chapter 24 phys

Chapter 24: Capacitance and Dielectrics

Introduction

• Understanding of capacitors and dielectrics is essential in physics and engineering.

• Focus on calculating capacitance, energy storage, and utilizing dielectrics in capacitors.

Objectives

• Understand capacitor nature, charge calculations.

• Analyze capacitor networks.

• Calculate energy stored in capacitors.

• Learn about dielectrics and their effects on capacitance.

• Understand dielectric polarization.

• Apply Gauss’s laws in the presence of dielectrics.

Capacitors and Capacitance

Definition

• A capacitor is formed by two conductors separated by an insulator or vacuum.

• Charging a capacitor involves adding charge to it.

• For a capacitor with charge Q:

  • Higher potential: +Q

  • Lower potential: -Q

Capacitance

• Capacitance (C) defined as:

  • C = Q/Vab

• SI unit for capacitance: Farad (F); 1 F = 1 C/V

• Represents the ability of a capacitor to store energy.

Parallel-Plate Capacitor

• Most common configuration; consists of two parallel plates.

• Capacitance formula:

  • C = (ϵ₀ A) / d

  • Where A = area of plates, d = separation distance, ϵ₀ = permittivity of free space.

Example Calculations

• iClicker examples:

  • Capacitance with given plate area and separation and a potential difference of 10 kV.

  • Charge on each plate given capacitance.

  • Electric field between capacitor plates.

Capacitor Circuits

Series Capacitors

• Capacitors in series share the same charge (Q).

• Total voltage across series equals the sum of individual voltages.

• Equivalent capacitance for series:

  • 1/Ceq = 1/C₁ + 1/C₂ + ...

Parallel Capacitors

• Capacitors in parallel share the same voltage (V).

• Charges in parallel add up; total charge across parallel equals sum of individual charges.

• Equivalent capacitance for parallel:

  • Ceq = C₁ + C₂ + ...

Energy Stored in Capacitors

• Energy formulas:

  • U = 1/2 C V²

  • U = 1/2 Q V

  • U = Q² / (2C)

• Explanation of the energy storage process.

Dielectrics

Definition and Function

• Dielectrics are non-conductive materials used between capacitor plates.

• Functions:

  • Separate conducting plates.

  • Increase potential difference capacity.

  • Avoid dielectric breakdown, which leads to conduction in strong fields.

• Inserting dielectrics increases capacitance.

Dielectric Constant

• Ratio of capacitance with dielectric (C) to capacitance without (C₀).

• Defined as:

  • K = C/C₀

• K is unitless; K = 1 for vacuum.

Electric-Field Energy Density

• Energy density in vacuum defined by:

  • u = 1/2 ϵ₀ E²

• Valid for any field configuration.

Gauss's Law in Dielectrics

Application

• Total charge enclosed within Gaussian surface impacted by polarization.

• Use of Gauss’s law provides equations to determine electric fields in dielectric cylinders.

Example Problems

• Calculating charge on capacitor plates and induced charges using dielectrics.

Summary of Important Formulas

• Capacitor formula: C = Q/V

• Energy stored formulas: U = 1/2 C V², U = 1/2 Q V, U = Q² / (2C)

• Series: 1/Ceq = 1/C₁ + 1/C₂

• Parallel: Ceq = C₁ + C₂

Conclusion

• Mastery of these concepts assists in solving real-world capacitor problems and circuits.

• Understanding capacitors and dielectrics is crucial for further studies in electronics and electrostatics.