Introduction

Relevance of the Topic

"Resistor Association" is one of the fundamental pillars of Electricity in Physics. The lesson aims to provide a solid foundation for understanding more complex circuits and the practical implications of resistance in electrical systems. The ability to analyze and calculate resistance in circuits with different arrangements of resistors is essential for the development of advanced skills in Physics and engineering, as well as practical knowledge in various everyday applications, such as lighting and electronic circuits.

Contextualization

Within the teaching plan, "Resistor Association" is part of the broader topic that covers Ohm's Law and Circuit Analysis, closing the initial cycle of understanding Electricity. This topic is a prelude to the understanding of more advanced topics, such as power in circuits, Joule's law, and Kirchhoff's laws. The concepts learned here will be the basis for the in-depth study of these and other Physics topics. This unit also prepares students for the application of complex mathematical concepts and techniques, such as series summation and the application of differential equations, in Physics and related areas.

By understanding "Resistor Association", students will be able to solve practical and theoretical problems involving resistance in circuits with different arrangements of resistors. This understanding can also be extended to comprehend other forms of resistance in Physics, such as resistance in conductive materials.

Theoretical Development

Components

• Electrical Resistance (R): Measured in ohms (Ω), electrical resistance is the opposition that a material offers to the flow of electric current through it. It is a characteristic of the material and the length and area of the conductor's cross-section.

• Ohm's Law (V = I.R): Ohm's Law is one of the most fundamental relationships in electricity. It states that the current passing through a conductor is directly proportional to the applied voltage and inversely proportional to the conductor's resistance.

• Series Circuit: In a circuit with resistors in series, the current is the same in all resistors, but the voltage is shared among them according to the resistance each one offers. The total resistance of the circuit is the sum of the individual resistances.

• Parallel Circuit: In a circuit with resistors in parallel, the voltage is the same across all resistors, but the current is shared among them according to the resistance each one offers. The total resistance of the circuit is the inverse of the sum of the inverses of the individual resistances.

Key Terms:

• Resistor: An electrical circuit device that consumes electrical energy by opposing the flow of current.

• Series Circuit: Arrangement of components where the same current passes through all elements.

• Parallel Circuit: Arrangement of components where the same voltage is applied to all elements.

Examples and Cases:

• Example of Series Circuit with 3 Resistors: Consider a circuit with a 12V battery and 3 resistors of 2Ω, 4Ω, and 6Ω in series. Using Ohm's Law and the properties of the series circuit, we can calculate the current (I) passing through the circuit and the voltage (V) dropping across each resistor.

  • The total circuit resistance is the sum of the individual resistances, R_total = 2Ω + 4Ω + 6Ω = 12Ω.
  • Using Ohm's Law (V = I.R), the current (I) is V/R_total = 12V/12Ω = 1A.
  • The voltage (V) across each resistor is I.R, so each resistor has a voltage of 2V, 4V, and 6V, respectively. It is noted that the sum of these voltages is the total battery voltage (12V), confirming Kirchhoff's Voltage Law.

• Example of Parallel Circuit with 3 Resistors: Let's consider a circuit with a 12V battery and 3 resistors of 2Ω, 4Ω, and 6Ω in parallel. Using Ohm's Law and the properties of the parallel circuit, we can calculate the total resistance (R_total) of the circuit and the current (I) flowing through the circuit.

  • The total resistance of resistors in parallel is the inverse of the sum of the inverses of the individual resistances: 1/R_total = 1/2Ω + 1/4Ω + 1/6Ω = 11/12Ω. Therefore, R_total = 12/11Ω.
  • Using Ohm's Law (V = I.R), the current (I) flowing through the circuit is V/R_total = 12V / (12/11)Ω = 11A.
  • The current flowing through each resistor is the same (Kirchhoff's Current Law), so each resistor will have a current of 11A. The voltage (V) across each resistor will be I.R, resulting in voltages of 22V, 44V, and 66V, respectively.

Detailed Summary

Key Points

• Resistance and Ohm's Law: Firstly, we understand the concept of electrical resistance (R), a property of materials that opposes the flow of current. Ohm's Law (V = I.R) is introduced to link resistance (R), current (I), and voltage (V) in a circuit.

• Series Circuits: The arrangement of resistors in a circuit affects both the current and voltage in each resistor. In the case of a series arrangement, these parameters are identical for all resistors. The total resistance (R_total) in a series circuit is obtained by summing the individual resistances (R_total = R1 + R2 + ... + Rn).

• Parallel Circuits: Unlike the series arrangement, in a parallel circuit, the voltage is the same across all resistors, while the current is divided among them. The total resistance (R_total) in a parallel circuit is the inverse of the sum of the inverses of the individual resistances (1/R_total = 1/R1 + 1/R2 + ... + 1/Rn).

Conclusions

• The ability to calculate and understand how resistors behave in different circuit arrangements (series and parallel) is crucial for the analysis of more complex circuits and for solving practical problems.

• By applying Ohm's Law and the principles of series and parallel circuits, we can determine the current in different parts of the circuit, the voltage across each resistor, and the total resistance of the circuit.

Exercises

1. Series Resistor Association: Given a circuit with a 10V battery and the following resistors: R1 = 2Ω, R2 = 3Ω, R3 = 5Ω, and R4 = 1Ω, calculate the current passing through the circuit and the voltage across each resistor.

2. Parallel Resistor Association: Suppose a circuit with a 20V battery and the following resistors: R1 = 5Ω, R2 = 10Ω, R3 = 20Ω, and R4 = 40Ω. Calculate the total resistance of the circuit, the current flowing through it, and the voltage across each resistor.

3. Circuit Design: Imagine you need to build a circuit with a total resistance of 50Ω, using only the available resistors: R1 = 8Ω, R2 = 12Ω, R3 = 18Ω, R4 = 32Ω, and R5 = 50Ω. How would you organize these resistors in a series or parallel circuit to achieve the desired resistance? What would be the current and voltage in the configuration you chose?