimentally verified in this lab.

Kirchhoff’s Laws¶

Kirchhoff’s Laws provide a more comprehensive method for analyzing circuits:

Kirchhoff’s Current Law (KCL) states that at any junction (node) in a circuit, the sum of currents entering must equal the sum of currents leaving:

$\sum I_{\text{in}} = \sum I_{\text{out}}$

This law is based on the conservation of charge and is essential for analyzing circuits with multiple branches.

Kirchhoff’s Voltage Law (KVL) states that the sum of the voltages around any closed loop in a circuit must be zero:

$\sum V = 0$

This law is derived from the conservation of energy, ensuring that all voltage drops and gains in a loop cancel out.

Internal Resistance of a Battery¶

Real batteries are not ideal voltage sources; they have an internal resistance $r$ , which affects the terminal voltage $V_t$ when a current $I$ is drawn:

$V_t = E - Ir$

where $E$ is the electromotive force (EMF) of the battery. In this lab, we will determine the internal resistance experimentally using different load resistors and a linear regression analysis.

Objectives¶

By the end of this lab, you will be able to:
- Measure and analyze the internal resistance of a battery.
- Investigate the behavior of resistors in series and parallel circuits.
- Apply Kirchhoff’s Current and Voltage Laws to analyze and verify circuit behavior.
- Use linear regression to determine internal resistance and electromotive force (EMF) from experimental data.
- Develop proficiency in using digital multimeters (DMMs) to measure voltage, current, and resistance in DC circuits.

Materials List¶

Digital Multimeter (DMM)
Batteries: 1.5V and 3.0V
Resistors: 100 Ω, 220 Ω, 330 Ω, 470 Ω
Breadboard
Connecting Cables and Cable Clips

Measurement 1: Determining the Internal Resistance of a Battery¶

You will measure the open-circuit voltage and terminal voltage of a battery while varying the load resistance.
Using Ohm’s Law and linear regression, the internal resistance of the battery will be determined.
A graph of terminal voltage vs. current will be plotted, and the internal resistance will be extracted from the slope.

Measurement 2: Investigating Resistors in Parallel¶

A parallel circuit will be assembled using four resistors.
The voltage across each resistor and the total circuit current will be measured.
The current division rule and the equation for equivalent resistance in parallel circuits will be verified.

*Figure 1:* Resistor parallel circuit on breadboard

*Figure 1:* Resistor parallel circuit connected to battery

Measurement 3: Verification of Kirchhoff’s Laws in a Two-Loop Circuit¶

A mixed circuit with two voltage sources and multiple resistors will be constructed.
You will measure the current through different branches and voltage drops across components.
The collected data will be used to verify Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL) by checking if the sum of currents at junctions and the sum of voltage drops around loops follow theoretical predictions.