Tru Physics

Chapter 10: Resistance and Resistivity

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10.1 Introduction to Resistance

Resistance is a property of a material that opposes the flow of electric current. It is a measure of how difficult it is for current to pass through a conductor. In this chapter, we will explore the concept of resistance and its relationship to current, voltage, and the physical properties of materials.

Resistance is color coded via the four different colored bands seen in the image. (Yellow, Blue, Red, Gold).
Resistors like the one above are extremely common is beginner digital electronics courses as well as hobby builds. Resistance is color coded via the four different colored bands (with the fourth corresponding to the percent tolerance). Typically resistors in everyday electronics like smartphones and televisions with be smaller and much more precise.

10.2 Definition of Resistance

The resistance (R) of a conductor is defined as the ratio of the voltage (V) across the conductor to the current (I) passing through it, as described by Ohm’s Law:

R = \dfrac{V}{I}

where

  • R is the resistance in ohms (\Omega),
  • V is the voltage in volts (V),
  • I is the current in amperes (A).

10.3 Resistivity

Resistivity, denoted by the Greek letter ‘rho’ (\rho), is an intrinsic property of a material that determines its resistance to the flow of electric current. It is dependent on the material’s atomic structure and temperature. The resistance of a conductor can be calculated using the following formula:

R = \rho \dfrac{L}{A}

where

  • R is the resistance in ohms (\Omega),
  • \rho is the resistivity in ohm-meters (\Omega \cdot \text{m}),
  • L is the length of the conductor in meters (m),
  • A is the cross-sectional area of the conductor in square meters (\text{m}^2).

10.4 Factors Affecting Resistivity

Several factors can affect a material’s resistivity, including:

  1. Material: Different materials have different resistivities due to their atomic structures.
  2. Temperature: Resistivity generally increases with temperature for conductors and decreases for semiconductors.
  3. Impurities: The presence of impurities in a material can significantly alter its resistivity.

10.5 Temperature Dependence of Resistance

The resistance of most materials changes with temperature. For conductors, the resistance typically increases with increasing temperature, while it decreases for semiconductors. The temperature dependence of resistance can be described by the following equation:

R(T) = R_0 \left[1 + \alpha(T - T_0)\right]

where

  • R(T) is the resistance at temperature T,
  • R_0 is the resistance at reference temperature T_0,
  • \alpha is the temperature coefficient of resistance,
  • T is the temperature in degrees Celsius (\text{°}C),
  • T_0 is the reference temperature in degrees Celsius (\text{°}C).

10.6 Power Dissipation in Resistors

When current flows through a resistor, some of the electrical energy is converted into heat due to the resistance of the material. This power dissipation (P) can be calculated using the following formulas:

P = I^2 R

or

P = \dfrac{V^2}{R}

where

  • P is the power in watts (W),
  • I is the current in amperes (A),
  • V is the voltage in volts (V),
  • R is the resistance in ohms (\Omega).

Chapter Summary

In this chapter, we discussed the concepts of resistance and resistivity, as well as their relationship to voltage and current. We also explored the factors affecting resistivity and the temperature dependence of resistance. Furthermore, we examined the power dissipation in resistors due to the flow of electric current. Understanding these principles is crucial for the study of electrical circuits and their components.

Continue to Chapter 11: The Electromotive Force

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