Lesson 5.2: Resistance, Resistivity and I–V Characteristics

Introduction

Welcome to Lesson 5.2! Today, we will dive into some fundamental concepts of electricity—the backbone of many modern technologies. By the end of this lesson, you will understand resistance, resistivity, and I-V characteristics of various materials.

Objectives

By the end of this lesson, you should be able to:

Explain Ohm's Law and the conditions under which it holds.
Describe resistance and resistivity, including the effects of length, area, and temperature.
Analyze the I-V characteristics of various components, including ohmic conductors, filament lamps, diodes, and thermistors.
Understand the concept of superconductivity and the temperature dependence of resistance.
Calculate resistance from resistivity and dimensions.

Section 1: Understanding Ohm's Law

Ohm's Law is a fundamental principle in electricity that states:

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

Where:

$I$ is the current in Amperes (A)
$V$ is the voltage in Volts (V)
$R$ is the resistance in Ohms (Ω)

Ohm's Law holds for materials known as ohmic conductors. These conductors have a constant resistance regardless of the voltage across them. For example, consider a simple copper wire connected to a battery. As the voltage increases, the current will also increase proportionally, indicating linear behavior.

However, not all materials obey Ohm's Law. Devices like diodes and thermistors behave differently, showing that resistance can change with temperature and voltage.

Real-world Example

Imagine you have a circuit with a light bulb connected to a battery. If you increase the voltage from 3 volts to 6 volts, and the light bulb is an ohmic conductor, you would see the current double! But what if you replaced the light bulb with a diode? The behavior will be different, and we’ll explore that in the next sections.

Section 2: Exploring Resistance and Resistivity

Resistance is a measure of how much a material opposes the flow of electric current. Factors affecting resistance include:

Length ($L$): Longer wires have more resistance.
Cross-sectional Area ($A$): Thicker wires have less resistance.
**Material (Resistivity

ho)**: Different materials have different resistivities.

The formula for resistance is given by:

$$R =

ho $\cdot$ $\frac{L}{A}$$$

Where:

$R$ is resistance (Ω)

ho is resistivity (Ω·m) of the material

$L$ is the length of the conductor (m)
$A$ is the cross-sectional area (m²)

Factors Affecting Resistance

Length: If you double the length of a wire, you double the resistance.
Area: If you double the cross-sectional area, you halve the resistance.
Temperature: For most materials, resistance increases with temperature. For example, if you heat a metal wire, the atoms vibrate more, making it harder for electrons to flow.
Superconductivity: At very low temperatures, certain materials exhibit zero electrical resistance, allowing current to flow without energy loss.

Section 3: I-V Characteristics of Different Components

Now, let’s look at the I-V characteristics of various components:

3.1 Ohmic Conductors

For ohmic materials, the relationship between current and voltage is linear. When you graph it, you get a straight line.

3.2 Filament Lamp

In filament lamps, the I-V characteristic is non-linear. At low voltages, the resistance is low, but as the filament heats up, resistance increases significantly.

3.3 Diode

Diodes only allow current to flow in one direction. Their I-V characteristic graph shows a rapid increase in current after a certain voltage threshold (the forward-bias region). Below this threshold, no significant current flows (the reverse-bias region).

3.4 Thermistor

Thermistors are temperature-sensitive resistors. When temperature increases, their resistance decreases significantly, leading to high current flow under higher temperatures.

Real-world Application

Think about how diodes are used in chargers and power supplies. They ensure that current flows only in the desired direction, preventing damage to electronic devices. Thermistors can be found in temperature sensors, helping to regulate air conditioning systems.

Section 4: Calculating Resistance from Resistivity

Let’s do a quick calculation! Suppose you have a copper wire with a length of 2 meters and a cross-sectional area of $1 \text{ mm}^2$ (which is $1 \times 10^{-6} \text{ m}^2$). The resistivity of copper is approximately $1.68 \times 10^{-8} \, \Omega·m$.

To find the resistance, you plug the values into the formula:

$$R =

ho $\cdot$ $\frac{L}{A}$ = $1.68 \times 10^{-8}$ $\cdot$ $\frac{2}{1 \times 10^{-6}}$ = 0.0336 \, $\Omega$$$

So, the resistance of the copper wire is approximately $0.0336 \, \Omega$!

Conclusion

In this lesson, we explored the concepts of resistance, resistivity, and how they apply in different scenarios. Understanding these principles is crucial for working with circuits and electronic devices. Remember that factors such as length, area, temperature, and material type significantly influence resistance and behavior in real-world applications.

Study Notes

Ohm's Law explains the linear relationship between voltage and current across ohmic materials.
Resistance depends on length, area, and resistivity.
Different materials show varying I-V characteristics: ohmic conductors have linear graphs, while non-ohmic devices do not.
Superconductors exhibit zero resistance at very low temperatures.
Resistance can be calculated using R =

ho $\cdot$ $\frac{L}{A}$.