Introduction

Relevance of the Theme

Welcome to the electrifying world of Resistivity and the 2nd Ohm's Law! With these tools, you will be equipped to understand and efficiently operate any resistive circuit. They are the core of the Physics of electrical currents and are the basis of numerous practical applications, from energy infrastructure to consumer electronics.

• Fundamentals of Electricity: These are the building blocks of electrical circuits. Understanding their function is essential to grasp more advanced topics in Physics and is the basis for topics in electrical and electronic engineering.

• Contextualization in Physics: Resistivity and the 2nd Ohm's Law are linked to the broader concept of electric current, which in turn is closely related to the concept of electric field. Therefore, studying resistivity and the 2nd Ohm's Law allows for a deeper exploration of fundamental laws and principles of Physics.

• Practical Relevance: These concepts are applied in numerous devices in our daily lives, as well as in important industrial sectors such as automation, telecommunications, electronics, and generation and transmission of electrical energy.

Contextualization

In this context, Resistivity and the 2nd Ohm's Law fit into a broader section dedicated to electrical circuits. They are usually addressed after introductory concepts of current and voltage, and precede the discussion on Joule's Law and the power dissipated in a resistor, which are subsequent topics in the study of the Physics of electricity.

• Resistivity: It is an inherent property of any material that opposes the flow of electric current through it. Understanding resistivity is essential to comprehend the behavior of circuits containing materials with different resistivities.

• 2nd Ohm's Law: It establishes that, in an electrical circuit where resistance is not constant, the electric current in a section of the circuit is equal to the variation of the electric voltage across this section in relation to the resistance, with resistance being the factor that modulates this relationship.

Resistivity and the 2nd Ohm's Law are interrelated in nature and are fundamental to understanding complex circuits. Let's explore them together!

Theoretical Development

Components

• Resistivity (ρ): The ability of a material to oppose the passage of electric current. Resistivity is an intrinsic characteristic of the material and is expressed in ohms per meter (Ω.m) in the International System of Units (SI). The higher the resistivity of a material, the more difficult the passage of electric current.

• Conductivity (σ): It is the measure of a material's ability to conduct electric current, being the inverse of resistivity (σ = 1/ρ). The unit of conductivity, in SI, is siemens per meter (S/m).

• Length (L): Distance that the electric current travels inside a material. It is measured in meters (m).

• Cross-Sectional Area (A): Surface through which the current flows. It is measured in square meters (m²).

• 2nd Ohm's Law: Formulated by Georg Simon Ohm, a German physicist from the 19th century, the 2nd Ohm's Law establishes that, in an electrical circuit where resistance is not constant, the electric current in a section of the circuit is equal to the variation of the electric voltage across this section in relation to the resistance, with resistance being the factor that modulates this relationship. Mathematically, it is represented by I = ∆V/R.

Key Terms

• Electric Current (I): It is the ordered flow of electrically charged particles. In the International System of Units (SI), it is measured in amperes (A).

• Electric Voltage (V): It is a potential energy per unit charge, available in an electric circuit to move electrons. In SI, it is measured in volts (V).

• Resistance (R): It is the opposition that a material offers to the passage of electric current. In SI, it is measured in ohms (Ω).

Examples and Cases

• Example 1 - Resistivity in Copper and Aluminum Wires: Consider a copper wire and an aluminum wire, both with a length of 1 meter and a diameter of 1mm. The resistivity of copper is 0.0000017 Ω.m, while that of aluminum is 0.00000282 Ω.m. Applying the resistance formula, R = ρL/A, where L is the length and A is the cross-sectional area, the resistance of the copper wire will be lower than that of the aluminum wire, indicating a greater capacity to conduct electric current.

• Example 2 - 2nd Ohm's Law with Variable Resistance: Let's consider a circuit composed of 3 resistors of 4Ω, 6Ω, and 8Ω in series, powered by a 12V voltage source. According to the 2nd Ohm's Law, the current in the circuit is constant and will be 1A. Applying the voltage formula, V = IR, in each resistor we will have voltages of 4V, 6V, and 8V, respectively, confirming the law.

With these examples, we hope you have a clearer view of what resistivity is and how it relates to the 2nd Ohm's Law. Now, let's practice!

Detailed Summary

Key Points

• Resistivity and Conductivity: These are fundamental properties of materials that determine how well they conduct electric current. Wires and components with low resistivity (high conductivity) are preferred in circuits to minimize energy losses.

• Ohmic Material Law: This states that the resistivity of a material is constant for a given temperature, regardless of voltage or current. Furthermore, voltage is directly proportional to current, with resistance being the proportionality factor.

• Relationship between Resistance, Length, and Cross-Sectional Area: The resistance of a conductor is directly proportional to its length and inversely proportional to the area of its cross-section. This relationship is expressed by the formula R = ρL/A, where R is the resistance, ρ is the resistivity, L is the length, and A is the cross-sectional area.

• Practical Implications of the 2nd Ohm's Law: The 2nd Ohm's Law allows predicting the behavior of current in circuits with variable resistances. In such circuits, currents are stronger in sections with lower resistance.

Conclusions

• Resistivity and the 2nd Ohm's Law are rudimentary yet vital components of the study of electricity. They establish the foundation for understanding the resistance of materials and currents in circuits.

• Resistivity is an intrinsic characteristic of materials and will essentially depend on their composition, temperature, and structural conditions.

• The 2nd Ohm's Law, in turn, provides us with a tool to analyze circuits where resistance varies, allowing us to calculate the current at any point in the circuit.

Suggested Exercises

1. Calculation of Resistance and Voltage in a Circuit Section: Given a circuit composed of a 5Ω resistor and a capacitor connected in series to a 10V voltage source, and knowing that the resistance of the capacitor varies over time, determine the voltage and resistance in the circuit section containing the resistor every 1 second after connecting the source.

2. Analysis of Resistivity of Different Metals: Consider an industrial application that requires a current of 1A to pass through a 10-meter-long wire. The wire can be made of copper or aluminum. Knowing that the resistivity of copper is 0.0000017 Ω.m and that of aluminum is 0.00000282 Ω.m, which of these two metals would be the best choice for the wire? Justify your answer.

3. Application of the 2nd Ohm's Law in a Complex Circuit: Considering the circuit below, where all resistors are ideal and the voltage source provides a potential difference of 12V, calculate the current passing through each of the resistors and the total current of the circuit.

   • R1 = 4Ω, R2 = 6Ω, R3 = 8Ω, R4 = 5Ω, R5 = 10Ω, R6 = 2Ω
   • Each resistor is connected to potential differences V1 = 9V, V2 = 6V, V3 = 3V, V4 = 4V, V5 = 2V respectively.
   • The direction of the current in the circuit is from A to B.