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.